\documentclass[reqno]{amsart} \usepackage{amssymb, mathrsfs, hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 116, pp. 1--43.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/116\hfil Pseudodifferential operators] {Pseudodifferential operators with generalized symbols and regularity theory} \author[C. Garetto, T. Gramchev, M. Oberguggenberger\hfil EJDE-2005/116\hfilneg] {Claudia Garetto, Todor Gramchev, Michael Oberguggenberger} \address{Claudia Garetto \hfill\break Institut f\"ur Technische Mathematik\\ Geometrie und Bauinformatik\\ Universit\"at Innsbruck\\ A - 6020 Innsbruck, Austria} \email{claudia@mat1.uibk.ac.at} \address{Todor Gramchev \hfill\break Dipartimento di Matematica e Informatica\\ Universit\`a di Cagliari, I - 09124 Cagliari, Italia} \email{todor@unica.it} \address{Michael Oberguggenberger \hfill\break Institut f\"ur Technische Mathematik \\ Geometrie und Bauinformatik\\ Universit\"at Innsbruck, A - 6020 Innsbruck, Austria} \email{michael@mat1.uibk.ac.at} \date{} \thanks{Submitted June 13, 2005. Published October 21, 2005.} \thanks{C. Garetto was supported by INDAM--GNAMPA, Italy. \hfill\break\indent T. Gramchev was supported by INDAM--GNAMPA, Italy and by grant PST.CLG.979347 \hfill\break\indent from NATO. \hfill\break\indent M. Oberguggenberger was supported by project P14576-MAT from FWF, Austria} \subjclass[2000]{35S50, 35S30, 46F10, 46F30, 35D10} \keywords{Pseudodifferential operators; small parameter; slow scale net; \hfill\break\indent algebras of generalized functions} \begin{abstract} We study pseudodifferential operators with amplitudes $a_\varepsilon (x,\xi)$ depending on a singular parameter $\varepsilon \to 0$ with asymptotic properties measured by different scales. We prove, taking into account the asymptotic behavior for $\varepsilon \to 0$, refined versions of estimates for classical pseudodifferential operators. We apply these estimates to nets of regularizations of exotic operators as well as operators with amplitudes of low regularity, providing a unified method for treating both classes. Further, we develop a full symbolic calculus for pseudodifferential operators acting on algebras of Colombeau generalized functions. As an application, we formulate a sufficient condition of hypoellipticity in this setting, which leads to regularity results for generalized pseudodifferential equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{Introduction} This paper is devoted to pseudodifferential equations of the form \[ A_\epsilon(x,D)u_\epsilon(x) = f_\epsilon(x), \] where $x \in \Omega \subset \mathbb{R}^n$, depending on a small parameter $\epsilon > 0$. Equations of this type arise, e. g., in the study of singularly perturbed partial differential equations, in semiclassical analysis, or when regularizing partial differential operators with non-smooth coefficients or pseudodifferential operators with irregular symbols. We take the point of view of asymptotic analysis: the regularity of the right hand side and of the solution as well as the mapping properties of the operator will be described by means of asymptotic estimates in terms of the parameter $\epsilon \to 0$. We will develop a full pseudodifferential calculus in this setting, with formal series expansions of symbols, construction of parametrices and deduction of regularity results. Our investigations will naturally lead us to introducing different scales of growth in the parameter $\epsilon$, rapid decay signifying negligibility and new classes of $\epsilon$-dependent amplitudes, symbols and operators acting on algebras of generalized functions. As another motivation we mention the recent results on the calculus for generalized functions and their applications in geometry and physics, cf. \cite{GFKS:01, GKOS:01, GKSV:02}. Before going into a detailed description of the contents of the paper and its relation to previous research, we wish to exhibit some of the essential effects by means of a number of motivating examples. \begin{example} \rm {\em Singularly perturbed differential equations}. The appearance of scales of growth and decay can be seen from two very simple equations on $\mathbb{R}$, \begin{equation}\label{singpert1} \Big(-\epsilon^2\frac{d^2}{dx^2} + 1\Big) u_\epsilon = f \end{equation} and \begin{equation}\label{singpert2} \Big(-\epsilon^2\frac{d^2}{dx^2} - 1\Big) v_\epsilon = f. \end{equation} Suppose, for simplicity, that $f \in \mathcal{E}'(\mathbb{R})$ is a distribution with compact support and that we want to solve the equations in ${\mathscr{S}}'(\mathbb{R})$, the space of tempered distributions. Let \[ U_\epsilon(x) = \frac{1}{2\epsilon} e^{-|x/\epsilon|}\,,\quad V_\epsilon(x) = \frac{1}{\epsilon} \sin\left(\frac{x}{\epsilon}\right)H(x) \] where $H$ denotes the Heaviside function. The (unique) solution of (\ref{singpert1}) in ${\mathscr{S}}'(\mathbb{R})$ is given by \[ u_\epsilon(x) = U_\epsilon \ast f(x), \] while (\ref{singpert2}) has the solutions \[ v_\epsilon(x) = V_\epsilon \ast f(x) + C_1 \sin\left(\frac{x}{\epsilon}\right) + C_2 \cos\left(\frac{x}{\epsilon}\right). \] The basic asymptotic scale - growth in powers of $\frac{1}{\epsilon}$ - enters the picture, when we regularize a given distribution $f \in \mathcal{E}'(\mathbb{R})$ by means of convolution: \begin{equation}\label{convolution} f_\epsilon(x) = f \ast \varphi_\epsilon(x), \end{equation} where $\varphi_\epsilon\in {\mathcal C}_c^\infty(\mathbb{R})$ is a mollifier of the form \begin{equation}\label{regularizer} \varphi_\epsilon(x) = \frac{1}{\epsilon}\varphi \left(\frac{x}{\epsilon}\right), \end{equation} with $\int \varphi(x)dx = 1$. Then the family of smooth, compactly supported functions $(f_\epsilon)_{\epsilon \in (0,1]}$ satisfies an asymptotic estimate of the type \begin{equation}\label{moderate} \forall \alpha \in \mathbb{N},\ \exists N \in \mathbb{N} : \sup_{x\in\mathbb{R}}|\partial ^\alpha f_\epsilon(x)| = O(\epsilon^{-N})\,. \end{equation} If we replace the right hand sides in (\ref{singpert1}) and (\ref{singpert2}) by a family of smooth functions $f_\epsilon$ enjoying the asymptotic property (\ref{moderate}) then an estimate of the same type (\ref{moderate}) holds for the solutions $u_\epsilon$ and $v_\epsilon$. On the other hand, a family of smooth functions $(f_\epsilon)_{\epsilon \in (0,1]}$ satisfying an estimate of the type \begin{equation}\label{negligible} \forall \alpha \in \mathbb{N},\ \forall q \in \mathbb{N},\ \sup_{x\in\mathbb{R}}|\partial ^\alpha f_\epsilon(x)| = O(\epsilon^{q}) \end{equation} as $\epsilon \to 0$, will be considered as asymptotically negligible. Clearly, if $f_\epsilon$ as right hand side in (\ref{singpert1}) or (\ref{singpert2}) is asymptotically negligible, so are the solutions $u_\epsilon$ and $v_\epsilon$ (with $C_1 = C_2 = 0$ in the latter case). The condition \begin{equation}\label{regular} \exists N \in \mathbb{N} :\ \forall \alpha \in \mathbb{N},\ \sup_{x\in\mathbb{R}}|\partial ^\alpha f_\epsilon(x)| = O(\epsilon^{-N}) \end{equation} signifies a regularity property of the family $(f_\epsilon)_{\epsilon \in (0,1]}$; it is known \cite{O:92} that if the regularizations (\ref{convolution}) of a distribution $f$ satisfy (\ref{regular}) then $f$ actually is an infinitely differentiable function. Now assume the right hand sides in (\ref{singpert1}) and (\ref{singpert2}) are given by compactly supported smooth functions satisfying the regularity property (\ref{regular}). We ask whether the corresponding solutions will inherit this property. This is true of the solution $u_\epsilon$ to (\ref{singpert1}), as can be seen by Fourier transforming the equation. It is not true of the solutions $v_\epsilon$ to (\ref{singpert2}); already the homogeneous part $C_1\sin\frac{x}{\epsilon} + C_2\cos\frac{x}{\epsilon}$ destroys the property. However, let us consider equation (\ref{singpert2}) with a different scaling in $\epsilon$, say \begin{equation}\label{singpert2a} \Big(-\omega(\epsilon)^2\frac{d^2}{dx^2} - 1\Big) v_\epsilon = f_\epsilon \end{equation} with $\omega(\epsilon) \to 0$. If $v_\epsilon$ is a solution, we may express the higher derivatives by means of its 0-th derivative and the derivatives of the right hand side: \begin{align*} -\frac{d^2}{dx^2}v_\epsilon &= \frac{1}{\omega(\epsilon)^2} f_\epsilon + \frac{1}{\omega(\epsilon)^2} v_\epsilon,\\ -\frac{d^4}{dx^4}v_\epsilon &= \frac{1}{\omega(\epsilon)^2} \frac{d^2}{dx^2} f_\epsilon + \frac{1}{\omega(\epsilon)^2} \frac{d^2}{dx^2}v_\epsilon \\ &= \frac{1}{\omega(\epsilon)^2} \frac{d^2}{dx^2} f_\epsilon - \frac{1}{\omega(\epsilon)^4} f_\epsilon - \frac{1}{\omega(\epsilon)^4} v_\epsilon, \end{align*} and so on. Thus if $f_\epsilon$ satisfies the regularity property (\ref{regular}) and the net $(\omega(\epsilon))_{\epsilon \in (0,1]}$ satisfies \begin{equation}\label{slowscale} \forall p \geq 0 : \biggl(\frac{1}{\omega(\epsilon)}\biggr)^p = O\biggl(\frac{1}{\epsilon}\biggr) \end{equation} as $\epsilon \to 0$ then every solution $(v_\epsilon)_{\epsilon \in (0,1]}$ satisfies (\ref{regular}) as well. We shall refer to property (\ref{slowscale}) by saying that $1/\omega(\epsilon)$ forms a {\em slow scale net}. This example not only shows the appearance of different asymptotic scales, but also that regularity results in terms of property (\ref{regular}) depend on lower order terms in the equation and/or the scales used to describe the asymptotic behavior as $\epsilon \to 0$. \end{example} \begin{example} \rm {\em Regularity of distributions expressed in terms of asymptotic estimates on the regularizations}. Let $f \in \mathscr{S}'(\mathbb{R}^n)$, $s\in \mathbb{R}$ and $\varphi_\epsilon$ a regularizer as in (\ref{regularizer}). The following assertions about Sobolev regularity hold: \begin{itemize} \item[(a)] If $f \in H^s(\mathbb{R}^n)$ then \begin{equation}\label{Sobolev} \forall \alpha \in \mathbb{N}^n : \|\partial ^\alpha f\ast \varphi_\epsilon\|_{L^2(\mathbb{R}^n)} = O\big(\epsilon^{-(|\alpha|-s)_+}\big) \end{equation} where $(\cdot)_+$ denotes the positive part of a real number. \item[(b)] Conversely, if $f \in {\mathcal E}'(\mathbb{R}^n)$ and (\ref{Sobolev}) holds, then $f \in H^t(\mathbb{R}^n)$ for all $t < s - n/2$. In addition, $f$ belongs to $H^s(\mathbb{R}^n)$ in case $s$ is a nonnegative integer. \end{itemize} Indeed, it is readily seen that $f$ belongs to $L^2(\mathbb{R}^n)$ if and only if $\|f\ast \varphi_\epsilon\|_{L^2(\mathbb{R}^n)} = O(1)$. Part (a), for $s < 0$, follows easily by Fourier transform, while for $s = k + \tau$ with $k \in \mathbb{N}, 0\leq \tau <1$ the observation that $f$ belongs to $H^s(\mathbb{R}^n)$ if and only if $\partial ^\alpha f$ is in $L^2(\mathbb{R}^n)$ for $|\alpha| \leq k$ and in $H^{\tau-1}(\mathbb{R}^n)$ for $|\alpha| = k+1$ may be used. Part (b) for $s < 0$ is derived along the lines of \cite[Thm. 25.2]{O:92} by showing that $(1+|\xi|)^s$ times the Fourier transform $\widehat{f}(\xi)$ is bounded. For $s \geq 0$, a similar observation as above concludes the argument. An analogous characterization for the Zygmund classes $\mathcal{C}_\ast^s(\mathbb{R}^n)$ has been proven by H\"ormann \cite{GH:02b}. Further, given a distribution $f \in \mathcal{D}'(\Omega)$, it was already indicated above that $f$ is a smooth function if and only if the regularizations $f\ast \varphi_\epsilon$ satisfy property (\ref{regular}) (suitably localized with the supremum taken on compact sets of $\Omega$). \end{example} \begin{example} \rm {\em Regularization of operators with non-smooth coefficients}. Consider a linear partial differential operator \[ A(x,D) = \sum_{|\alpha|\leq m} a_\alpha(x)D^\alpha. \] If its coefficients are distributions, we may form the regularized operator \[ A_\epsilon(x,D) = \sum_{|\alpha|\leq m} a_\alpha\ast \varphi_\epsilon(x)D^\alpha. \] The regularized coefficients will satisfy an estimate of type (\ref{moderate}), at least locally on compact sets, and the action of $A_\epsilon(x,D)$ on nets $(u_\epsilon)_{\epsilon \in (0,1]}$ preserves the asymptotic properties (\ref{moderate}) and (\ref{negligible}); that is, if $(u_\epsilon)_{\epsilon \in (0,1]}$ enjoys either of these properties, so does $(A_\epsilon(\cdot,D)u_\epsilon)_{\epsilon \in (0,1]}$. However, the regularity property (\ref{regular}) will not be preserved in general, unless the regularization of the coefficients is performed with a slow scale mollifier, that is, by convolution with $\varphi_{\omega(\epsilon)}$ where $\omega(\epsilon)^{-1}$ is a slow scale net. For example, consider the multiplication operator \[ M_\epsilon(x,D) u(x) = \varphi_{\omega(\epsilon)} u(x). \] Then $(M_\epsilon(x,D))_{\epsilon \in (0,1]}$ maps the space of nets enjoying regularity property (\ref{regular}) into itself if and only if $\omega(\epsilon)^{-1}$ is a slow scale net. Indeed, the sufficiency of the slow scale condition is quite clear. To prove its necessity, take a fixed smooth function $u$ identically equal to one near $x=0$. Then the derivatives $\partial ^\alpha (M_\epsilon u)$ have a uniform asymptotic bound $O(\epsilon^{-N})$ independently of $\alpha \in \mathbb{N}^n$ if and only if $\omega(\epsilon)^{-|\alpha|} = O(\epsilon^{-N})$ for all $\alpha$; that is, if and only if $\omega(\epsilon)^{-1}$ is a slow scale net. \end{example} \begin{example} \rm {\em $L^2$-estimates for pseudodifferential operators in exotic classes}. Consider first a symbol $a(x,\xi)$ in the H\"ormander class $S_{1,0}^0(\mathbb{R}^{2n})$ (for simplicity, we restrict our discussion to global zero order symbols here). It is well known that the corresponding operator $a(x,D)$ maps $L^2(\mathbb{R}^n)$ continuously into itself, with operator norm depending on a finite number of derivatives of $a(x,\xi)$. More precisely, an estimate of the following form holds (see e. g. \cite[Sect. 2.4, Thm. 4.1]{Kumano-go:81} and \cite[Sect. 18.1]{Hoermander:V3}, see also \cite{Ho:86} and the references therein): \begin{equation}\label{L2estimate} \|a(x,D) u\|_{L^2(\mathbb{R}^n)}^2 \leq c_0^2\|u\||_{L^2(\mathbb{R}^n)}^2 + c_1^2 p_l^2(a)\|u\||_{L^2(\mathbb{R}^n)}^2 \end{equation} where $c_0$ is a strict upper bound for the $L^\infty$-norm of the symbol $a$ on $\mathbb{R}^{2n}$ and $p_l$ signifies the norm \[ p_l(a) = \max_{|\alpha + \beta|\leq l} \sup_{(x,\xi)\in\mathbb{R}^{2n}} |\partial _\xi^\alpha\partial _x^\beta a(x,\xi)|\langle\xi\rangle^{|\alpha|}. \] In estimate (\ref{L2estimate}), $l$ is an integer depending on the type of symbol, but generically is strictly greater than zero. However, if $a(x,\xi)$ is positively homogeneous of order zero with respect to $\xi$ the $L^2$-continuity holds provided $\partial _\xi^\alpha a(x,\xi)$ is bounded in $\mathbb{R}^n \times \mathbb{S}^{n-1}$, cf. \cite{Coifman-Meyer:97}. If the symbol $a(x,\xi)$ belongs to the exotic class $S_{1,1}^0(\mathbb{R}^{2n})$ then $a(x,D)$ will not map $L^2(\mathbb{R}^n)$ into itself, in general \cite[Thm. 9]{Coifman-Meyer:78}, \cite{Rodino:76}. However, if we regularize by convolution in the $x$-variable, \[ a_\epsilon(x,\xi) = a(\cdot,\xi)\ast \varphi_\epsilon(x), \] we get a family of symbols each of which belongs to the class $S_{1,0}^0(\mathbb{R}^{2n})$, thus maps $L^2(\mathbb{R}^n)$ continuously into itself, but with an operator norm that behaves asymptotically like $\epsilon^{-N}$ where $N$ is some integer less or equal to $l$ in (\ref{L2estimate}); note that the convolution with the mollifier $\varphi_\epsilon$ does not increase the constant $c_0$. Classically, there is a pseudodifferential calculus (including e. g. composition) for symbols in $S_{1,0}^0(\mathbb{R}^{2n})$, but not for exotic classes, like $S_{1,1}^0(\mathbb{R}^{2n})$, in general. The regularization approach bridges this gap: we will develop a full pseudodifferential calculus for classes of regularized symbols in this paper. Estimate (\ref{L2estimate}) remains valid with uniform finite bounds in $\epsilon$ for symbols $a_\epsilon(x,\xi)$ obtained by convolution from symbols in $S_{1,0}^0(\mathbb{R}^{2n})$, but we will have to face asymptotic growth as $\epsilon \to 0$ in exchange for the lack of $L^2$-continuity in the case of exotic symbols. \end{example} \begin{remark} \rm {\em Algebras of generalized functions}. The families of smooth functions $(u_\epsilon)_{\epsilon\in (0,1]}$ satisfying estimate (\ref{moderate}), globally or possibly only on compact sets, form a differential algebra; the nets $(u_\epsilon)_{\epsilon\in (0,1]}$ of negligible elements form a differential ideal therein. The space of distributions can be embedded into the corresponding factor algebra by means of cut-off and convolution, with a consistent notion of derivatives. The fact that for smooth functions $f$, the net $(f - f\ast\varphi_\epsilon)_{\epsilon\in (0,1]}$ is negligible for suitably chosen regularizers $\varphi_\epsilon$ was discovered by Colombeau \cite{Colombeau:84, Colombeau:85}; thus the multiplication in the factor algebra is also consistent with the product of smooth functions. The factor algebras of nets satisfying (\ref{moderate}) modulo negligible nets is a suitable framework for studying families of pseudodifferential operators and the asymptotic behavior of their action on functions or generalized functions. We note that a condition similar to the asymptotic negligibility (\ref{negligible}) was considered by Maslov et al. \cite{Maslov-Tsupin:79b, Maslov-Tsupin:79a} earlier in the context of asymptotic solutions to partial differential equations. In introducing factor spaces of families of amplitudes and symbols (modulo negligible ones) as well, we will succeed in this paper to establish a full symbolic calculus of operators acting on generalized functions. This is a new contribution to the field of non-smooth operators. Our essential tools for describing the mapping properties and regularity results will be asymptotic estimates and scales of growth. \end{remark} We now describe the contents of the paper in more detail. Section 2 serves to introduce the basic notions - asymptotic properties defining the algebras of generalized functions on which our operators will act, the notion of regularity intrinsic to these algebras (the so-called $\mathcal{G}^\infty$-regularity, indicated in (\ref{regular})), some new technical results needed, and a basic theory of integral operators with generalized kernels. In Section 3 we start our theory by studying oscillatory integrals with smooth phase functions and generalized symbols, introduced as equivalence classes of certain nets of smooth symbols modulo negligible ones. Section 4 employs these techniques to introducing and studying pseudodifferential operators with generalized amplitudes, their mapping properties, pseudo-locality with respect to the notion of $\mathcal{G}^\infty$-regularity mentioned above, and their kernels in the sense of the algebras of generalized functions. The full symbolic calculus of our class of generalized pseudodifferential operators is developed in Section 5. It starts with formal series and asymptotic expansions of equivalence classes of symbols, proceeds with the construction of symbols for (generalized) pseudodifferential operators, their transposes and their compositions. The paper culminates in the regularity theory presented in Section 6. We generalize the notion of hypoellipticity to our class of symbols and construct parametrices for these symbols. We show that the solutions to the corresponding pseudodifferential equations are $\mathcal{G}^\infty$-regular in those regions where the right hand sides are $\mathcal{G}^\infty$-regular. Here the importance of different scales of asymptotic growth becomes apparent. What concerns previous literature on the subject, we mention that $\mathcal{G}^\infty$-regularity was introduced in \cite{O:92} where it was already applied to prove regularity results for solutions to classical constant coefficient partial differential equations. Completely new effects arise when the coefficients are allowed to be generalized constants, depending on the parameter $\epsilon > 0$. These effects and a regularity theory for such operators was developed in \cite{HO:02}, see also \cite{NP:98}, and extended to the case of partial differential operators with generalized, non-constant coefficients in \cite{HOP:02}. The study of pseudodifferential operators in the setting of algebras of generalized functions was started in \cite{O-pseudo:90}, developed in a rudimentary version in \cite{NPS:98, Pilipovic:94}. A full version with nets of symbols and a full symbolic calculus, albeit for global symbols and in the algebra of tempered generalized functions is due to \cite{Garetto:03}. Our contribution is the first in the literature containing a full local symbolic calculus of generalized pseudodifferential operators, equivalence classes of symbols, strong $\mathcal{G}^\infty$-regularity results and the incorporation of different scales (the necessity of which was demonstrated in \cite{HO:02}). For microlocal notions of $\mathcal{G}^\infty$-regularity we refer to \cite{Hoermann:99, Hoermann:03, HK:01, NPS:98, Scarpalezos:00}. Motivating examples from semiclassical analysis can be found in \cite{Boggiatto-Rodino:03, Robert:87}. Further studies of kernel operators in Colombeau algebras including topological investigations are carried out in \cite{Delcroix:04, Garetto:05a, Garetto:05b}. \section{Basic notions} In this section we recall the definitions and results needed from the theory of Colombeau generalized functions. For details of the constructions we refer to \cite{Biagioni:90, Colombeau:85, Colombeau:92,GKOS:01,NPS:98,O:92}. In the sequel we denote by ${\mathcal{E}}[\Omega]$, $\Omega$ an open subset of $\mathbb{R}^n$, the algebra of all the sequences $(u_\epsilon)_{\epsilon\in(0,1]}$ (for short, $(u_\epsilon)_\epsilon$) of smooth functions $u_\epsilon\in\mathcal{C}^\infty(\Omega)$. \begin{definition} \label{defmod} \rm ${\mathcal{E}}_M(\Omega)$ is the differential subalgebra of the elements $(u_\epsilon)_\epsilon\in\mathcal{E}[\Omega]$ such that for all $K\Subset\Omega$, for all $\alpha\in\mathbb{N}^n$ there exists $N\in\mathbb{N}$ with the following property: \[ \sup_{x\in K}|\partial^\alpha u_\epsilon(x)|=O(\epsilon^{-N}) \quad \mbox{as } \epsilon\to 0. \] \end{definition} \begin{definition} \label{defneg} \rm We denote by $\mathcal{N}(\Omega)$ the differential subalgebra of the elements $(u_\epsilon)_\epsilon$ in $\mathcal{E}[\Omega]$ such that for all $K\Subset\Omega$, for all $\alpha\in\mathbb{N}^n$ and $q\in\mathbb{N}$ the following property holds: \[ \sup_{x\in K}|\partial^\alpha u_\epsilon(x)|=O(\epsilon^q) \quad \mbox{as } \epsilon\to 0. \] \end{definition} The elements of $\mathcal{E}_M(\Omega)$ and $\mathcal{N}(\Omega)$ are called moderate and negligible, respectively. The factor algebra $\mathcal{G}(\Omega):=\mathcal{E}_M(\Omega)/\mathcal{N}(\Omega)$ is the algebra of generalized functions on $\Omega$. As shown e. g. in \cite{O:92}, suitable regularizations and the sheaf properties of $\mathcal{G}(\Omega)$, allow us to define an embedding $\imath$ of $\mathcal{D}'(\Omega)$ into $\mathcal{G}(\Omega)$ extending the constant embedding $\sigma:f\to(f)_\epsilon+\mathcal{N}(\Omega)$ of $\mathcal{C}^\infty(\Omega)$ into $\mathcal{G}(\Omega)$. In the computations of this paper, the following characterization of $\mathcal{N}(\Omega)$ as a subspace of $\mathcal{E}_M(\Omega)$, proved in Theorem 1.2.3 of \cite{GKOS:01}, will be very useful. \begin{proposition} \label{carneg} $(u_\epsilon)_\epsilon\in\mathcal{E}_M(\Omega)$ is negligible if and only if \[ \forall K\Subset\Omega,\ \forall q\in\mathbb{N},\quad \sup_{x\in K}|u_\epsilon(x)|=O(\epsilon^q)\quad as\quad \epsilon\to 0. \] \end{proposition} We consider now some particular subalgebras of $\mathcal{G}(\Omega)$. \begin{definition} \label{defk} \rm Let $\mathbb{K}$ be the field $\mathbb{R}$ or $\mathbb{C}$. We set \[ \begin{split} \mathcal{E}_{M}&=\{ (r_\epsilon)_\epsilon\in\mathbb{K}^{(0,1]}: \exists N\in\mathbb{N}: |r_\epsilon|=O(\epsilon^{-N})\mbox{ as } \epsilon\to 0\},\\ \mathcal{N}&=\{ (r_\epsilon)_\epsilon\in\mathbb{K}^{(0,1]}: \forall q\in\mathbb{N}: \quad |r_\epsilon|=O(\epsilon^{q})\mbox{ as }\epsilon\to 0\}. \end{split} \] $\widetilde{\mathbb{K}}:= {\mathcal{E}_M}/{\mathcal{N}}$ is called the ring of generalized numbers. \end{definition} In the case of $\mathbb{K}=\mathbb{R}$ we get the algebra $\widetilde{\mathbb{R}}$ of real generalized numbers and for $\mathbb{K}=\mathbb{C}$ the algebra $\widetilde{\mathbb{C}}$ of complex generalized numbers. $\widetilde{\mathbb{R}}$ can be endowed with the structure of a partially ordered ring (for $r,s\in\widetilde{\mathbb{R}}$, $r\le s$ if and only if there are representatives $(r_\epsilon)_\epsilon$ and $(s_\epsilon)_\epsilon$ with $r_\epsilon\le s_\epsilon$ for all $\epsilon\in(0,1]$). $\widetilde{\mathbb{C}}$ is naturally embedded in $\mathcal{G}(\Omega)$ and it can be considered as the ring of constants of $\mathcal{G}(\Omega)$ if $\Omega$ is connected. Moreover, using $\widetilde{\mathbb{C}}$, we can define a concept of generalized point value for the generalized functions of $\mathcal{G}(\Omega)$. In the sequel we recall the crucial steps of this construction, referring to Section 1.2.4 in \cite{GKOS:01} and to \cite{OK:99} for the proofs. \begin{definition} \label{defpoint} \rm On \[ \Omega_M=\{(x_\epsilon)_\epsilon\in\Omega^{(0,1]}: \exists N\in\mathbb{N},\quad |x_\epsilon|=O(\epsilon^{-N})\quad \mbox{as } \epsilon\to 0\}, \] we introduce an equivalence relation given by \[ (x_\epsilon)_\epsilon\sim(y_\epsilon)_\epsilon\ \Leftrightarrow\ \forall q\in\mathbb{N},\ |x_\epsilon-y_\epsilon| =O(\epsilon^q)\quad \mbox{as }\epsilon\to 0 \] and denote by $\widetilde{\Omega}:=\Omega_M/\sim$ the set of generalized points. Moreover, if $[(x_\epsilon)_\epsilon]$ is the class of $(x_\epsilon)_\epsilon$ in $\widetilde{\Omega}$ then the set of compactly generalized points is \[ \widetilde{\Omega}_c=\{\tilde{x}=[(x_\epsilon)_\epsilon]\in\widetilde{\Omega} : \exists K\Subset\Omega,\ \exists \eta>0: \forall\epsilon\in(0,\eta],\ x_\epsilon\in K\}. \] \end{definition} Obviously if the $\widetilde{\Omega}_c$-property holds for one representative of $\tilde{x}\in\widetilde{\Omega}$ then it holds for every representative. Also, for $\Omega=\mathbb{R}$ we have that the factor ${\mathbb{R}}_M/\sim$ is the usual algebra of real generalized numbers. In the following $(u_\epsilon)_\epsilon$ and $(x_\epsilon)_\epsilon$ are arbitrary representatives of $u\in\mathcal{G}(\Omega)$ and $\tilde{x}\in\widetilde{\Omega}_c$, respectively. It is clear that the generalized point value of $u$ at $\tilde{x}$, \begin{equation}\label{propoint} u(\tilde{x}):=(u_\epsilon(x_\epsilon))_\epsilon+\mathcal{N} \end{equation} is a well-defined element of $\widetilde{\mathbb{C}}$. An interesting application of this notion is the characterization of generalized functions by their generalized point values. \begin{proposition} \label{carpoint} Let $u\in\mathcal{G}(\Omega)$. Then $u=0$ if and only if $u(\tilde{x})=0$ for all $\tilde{x}\in\widetilde{\Omega}_c$. \end{proposition} We continue now our study of $\mathcal{G}(\Omega)$ with the notions of support and generalized singular support. \begin{definition} \label{defmodc} \rm We denote by $\mathcal{E}_{c,M}(\Omega)$ the set of all the elements $(u_\epsilon)_\epsilon\in\mathcal{E}_{M}(\Omega)$ such that there exists $K\Subset\Omega$ with $\mathop{\rm supp}u_\epsilon\subseteq K$ for all $\epsilon\in(0,1]$. \end{definition} \begin{definition} \label{defnegc} \rm We denote by $\mathcal{N}_c(\Omega)$ the set of all the elements $(u_\epsilon)_\epsilon\in\mathcal{N}(\Omega)$ such that there exists $K\Subset\Omega$ with $\mathop{\rm supp}u_\epsilon\subseteq K$ for all $\epsilon\in(0,1]$. \end{definition} $\mathcal{G}_c(\Omega):=\mathcal{E}_{c,M}(\Omega)/\mathcal{N}_c(\Omega)$ is the algebra of compactly supported generalized functions. Since the map $l:\mathcal{G}_c(\Omega)\to\mathcal{G}(\Omega):(u_\epsilon)_\epsilon +\mathcal{N}_c(\Omega)\to (u_\epsilon)_\epsilon+\mathcal{N}(\Omega)$ is injective, $\mathcal{G}_c(\Omega)$ is a subalgebra of $\mathcal{G}(\Omega)$ containing $\mathcal{E}'(\Omega)$ as a subspace and $\mathcal{C}^\infty_c(\Omega)$ as a subalgebra. Recalling that for $u\in\mathcal{G}(\Omega)$ and $\Omega'$ an open subset of $\Omega$, $u\vert_{\Omega'}$ is the generalized function in $\mathcal{G}(\Omega')$ having as representative $(u_\epsilon\vert_{\Omega'})_\epsilon$, it is possible to define the support of $u$, setting \[ \Omega\setminus {\mathop{\rm supp}}\,u=\{x\in\Omega:\ \exists V(x)\subset\Omega,\text{ open}, x\in V(x):\ u\vert_{V(x)}=0\}. \] The map $l$ identifies $\mathcal{G}_c(\Omega)$ with the set of generalized functions in $\mathcal{G}(\Omega)$ with compact support. It is sufficient to observe that if $u\in\mathcal{G}(\Omega)$ with supp\,$u$ $\Subset\Omega$, and $\psi\in\mathcal{C}^\infty_c(\Omega)$ is identically equal to 1 in a neighborhood of supp\,$u$ then $\psi u:=(\psi u_\epsilon)_\epsilon+\mathcal{N}_c(\Omega)$ belongs to $\mathcal{G}_c(\Omega)$ and $u=l(\psi u)$. If we consider an open subset $\Omega'$ of $\Omega$, the map ${\mathcal{G}}_c(\Omega')\to\mathcal{G}_c(\Omega)$: $(u_\epsilon)_\epsilon+\mathcal{N}_c(\Omega') \to(u_\epsilon)_\epsilon+\mathcal{N}_c(\Omega)$ allows us to embed ${\mathcal{G}}_c(\Omega')$ into $\mathcal{G}_c(\Omega)$. A generalized function $u\in\mathcal{G}(\Omega)$ can be integrated over a compact subset of $\Omega$, using the definition \[ \int_K u(x)dx:=\Bigl(\int_K u_\epsilon(x)dx\Bigr)_\epsilon+\mathcal{N}; \] in particular, a generalized function $u\in\mathcal{G}_c(\Omega)$ can be integrated over $\Omega$ by means of the prescription \[ \int_\Omega u(x)dx:=\Bigl(\int_V u_\epsilon(x)dx\Bigr)_\epsilon+\mathcal{N}, \] where $V$ is any compact set containing ${\rm{supp}}\, u$ in its interior. \begin{definition} \label{defmodreg} \rm We denote by $\mathcal{E}^\infty_M(\Omega)$ the set of all the elements $(u_\epsilon)_\epsilon\in\mathcal{E}[\Omega]$ such that for all $K\Subset\Omega$ there exists $N\in\mathbb{N}$ with the following property: \[ \forall\alpha\in\mathbb{N}^n:\quad \sup_{x\in K}|\partial^\alpha u_\epsilon(x)| =O(\epsilon^{-N})\quad as\quad \epsilon\to 0. \] \end{definition} $\mathcal{G}^\infty(\Omega):=\mathcal{E}^\infty_M(\Omega)/ \mathcal{N}(\Omega)$ is the algebra of regular generalized functions. Theorem 25.2 in \cite{O:92} shows that $\mathcal{G}^\infty(\Omega)\cap \mathcal{D}'(\Omega) =\mathcal{C}^\infty(\Omega)$. Finally, if $\mathcal{E}^\infty_{c,M}(\Omega):=\mathcal{E}^\infty_M(\Omega) \cap\mathcal{E}_{c,M}(\Omega)$, $\mathcal{G}^\infty_c(\Omega):=\mathcal{E}^\infty_{c,M}(\Omega)/ \mathcal{N}_c(\Omega)$ is the algebra of regular compactly supported generalized functions, and $\mathcal{G}^\infty_c(\Omega)\cap\mathcal{E}'(\Omega) =\mathcal{C}^\infty_c(\Omega)$. As above, it is possible to define the generalized singular support of $u\in\mathcal{G}(\Omega)$ setting \[ \Omega\setminus {\rm{{sing\,supp}_g}}u =\{x\in\Omega: \exists V(x)\subset\Omega,\ \text{open},\ x\in V(x): \ u\vert_{V(x)}\in\mathcal{G}^\infty(V(x))\}. \] Using the sheaf properties of $\mathcal{G}^\infty(\Omega)$ we can identify the algebra of regular generalized functions with the set of generalized functions in $\mathcal{G}(\Omega)$ having empty generalized singular support. In the same way $\mathcal{G}^\infty_c(\Omega)$ is the set of generalized functions in $\mathcal{G}(\Omega)$ with compact support and empty generalized singular support. \begin{definition} \label{defgs} \rm Let ${\mathscr{S}}[\mathbb{R}^n]:={{\mathscr{S}}(\mathbb{R}^n)}^{(0,1]}$. The elements of \begin{align*} &\mathcal{E}_\mathscr{S}(\mathbb{R}^n)\\ &=\big\{(u_\epsilon)_\epsilon \in{\mathscr{S}}[\mathbb{R}^n]:\ \forall\alpha,\beta\in\mathbb{N}^n,\ \exists N\in\mathbb{N}:\ \sup_{x\in\mathbb{R}^n}|x^\alpha\partial^\beta u_\epsilon(x)| =O(\epsilon^{-N}) \text{ as } \epsilon\to 0\big\} \end{align*} are called ${\mathscr{S}}$-moderate. The elements of \begin{align*} &\mathcal{E}_\mathscr{S}^\infty(\mathbb{R}^n)\\ &=\big\{(u_\epsilon)_\epsilon \in{\mathscr{S}}[\mathbb{R}^n]:\ \exists N\in\mathbb{N}:\ \forall\alpha,\beta\in\mathbb{N}^n,\ \sup_{x\in\mathbb{R}^n}|x^\alpha\partial^\beta u_\epsilon(x)| =O(\epsilon^{-N}) \text{ as }\epsilon\to 0\} \end{align*} are called ${\mathscr{S}}$-regular. The elements of \begin{align*} &\mathcal{N}_\mathscr{S}(\mathbb{R}^n)\\ &=\big\{(u_\epsilon)_\epsilon\in{\mathscr{S}}[\mathbb{R}^n]:\ \forall\alpha,\beta\in\mathbb{N}^n,\ \forall q\in\mathbb{N}:\ \sup_{x\in\mathbb{R}^n}|x^\alpha\partial^\beta u_\epsilon(x)| =O(\epsilon^{q})\text{ as } \epsilon\to 0\big\} \end{align*} are called $\mathscr{S}$-negligible. \end{definition} The factor algebra $\mathcal{G}_\mathscr{S}(\mathbb{R}^n) :=\mathcal{E}_\mathscr{S}(\mathbb{R}^n)/\mathcal{N}_\mathscr{S}(\mathbb{R}^n)$ is the algebra of ${\mathscr{S}}$-generalized functions while its subalgebra $\mathcal{G}_\mathscr{S}^\infty(\mathbb{R}^n) :=\mathcal{E}_\mathscr{S}^\infty(\mathbb{R}^n)/ \mathcal{N}_\mathscr{S}(\mathbb{R}^n)$ is called the algebra of ${\mathscr{S}}$-regular generalized functions. Obviously, $\mathcal{G}_c(\Omega)\subseteq\mathcal{G}_\mathscr{S}(\mathbb{R}^n)$ and $\mathcal{G}^\infty_c(\Omega)\subseteq\mathcal{G} _\mathscr{S}^\infty(\mathbb{R}^n)$. For $u\in\mathcal{G}_\mathscr{S}(\mathbb{R}^n)$ there is a natural definition of Fourier transform, given by $\widehat{u}:=(\widehat{u_\epsilon})_\epsilon+\mathcal{N}_\mathscr{S} (\mathbb{R}^n)$. The Fourier transform maps $\mathcal{G}_\mathscr{S}(\mathbb{R}^n)$ into $\mathcal{G}_\mathscr{S}(\mathbb{R}^n)$, $\mathcal{G}_\mathscr{S}^\infty(\mathbb{R}^n)$ into $\mathcal{G}_\mathscr{S}^\infty(\mathbb{R}^n)$, $\mathcal{G}_c(\Omega)$ into $\mathcal{G}_\mathscr{S}(\mathbb{R}^n)$ and $\mathcal{G}^\infty_c(\Omega)$ into $\mathcal{G}_\mathscr{S}^\infty(\mathbb{R}^n)$. In the sequel, given $\varphi\in\mathcal{C}^\infty_c(\mathbb{R}^n)$, $\tilde{t}\in\widetilde{\Omega}_c$ and $\tilde{\tau}\in\widetilde{\mathbb{R}}_c$, $0\le\tilde{\tau}$ invertible, we denote by $\varphi_{\tilde{t}, \tilde{\tau}}\in\mathcal{G}_c(\mathbb{R}^n)$ the generalized function \[ \varphi_{\tilde{t},\tilde{\tau}}(x) =\varphi\big(\frac{x-\tilde{t}}{\tilde{\tau}}\big). \] Further, we let \[ T_\Omega(\varphi)=\{\varphi_{\tilde{t},\tilde{\tau}}:\ \tilde{\tau}\in\widetilde{\mathbb{R}}_c,\ 0\le\tilde{\tau}\ \text{invertible},\ \tilde{t}\in\widetilde{\Omega}_c,\ \mathop{\rm supp}(\varphi_{\tilde{t}, \tilde{\tau}})\subset\Omega\} \] \begin{proposition} \label{propmo} Let $u\in\mathcal{G}(\Omega)$. If there is $\varphi\in\mathcal{C}^\infty_c(\mathbb{R}^n)$, $\varphi\ge 0$, $\int\varphi(x)dx=1$ such that \[ \int u(x)v(x)dx=0\quad \text{in } \widetilde{\mathbb{C}} \] for all $v\in T_\Omega(\varphi)$ then $u=0$ in $\mathcal{G}(\Omega)$. \end{proposition} \begin{proof} We may assume that $u$ is real valued. If $u\neq 0$ then there exist a representative $(u_\epsilon)_\epsilon$ of $u$, a natural number $q$ and a sequence $\epsilon_k\to 0$ such that \[ |u_{\epsilon_k}(t_{\epsilon_k})|\ge \epsilon_k^q \] for all $k\in\mathbb{N}$. On the other hand, there is $N\in\mathbb{N}$ such that \[ \sup_{x\in K}|\nabla u_\epsilon(x)|\le \epsilon^{-N} \] for sufficiently small $\epsilon\in(0,1]$, where $K$ is a compact subset of $\Omega$ containing $(t_\epsilon)_\epsilon$ in its interior. Then \[ \begin{split} |u_{\epsilon_k}(x)|&=|u_{\epsilon_{k}}(t_{\epsilon_k})+(x-t_{\epsilon_k}) \cdot\int_0^1\nabla u_{\epsilon_k}(t_{\epsilon_k}+\sigma(x-t_{\epsilon_k})) d\sigma|\\ &\ge \epsilon_k^q-|x-t_{\epsilon_k}|\epsilon_k^{-N}\ge \frac{1}{2} \epsilon_k^q \end{split} \] provided $|x-t_{\epsilon_k}|\le \frac{1}{2}\epsilon_k^{N+q}$. Noting that \[ \varphi\big(\frac{x-t_\epsilon}{\epsilon^{N+q+1}}\big)=0 \] when $|x-t_\epsilon|>\frac{1}{2}\epsilon^{N+q}$ eventually and that $u_{\epsilon_k}(x)$ does not change sign for $|x-t_{\epsilon_k}| \le\frac{1}{2}\epsilon^{N+q}$, we see that \[ \Bigl|\int u_{\epsilon_k}(x)\varphi \Big(\frac{x-t_{\epsilon_k}}{\epsilon_k^{N+q+1}}\Big)dx\Bigr| \ge \frac{1}{2}\epsilon_k^{q+n(N+q+1)}. \] Thus, with $\tilde{t}$ as above and $\tau_\epsilon=\epsilon^{N+q+1}$, we have that \[ \int u(x)\varphi_{\tilde{t},\tilde{\tau}}(x)dx\neq 0\quad \text{in } \widetilde{\mathbb{C}} \] contradicting the hypothesis. \end{proof} In the sequel, we denote by $L(\mathcal{G}_c(\Omega),\widetilde{\mathbb{C}})$ the space of all $\widetilde{\mathbb{C}}$-linear maps from $\mathcal{G}_c(\Omega)$ into $\widetilde{\mathbb{C}}$. It is clear that every $u\in\mathcal{G}(\Omega)$ defines an element of $L(\mathcal{G}_c(\Omega),\widetilde{\mathbb{C}})$, setting $\jmath(u)(v)=\int_\Omega u(x)v(x)dx$ for $v\in\mathcal{G}_c(\Omega)$. As an immediate consequence of Proposition \ref{propmo} we have that the map $\jmath:\mathcal{G}(\Omega)\to L(\mathcal{G}_c(\Omega),\widetilde{\mathbb{C}}):u\to\jmath(u)$ is injective. Our interest in $L(\mathcal{G}_c(\Omega),\widetilde{\mathbb{C}})$ is motivated by some specific properties. We begin by defining the restriction of $T\in L(\mathcal{G}_c(\Omega),\widetilde{\mathbb{C}})$ to an open subset $\Omega'$ of $\Omega$, as the $\widetilde{\mathbb{C}}$-linear map \[ T_{\vert_{\Omega'}}:\mathcal{G}_c(\Omega')\to\widetilde{\mathbb{C}}: u\to T((u_\epsilon)_\epsilon+\mathcal{N}_c(\Omega)). \] By adapting the classical proof concerning the sheaf properties of $\mathcal{D}'(\Omega)$, we obtain the following result. \begin{proposition} \label{propsheaf} $L(\mathcal{G}_c(\Omega),\widetilde{\mathbb{C}})$ is a sheaf. \end{proposition} Thus it makes sense to define the support of $T\in L(\mathcal{G}_c(\Omega),\widetilde{\mathbb{C}})$, $\mathop{\rm supp}T$, as the complement of the largest open set $\Omega'$ such that $T_{\vert_{\Omega'}}=0$. \begin{proposition} \label{propsing} For all $u\in\mathcal{G}(\Omega)$, $\mathop{\rm supp}u = \mathop{\rm supp}\jmath(u)$. \end{proposition} \begin{proof} The inclusion $\Omega\setminus \mathop{\rm supp}u\subseteq\Omega\setminus \mathop{\rm supp}\jmath(u)$ is immediate. Now let $x_0\in\Omega\setminus \mathop{\rm supp}\jmath(u)$. There exists an open neighborhood $V$ of $x_0$ such that for all $v\in\mathcal{G}_c(V)$, $\jmath(u)(v)=0$. Therefore, from Proposition \ref{propmo}, $u_{\vert_V}=0$ in $\mathcal{G}(V)$ and $x_0\in\Omega\setminus \mathop{\rm supp}u$. \end{proof} We conclude this section with a discussion of operators defined by integrals. In the sequel, $\pi_1$ and $\pi_2$ are the usual projections of $\Omega\times\Omega$ on $\Omega$. \begin{proposition} \label{propkernel} Let us consider the expression \begin{equation} \label{nuc} Ku(x) = \int_\Omega k(x,y)u(y)dy. \end{equation} \begin{itemize} \item[i)] If $k\in\mathcal{G}(\Omega\times\Omega)$ then \eqref{nuc} defines a linear map $K:\mathcal{G}_c(\Omega)\to\mathcal{G}(\Omega)$: $u\to Ku$, where $Ku$ is the generalized function with representative\\ $\left(\int_{\Omega}k_\epsilon(x,y)u_\epsilon(y)dy\right)_\epsilon$; \item[ii)] if $k\in\mathcal{G}^\infty(\Omega\times\Omega)$ then $K$ maps $\mathcal{G}_c(\Omega)$ into $\mathcal{G}^\infty(\Omega)$; \item[iii)] if $k\in\mathcal{G}_c(\Omega\times\Omega)$ then $K$ maps $\mathcal{G}(\Omega)$ into $\mathcal{G}_c(\Omega)$; \item[iv)] if $k\in\mathcal{G}_c^\infty(\Omega\times\Omega)$ then $K$ maps $\mathcal{G}(\Omega)$ into $\mathcal{G}^\infty_c(\Omega)$; \item[v)] if $k\in\mathcal{G}(\Omega\times\Omega)$ and $\pi_1,\pi_2:{\rm{supp}}\,k\to\Omega$ are proper then ${\rm{supp}}(Ku)\Subset\Omega$ for all $u\in\mathcal{G}_c(\Omega)$ and $K$ can be uniquely extended to a linear map from $\mathcal{G}(\Omega)$ into $\mathcal{G}(\Omega)$ such that for all $u\in\mathcal{G}(\Omega)$ and $v\in\mathcal{G}_c(\Omega)$ \begin{equation} \label{prop0} \int_\Omega Ku(x)v(x)\,dx=\int_\Omega u(y){\ }^tKv(y)\,dy \end{equation} where ${\ }^tKv(y)=\int_\Omega k(x,y)v(x)\,dx$; \item[vi)] if $k\in\mathcal{G}^\infty(\Omega\times\Omega)$ and $\pi_1,\pi_2:\mathop{\rm supp}k\to\Omega$ are proper then the extension defined above maps $\mathcal{G}(\Omega)$ into $\mathcal{G}^\infty(\Omega)$. \end{itemize} \end{proposition} The conditions on $\pi_1$ and $\pi_2$ of $v)$ and $vi)$ say that $\mathop{\rm supp}k$ is a proper subset of $\Omega\times\Omega$. \begin{proof} We give only some details of the proof of the fifth statement. The inclusion \begin{equation} \label{prop1} \mathop{\rm supp}(Ku)\subseteq\pi_1(\pi_2^{-1}(\mathop{\rm supp}u) \cap\mathop{\rm supp}k),\quad u\in\mathcal{G}_c(\Omega), \end{equation} leads to ${\rm{supp}}(Ku)\Subset\Omega$, under the assumption that $\pi_1,\pi_2:{\rm{supp}} k\to\Omega$ are proper maps. Let $V_1\subset V_2\subset\dots$ be an exhausting sequence of relatively compact open sets and $F_j=\pi_2(\pi_1^{-1}(\overline{V_j})\cap\mathop{\rm supp}k)$. From \eqref{prop1} it follows that \begin{equation} \label{prop2} \mathop{\rm supp}u\subseteq\Omega\setminus F_j\quad \Rightarrow\quad \mathop{\rm supp}(Ku)\subseteq\Omega\setminus\overline{V_j},\quad u\in\mathcal{G}_c(\Omega). \end{equation} Let $u\in\mathcal{G}(\Omega)$. We define $K_ju\in\mathcal{G}(V_j)$ by $K(\psi_ju)_{\vert_{V_j}}$ where $\psi_j\in\mathcal{C}^\infty_c(\Omega)$, $\psi_j\equiv 1$ in an open neighborhood of $F_j$. By the sheaf property of $\mathcal{G}(\Omega)$, there exists a generalized function $Ku$ such that $Ku_{\vert_{V_j}}=K_ju$, provided the family $\{K_ju\}_{j\in\mathbb{N}}$ is coherent. But from \eqref{prop2} we have that \[ (K_ju-K_iu)_{\vert_{V_i}}=K((\psi_j-\psi_i)u)_{\vert_{V_i}}=0 \] for $i0:\\ \forall y\in K,\ \forall\xi\in\mathbb{R}^p,\ \forall\epsilon\in(0,\eta],\ |\partial^\alpha_\xi\partial^\beta_y a_\epsilon(y,\xi)|\le c\langle\xi \rangle^{m-\rho|\alpha|+\delta|\beta|}\epsilon^{-N}, \end{gather*} where $\langle\xi\rangle=\sqrt{1+|\xi|^2}$. \end{definition} The subscript $M$ underlines the moderate growth property, i.e., the bound of type $\epsilon^{-N}$ as $\epsilon \to 0$. \begin{definition} \label{defnrhop} \rm An element of $\mathcal{N}^{m}_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ is a net $(a_\epsilon)_\epsilon\in\mathcal{S}^m_{\rho,\delta}[\Omega\times\mathbb{R}^p]$ satisfying the following requirement: \begin{gather*} \forall\alpha\in\mathbb{N}^p,\ \forall\beta\in\mathbb{N}^n,\ \forall K\Subset\Omega,\ \forall q\in\mathbb{N},\ \exists\eta\in(0,1], \ \exists c>0 :\\ \forall y\in K,\ \forall\xi\in\mathbb{R}^p,\ \forall\epsilon\in(0,\eta],\ |\partial^\alpha_\xi\partial^\beta_y a_\epsilon(y,\xi)|\le c\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}\epsilon^{q}. \end{gather*} Nets of this type are called negligible. \end{definition} \begin{definition} \label{defsrhop} \rm A (generalized) symbol of order $m$ and type $(\rho,\delta)$ is an element of the factor space $\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p) :=\mathcal{S}^{m}_{\rho,\delta,\rm{M}}(\Omega\times\mathbb{R}^p) /\mathcal{N}^{m}_{\rho,\delta}(\Omega\times\mathbb{R}^p)$. \end{definition} In the following we denote an arbitrary representative of $a\in\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ by $(a_\epsilon)_\epsilon$. \begin{definition} \label{defsrhopreg} \rm An element $a\in\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ is called regular if it has a representative $(a_\epsilon)_\epsilon$ with the following property: \begin{equation} \label{regu} \begin{gathered} \forall K\Subset\Omega,\ \exists N\in\mathbb{N}:\ \forall\alpha\in\mathbb{N}^p,\ \forall\beta\in\mathbb{N}^n,\ \exists\eta\in(0,1],\ \exists c>0: \\ \forall y\in K,\ \forall\xi\in\mathbb{R}^p,\ \forall\epsilon\in(0,\eta],\ |\partial^\alpha_\xi\partial^\beta_y a_\epsilon(y,\xi)|\le c\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}\epsilon^{-N}. \end{gathered} \end{equation} We denote by ${\widetilde{\mathcal{S}}}^m_{\rho,\delta,\rm{rg}} (\Omega\times\mathbb{R}^p)$ the subspace of regular elements of $\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$. \end{definition} If the property \eqref{regu} is true for one representative of $a$, it holds for every representative. Consequently, if ${\mathcal{S}}^m_{\rho,\delta,\rm{rg}}(\Omega\times\mathbb{R}^p)$ is the space defined by \eqref{regu}, we can introduce ${\widetilde{\mathcal{S}}}^m_{\rho,\delta,\rm{rg}}(\Omega\times\mathbb{R}^p)$ as the factor space ${\mathcal{S}}^m_{\rho,\delta,\rm{rg}}(\Omega\times\mathbb{R}^p) /\mathcal{N}^{m}_{\rho,\delta}(\Omega\times\mathbb{R}^p)$. It is easy to prove that $(a_\epsilon)_\epsilon\in\mathcal{S}^{m}_{\rho,\delta, \rm{M}}(\Omega\times\mathbb{R}^p)$ implies $(\partial^\alpha_\xi\partial^\beta_x a_\epsilon)_\epsilon\in \mathcal{S}^{m-\rho|\alpha|+ \delta|\beta|}_{\rho,\delta,{\rm{M}}}(\Omega\times\mathbb{R}^p)$ and if $(a_\epsilon)_\epsilon\in\mathcal{S}^{m_1}_{\rho,\delta,{\rm{M}}} (\Omega\times\mathbb{R}^p)$, $(b_\epsilon)_\epsilon\in\mathcal{S}^{m_2}_{\rho,\delta,{\rm{M}}} (\Omega\times\mathbb{R}^p)$ then $(a_\epsilon+b_\epsilon)_\epsilon\in\mathcal{S}^{\max(m_1,m_2)} _{\rho,\delta,{\rm{M}}}(\Omega\times\mathbb{R}^p)$ and $(a_\epsilon b_\epsilon)_\epsilon\in \mathcal{S}^{m_1+m_2}_{\rho,\delta,{\rm{M}}}(\Omega\times\mathbb{R}^p)$. Since the results concerning derivatives and sums hold with $\mathcal{S}_{\rho,\delta,{\rm{rg}}}$ and $\mathcal{N}_{\rho,\delta}$ in place of $\mathcal{S}_{\rho,\delta,{\rm{M}}}$, we can define derivatives and sums on the corresponding factor spaces. Moreover, $(a_\epsilon)_\epsilon\in\mathcal{S}^{m_1}_{\rho,\delta,{\rm{M}}} (\Omega\times\mathbb{R}^p)$ and $(b_\epsilon)_\epsilon\in\mathcal{N}^{m_2}_{\rho,\delta} (\Omega\times\mathbb{R}^p)$ imply $(a_\epsilon b_\epsilon)_\epsilon\in\mathcal{N}^{m_1+m_2}_{\rho,\delta} (\Omega\times\mathbb{R}^p)$, thus we obtain that the product is a well-defined map from the space $\widetilde{\mathcal{S}}^{m_1}_{\rho,\delta}(\Omega\times\mathbb{R}^p)\times \widetilde{\mathcal{S}}^{m_2}_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ into $\widetilde{\mathcal{S}}^{m_1+m_2}_{\rho,\delta}(\Omega\times\mathbb{R}^p)$. Similarly, it is well-defined as a map from ${\widetilde{\mathcal{S}}}^{m_1}_{\rho,\delta,{\rm{rg}}} (\Omega\times\mathbb{R}^p)\times {\widetilde{\mathcal{S}}}^{m_2}_{\rho,\delta,{\rm{rg}}} (\Omega\times\mathbb{R}^p)$ into ${\widetilde{\mathcal{S}}}^{m_1+m_2}_{\rho,\delta,{\rm{rg}}} (\Omega\times\mathbb{R}^p)$. The classical space $S^{m}_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ is contained in ${\widetilde{\mathcal{S}}}^m_{\rho,\delta,\rm{rg}} (\Omega\times\mathbb{R}^p)$. Let us now study the dependence of $\widetilde{\mathcal{S}}^m_{\rho,\delta} (\Omega\times\mathbb{R}^p)$ on the open set $\Omega\subseteq\mathbb{R}^n$. We can define the restriction of $a\in\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ to an open subset $\Omega'$ of $\Omega$ by setting \[ a_{\vert_{\Omega'}}:=({a_\epsilon}_{\vert_{\Omega'}}) +\mathcal{N}^{m}_{\rho,\delta}(\Omega'\times\mathbb{R}^p). \] Following the same arguments adopted in the proof of Theorem 1.2.4 in \cite{GKOS:01}, we obtain that $\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ is a sheaf with respect to $\Omega$. This fact allows us to define $\mathop{\rm supp}_y\,a$ as the complement of the largest open set $\Omega'\subseteq\Omega$ such that $a_{\vert_{\Omega'}}=0$. We assume from now on that $\rho>0$ and $\delta<1$ and return to the meaning of the integral \begin{equation} \label{inta} \int_{K\times\mathbb{R}^p}\hskip-10pt e^{i\phi(y,\xi)} a_\epsilon(y,\xi)\, dy\,d\xi. \end{equation} Obviously, if $(a_\epsilon)_\epsilon\in\mathcal{S}^{m}_{\rho,\delta,\rm{M}} (\Omega\times\mathbb{R}^p)$ then \eqref{inta} makes sense as an oscillatory integral for every $\epsilon\in(0,1]$. Since our aim is to estimate its asymptotic behavior with respect to $\epsilon$, we state a lemma obtained as a simple adaptation of the reasoning presented in \cite[p.122-123]{Dieudonne:78}, \cite[p.88-89]{Hoermander:71}, \cite[p.4-5]{Shubin:87}. We recall that given the phase function $\phi$, there exists an operator \[ L=\sum_{i=1}^p a_i(y,\xi)\frac{\partial}{\partial\xi_i} +\sum_{k=1}^n b_k(y,\xi)\frac{\partial}{\partial y_k}+c(y,\xi) \] such that $a_i(y,\xi)\in S^0(\Omega\times\mathbb{R}^p)$, $b_k(y,\xi)\in S^{-1}(\Omega\times\mathbb{R}^p)$, $c(y,\xi)\in S^{-1}(\Omega\times\mathbb{R}^p)$, and such that ${\ }^tLe^{i\phi}=e^{i\phi}$, where ${\ }^tL$ is the formal adjoint. \begin{lemma} \label{leml} Let $s=\min\{\rho,1-\delta\}$ and $j\in\mathbb{N}$. Then the following statements hold: \begin{itemize} \item[i)] if $(a_\epsilon)_\epsilon\in\mathcal{S}^{m}_{\rho,\delta,\rm{M}} (\Omega\times\mathbb{R}^p)$ then $(L^ja_\epsilon)_\epsilon\in \mathcal{S}^{m-js}_{\rho,\delta,{\rm{M}}}(\Omega\times\mathbb{R}^p)$; \item[ii)] i) is valid with $\mathcal{S}_{\rho,\delta,\rm{rg}}$ in place of $\mathcal{S}_{\rho,\delta,{\rm{M}}}$; \item[iii)] i) is valid with $\mathcal{N}_{\rho,\delta}$ in place of $\mathcal{S}_{\rho,\delta,{\rm{M}}}$. \end{itemize} \end{lemma} For completeness we recall that for $m-js<-n$ and $\chi\in\mathcal{C}^\infty_c(\mathbb{R}^p)$ identically equal to 1 in a neighborhood of the origin, the oscillatory integral $I_{\phi,K}(a_\epsilon)$, at fixed $\epsilon$, can be defined by either of the two expressions on the right hand-side of \eqref{osc}: \begin{equation} \label{osc} \begin{split} I_{\phi,K}(a_\epsilon) &:=\int_{K\times\mathbb{R}^p} e^{i\phi(y,\xi)}a_\epsilon(y,\xi)\,dy\,d\xi\\ &=\int_{K\times\mathbb{R}^p}\hskip-20pt e^{i\phi(y,\xi)}L^j a_\epsilon(y,\xi)\, dy\,d\xi\\ &=\lim_{h\to 0^+}\int_{K\times\mathbb{R}^p} e^{i\phi(y,\xi)}a_\epsilon(y,\xi)\chi(h\xi)\, dy\,d\xi, \end{split} \end{equation} where the equalities hold for all $\epsilon\in(0,1]$. \begin{proposition} \label{propiok} Let $K$ be a compact set contained in $\Omega$. Let $\phi$ be a phase function on $\Omega\times\mathbb{R}^p$ and $a$ an element of $\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$. The oscillatory integral \[ I_{\phi,K}(a):=\int_{K\times\mathbb{R}^p}\hskip-20pt e^{i\phi(y,\xi)}a(y,\xi)\, dy\,d\xi := (I_{\phi,K}(a_\epsilon))_\epsilon +\mathcal{N} \] is a well-defined element of $\widetilde{\mathbb{C}}$. \end{proposition} \begin{proof} From Lemma \ref{leml}, if $(a_\epsilon)_\epsilon\in\mathcal{S}^{m}_{\rho,\delta,\rm{M}} (\Omega\times\mathbb{R}^p)$ then $(L^ja_\epsilon)_\epsilon\in\mathcal{S}^{m-js}_{\rho,\delta,{\rm{M}}} (\Omega\times\mathbb{R}^p)$ for every $j\in\mathbb{N}$. Taking $m-js<-n$, it is easy to see that $(I_{\phi,K}(a_\epsilon))_\epsilon\in\mathcal{E}_M$. Analogously, if $(a_\epsilon)_\epsilon$ is negligible, we have that $(I_{\phi,K}(a_\epsilon))_\epsilon\in\mathcal{N}$. \end{proof} \begin{definition} \label{defoi} \rm Let $a\in\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ with $\mathop{\rm supp}_y\,a\Subset\Omega$. We define the (generalized) oscillatory integral \[ I_{\phi}(a):=\int_{\Omega\times\mathbb{R}^p}\hskip-20pt e^{i\phi(y,\xi)}a(y,\xi)\, dy\,d\xi :=\int_{K\times\mathbb{R}^p}\hskip-20pt e^{i\phi(y,\xi)}a(y,\xi)\, dy\,d\xi, \] where $K$ is any compact subset of $\Omega$ containing $\mathop{\rm supp}_y\,a$ in its interior. \end{definition} It remains to show that this definition does not depend on the choice of $K$. Let $K_1,K_2\Subset\Omega$ with $\mathop{\rm supp}_y a\subseteq \mathop{\rm int}K_1\cap\mathop{\rm int}K_2$ and put $K_3=K_1\cup K_2$. But for $i=1,2$ and $j$ large enough \begin{align*} &\Bigl|\int_{K_3\times\mathbb{R}^p}\hskip-18pt e^{i\phi(y,\xi)}a_\epsilon(y,\xi)\, dy\,d\xi-\int_{K_i\times\mathbb{R}^p}\hskip-18pt e^{i\phi(y,\xi)}a_\epsilon(y,\xi)\, dy\,d\xi\Bigr|\\ &\le \int_{K_3\setminus\mathop{\rm int}K_i\times\mathbb{R}^p}|L^j a_\epsilon(y,\xi)|dy\,d\xi=O(\epsilon^q) \end{align*} for arbitrary $q\in\mathbb{N}$ since $K_3\setminus\mathop{\rm int}K_i$ is a compact subset of $\Omega\setminus\mathop{\rm supp}_y\,a$, as desired. It is clear that for each compact set $K$ containing ${\mathop{\rm supp}}_y\,a$ in its interior, we can find representatives $(a_\epsilon)_\epsilon$ with $\mathop{\rm supp}_y\,a_\epsilon\subset K$ for all $\epsilon$. For such a representative of $I_\phi(a)$, its components are defined by the classical oscillatory integral $\int_{\Omega\times\mathbb{R}^p}e^{i\phi(y,\xi)}a_\epsilon(y,\xi)\,dy\, d\llap{\raisebox{.9ex}{$\scriptstyle-\!$}}\xi$. \begin{remark} \label{remoi} \rm A particular example of a generalized oscillatory integral on $\Omega\times\mathbb{R}^p$ is given by \[ I_{\phi}(au):=\int_{\Omega\times\mathbb{R}^p}\hskip-10pt e^{i\phi(y,\xi)}a(y,\xi)u(y)\, dy\,d\xi, \] where $a\in\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ and $u\in\mathcal{G}_c(\Omega)$. We observe that the map $I_\phi(a):\mathcal{G}_c(\Omega)\to\widetilde{\mathbb{C}}:u\to I_\phi(au)$ is well-defined and belongs to $L(\mathcal{G}_c(\Omega),\widetilde{\mathbb{C}})$. \end{remark} We consider now phase functions and symbols depending on an additional parameter. We want to study oscillatory integrals of the form \[ I_{\phi,K}(a)(x):=\int_{K\times\mathbb{R}^p}\hskip-20pt e^{i\phi(x,y,\xi)}a(x,y,\xi)\, dy\,d\xi, \] where $x\in\Omega'$, an open subset of $\mathbb{R}^{n'}$. Obviously, if for any fixed $x\in\Omega'$, $\phi(x,y,\xi)$ is a phase function with respect to the variables $(y,\xi)$ and $a(x,y,\xi)$ belongs to $\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$, the oscillatory integral $I_{\phi,K}(a)(x)$ defines a map from $\Omega'$ to $\widetilde{\mathbb{C}}$. The smooth dependence of this map on the parameter $x$ is investigated in the following Remark \ref{remoipam} and in Proposition \ref{propw}. \begin{remark} \label{remoipam} \rm Let $\phi(x,y,\xi)$ be a real valued continuous function on $\Omega'\times\Omega\times\mathbb{R}^p$, smooth on $\Omega'\times\Omega\times\mathbb{R}^p\setminus\{0\}$ such that for all $x\in\Omega'$, $\phi(x,y,\xi)$ is a phase function with respect to $(y,\xi)$. As in Lemma \ref{leml}, we have that for all $j\in\mathbb{N}$, $(a_\epsilon)_\epsilon\in\mathcal{S}^m_{\rho,\delta,{\rm{M}}}(\Omega'\times \Omega\times\mathbb{R}^p)$ implies $(L^j_xa_\epsilon(x,y,\xi))_\epsilon\in \mathcal{S}^{m-js}_{\rho,\delta,{\rm{M}}}(\Omega'\times\Omega\times\mathbb{R}^p)$. The same result holds with $\mathcal{S}_{\rho,\delta,{\rm{rg}}}$ in place of $\mathcal{S}_{\rho,\delta,\rm{M}}$ and $\mathcal{N}_{\rho,\delta}$ in place of $\mathcal{S}_{\rho,\delta,\rm{M}}$ (this follows easily along the lines of \cite[p.124-125]{Dieudonne:78} and \cite[p.90]{Hoermander:71}). \end{remark} \begin{proposition} \label{propw} Let $\phi(x,y,\xi)$ be as in Remark \ref{remoipam}. \begin{itemize} \item[i)] If $a(x,y,\xi)\in\widetilde{\mathcal{S}}^{m}_{\rho,\delta}(\Omega'\times\Omega\times\mathbb{R}^p)$ then for all $K\Subset\Omega$ \[ w_K(x):=\int_{K\times\mathbb{R}^p}\hskip-20pt e^{i\phi(x,y,\xi)}a(x,y,\xi)\, dy\,d\xi \] belongs to $\mathcal{G}(\Omega')$. \item[ii)] If $a(x,y,\xi)\in{\widetilde{\mathcal{S}}}^m_{\rho,\delta,{\rm{rg}}}(\Omega'\times\Omega\times\mathbb{R}^p)$ then $w_K\in\mathcal{G}^\infty(\Omega')$ for all $K\Subset\Omega$. \item[iii)] If in addition $\phi$ is a phase function in $(x,y,\xi)$ then for all $K'\Subset\Omega'$ \[ \int_{K'}w_K(x)\,dx=\int_{K'\times K\times\mathbb{R}^p}\hskip-20pt e^{i\phi(x,y,\xi)}a(x,y,\xi)\, dx\,dy\,d\xi. \] \end{itemize} \end{proposition} \begin{proof} An arbitrary representative of $w_K$ is given by the oscillatory integral \[ (w_{K,\epsilon}(x))_\epsilon:=\Bigl(\int_{K\times\mathbb{R}^p}\hskip-20pt e^{i\phi(x,y,\xi)}a_\epsilon(x,y,\xi)\, dy d\xi\Bigr)_\epsilon. \] From Remark \ref{remoipam} it follows that $\big(\int_{K\times\mathbb{R}^p} e^{i\phi(x,y,\xi)}a_\epsilon(x,y,\xi)\, dy d\xi\big)_\epsilon\in\mathcal{E}[\Omega']$. At this point by computing the $x$-derivatives of the expression $e^{i\phi(x,y,\xi)}L^j_xa_\epsilon(x,y,\xi)$ for $\xi\neq 0$, we conclude that \begin{equation} \label{eqlj} \begin{gathered} \forall\alpha\in\mathbb{N}^{n'},\ \forall K'\Subset \Omega',\ \exists N\in\mathbb{N},\ \exists\eta\in(0,1]:\ \forall x\in K',\ \forall y\in K,\\ \forall\xi\in\mathbb{R}^p\setminus\{0\},\ \forall\epsilon\in(0,\eta],\ |\partial^\alpha_x(e^{i\phi(x,y,\xi)}L^j_xa_\epsilon(x,y,\xi))|\le \langle\xi\rangle^{m-js+|\alpha|}\epsilon^{-N}. \end{gathered} \end{equation} Now if $m-js+|\alpha|<-n$ then we obtain for $x\in K'$ and $\epsilon\in(0,\eta]$, \[ |\partial^\alpha_xw_{K,\epsilon}(x)|\le\epsilon^{-N}. \] Therefore $(w_{K,\epsilon})_\epsilon\in\mathcal{E}_M(\Omega')$. Obviously if $(a_\epsilon)_\epsilon\in\mathcal{N}^m_{\rho,\delta}(\Omega' \times\Omega\times\mathbb{R}^p)$ then $(w_{K,\epsilon})_\epsilon\in\mathcal{N}(\Omega')$. If $a\in{\widetilde{\mathcal{S}}}^{m}_{\rho,\delta,\rm{rg}} (\Omega'\times\Omega\times\mathbb{R}^p)$ the exponent $N$ in \eqref{eqlj} does not depend on the derivatives and then $(w_{K,\epsilon})_\epsilon\in\mathcal{E}^\infty_M(\Omega')$. This result completes the proof of the first two assertions. The proof of the third point is a consequence of the analogous statement in \cite[(23.17.6)]{Dieudonne:78}, applied to representatives. \end{proof} \begin{remark} \label{remrep} \rm Combining Proposition \ref{propw} with Definition \ref{defoi}, we obtain the following results: \begin{itemize} \item[i)] if $\phi$ is a phase function with respect to $(y,\xi)$ and there exists a compact set $K$ of $\Omega$ such that for all $x\in\Omega'$, $\mathop{\rm supp}_y\,a(x,\cdot,\cdot)\hskip-2pt\subseteq K$ then the oscillatory integral $\int_{\Omega\times\mathbb{R}^p}e^{i\phi(x,y,\xi)}a(x,y,\xi)\,dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi$ defines a generalized function belonging to $\mathcal{G}(\Omega')$; \item[ii)] if $\phi$ is a phase function with respect to $(y,\xi)$ and $(x,\xi)$ and $\mathop{\rm supp}_{x,y}\,a\Subset\Omega'\times\Omega$ then the two oscillatory integrals $\int_{\Omega\times\mathbb{R}^p}e^{i\phi(x,y,\xi)}a(x,y,\xi)\,dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi$ and $\int_{\Omega'\times\mathbb{R}^p}e^{i\phi(x,y,\xi)}a(x,y,\xi)\,dx\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi$ belong to $\mathcal{G}(\Omega')$ and $\mathcal{G}(\Omega)$ respectively. Moreover \[ \begin{split} \int_{\Omega'\times\Omega\times\mathbb{R}^p}\hskip-20pt e^{i\phi(x,y,\xi)}a(x,y,\xi)\,dx\,dy\, d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi &=\int_{\Omega'}\int_{\Omega\times\mathbb{R}^p}\hskip-20pt e^{i\phi(x,y,\xi)}a(x,y,\xi)\,dy\, d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi\, dx\\ &= \int_{\Omega}\int_{\Omega'\times\mathbb{R}^p}\hskip-20pt e^{i\phi(x,y,\xi)}a(x,y,\xi)\,dx\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}} \xi\, dy. \end{split} \] \end{itemize} \end{remark} \begin{remark} \label{remcphi} \rm We recall that for each phase function $\phi(x,\xi)$ \[ C_\phi:=\{(x,\xi)\in\Omega\times\mathbb{R}^p\setminus\{0\}: \nabla_\xi\phi(x,\xi)=0\} \] is a cone-shaped subset of $\Omega\times\mathbb{R}^p\setminus\{0\}$. Let $\pi:\Omega\times\mathbb{R}^p\setminus\{0\}\to\Omega$ be the projection onto $\Omega$ and put $S_\phi:=\pi C_\phi$, $R_\phi:=\Omega\setminus S_\phi$. Interpreting $x\in\Omega$ as a parameter we have from Proposition \ref{propw} that \[ w(x):=\int_{\mathbb{R}^p}e^{i\phi(x,\xi)}a(x,\xi) d\xi=\Bigl(\int_{\mathbb{R}^p}e^{i\phi(x,\xi)}a_\epsilon(x,\xi) d\xi\Bigr)_\epsilon +\mathcal{N} \] makes sense as an oscillatory integral for $x\in R_\phi$. More precisely, we have that \begin{itemize} \item[i)] if $a\in\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$ then $w\in\mathcal{G}(R_\phi)$; \item[ii)] if $a\in\widetilde{\mathcal{S}}^m_{\rho,\delta} (\Omega\times\mathbb{R}^p)$ and ${\rm{supp}}_x\,a\subseteq \Omega\setminus\Omega'$, where $\Omega'$ is an open neighborhood of $S_\phi$, then $w$ can be extended to a generalized function on $\Omega$ with support contained in $\Omega\setminus\Omega'$; \item[iii)] i) and ii) hold with $\mathcal{S}_{\rho,\delta,{\rm{rg}}}$ in place of $\mathcal{S}_{\rho,\delta}$ and $\mathcal{G}^\infty$ in place of $\mathcal{G}$; \item[iv)] if $a\in\widetilde{\mathcal{S}}^m_{\rho,\delta} (\Omega\times\mathbb{R}^p)$ then for all $u\in\mathcal{G}_c(R_\phi)$ \begin{equation} \label{eqw} \int_{\Omega\times\mathbb{R}^p}\hskip-20pt e^{i\phi(x,\xi)}a(x,\xi)u(x)\, dx\,d\xi=\int_\Omega w(x)u(x)\,dx. \end{equation} \item[v)] under the hypothesis of the second statement, \eqref{eqw} holds for all $u\in\mathcal{G}_c(\Omega)$. \end{itemize} \end{remark} We just give some details concerning the proof of the assertion $ii)$ and $v)$. Let $\{\Omega'_j\}_{j\in\mathbb{N}\setminus \{0\}}$ be an open covering of $\Omega'$ such that $\Omega'_j$ is relatively compact and $\Omega'_j\subseteq\overline{\Omega'_j}\subseteq\Omega'_{j+1}$ for all $j$. Choosing cut-off functions $\{\psi_j\}_{j\in\mathbb{N}\setminus \{0\}}$ such that $\psi_j\in\mathcal{C}^\infty_c(\Omega')$ and $\psi_j\equiv 1$ in a neighborhood of $\overline{\Omega'_j}$, we observe that $((1-\psi_j(x))a_\epsilon(x,\xi))_\epsilon$ is a representative of $a$ identically equal to 0 on $\Omega'_j$ for all $\epsilon\in(0,1]$. At this point we see that \begin{gather*} w_0(x):=\Bigl(\Bigl(\int_{\mathbb{R}^n}e^{i\phi(x,\xi)}a_\epsilon(x,\xi)\, d\xi\Bigr)\big|_{R_\phi} \Bigr)_\epsilon +\mathcal{N}(R_\phi),\\ w_j(x):=\Bigl(\Bigl(\int_{\mathbb{R}^n}e^{i\phi(x,\xi)}(1-\psi_j(x)) a_\epsilon(x,\xi)\,d\xi\Bigr)\big|_{\Omega'_j} \Bigr)_\epsilon +\mathcal{N}(\Omega'_j),\quad j\ge 1 \end{gather*} is a coherent family of generalized functions which defines $w\in\mathcal{G}(\Omega)$ such that $w_0$ and $w_j$ are its restrictions to $R_\phi$ and $\Omega'_j$ respectively. Consequently, $\mathop{\rm supp}w\subseteq R_\phi\setminus\Omega'\equiv\Omega\setminus\Omega'$. Now for any $u\in\mathcal{G}_c(\Omega)$, $\mathop{\rm supp}_x (au)\cup\mathop{\rm supp}(wu)\subseteq (R_\phi\setminus\Omega')\cap\mathop{\rm supp}u\Subset R_\phi$. Taking $\psi\in\mathcal{C}^\infty_c(R_\phi)$ identically $1$ in a neighborhood of $(R_\phi\setminus\Omega')\cap\mathop{\rm supp}u$ we have that $wu=wu\psi$ in $\mathcal{G}(\Omega)$ and $au=au\psi$ in $\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega\times\mathbb{R}^p)$. Since $\psi u\in\mathcal{G}_c(R_\phi)$, $iv)$ gives \[ \begin{split} \int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i\phi(x,\xi)}a(x,\xi)u(x)\,dx\, d\xi &=\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i\phi(x,\xi)}a(x,\xi)u(x) \psi(x)\,dx\, d\xi\\ &=\int_{\Omega}\hskip-3pt w(x)u(x)\psi(x)\, dx=\int_\Omega\hskip-3pt w(x)u(x)\,dx. \end{split} \] \section{Pseudodifferential operators with generalized amplitudes} As mentioned in the Introduction and as will be seen shortly, we will need different asymptotic scales. This requires an extension of Definition \ref{defsMrhop} and \ref{defnrhop} which we now state. \begin{definition} \label{defsMmup} \rm Let $m,\mu,\rho,\delta$ be real numbers, $\rho,\delta\in[0,1]$. Let $\omega$ be a real valued function on the interval $(0,1]$, $\omega>0$, such that for some $r$ in $\mathbb{R}$, for some $C>0$ and for all $\epsilon\in(0,1]$, $\omega(\epsilon)\ge C\epsilon^r$. We denote by ${\mathcal{S}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p)$, $\Omega'$ an open subset of $\mathbb{R}^{n'}$, the set of all $(a_\epsilon)_\epsilon\in\mathcal{S}^m_{\rho,\delta}[\Omega' \times\mathbb{R}^p]$ such that the following statement holds: \begin{equation} \label{mod} \begin{gathered} \forall K\Subset\Omega',\ \exists N\in\mathbb{N}: \forall\alpha\in\mathbb{N}^p,\ \forall\beta\in\mathbb{N}^{n'},\ \exists\eta\in(0,1],\ \exists c>0: \forall x\in K,\ \forall\xi\in\mathbb{R}^p,\\ \forall\epsilon\in(0,\eta],\ |\partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi)|\le c \langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}\epsilon^{-N} \omega(\epsilon)^{-(|\beta|-\mu)_+}. \end{gathered} \end{equation} \end{definition} The exponent $-(|\beta|-\mu)_+=-\max\{0,|\beta|-\mu\}$ reflects differentiability up to order $\mu$ in the case when $a_\epsilon$ is obtained from a non-smooth, classical symbol by convolution with a mollifier with scale $\omega(\epsilon)$. \begin{definition} \label{defnmup} \rm An element of ${\mathcal{N}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p)$ is a net in $\mathcal{S}^m_{\rho,\delta}[\Omega'\times\mathbb{R}^p]$ fulfilling the following condition: \begin{equation} \label{neg} \begin{gathered} \forall K\Subset\Omega',\ \forall\alpha\in\mathbb{N}^p,\ \forall\beta\in\mathbb{N}^{n'},\ \forall q\in\mathbb{N},\ \exists\eta\in(0,1],\ \exists c>0:\ \forall x\in K,\\ \forall\xi\in\mathbb{R}^p,\ \forall\epsilon\in(0,\eta],\ |\partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi)|\le c \langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}\epsilon^q \omega(\epsilon)^{-(|\beta|-\mu)_+}. \end{gathered} \end{equation} Nets with this property are called negligible. \end{definition} The factor space ${\mathcal{S}}^{m,\mu}_{\rho,\delta,\omega} (\Omega'\times\mathbb{R}^p)/{\mathcal{N}}^{m,\mu}_{\rho,\delta,\omega} (\Omega'\times\mathbb{R}^p)$ will be denoted by ${\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega'\times\mathbb{R}^p)$. Related to Definitions \ref{defsMrhop} and \ref{defnrhop}, we note that ${\mathcal{S}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p) \subseteq\mathcal{S}^m_{\rho,\delta,{\rm{M}}}(\Omega'\times\mathbb{R}^p)$, ${\mathcal{N}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p) \subseteq\mathcal{N}^m_{\rho,\delta}(\Omega'\times\mathbb{R}^p)$ and ${\mathcal{S}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p) \cap\mathcal{N}^m_{\rho,\delta}(\Omega'\times\mathbb{R}^p) \subseteq{\mathcal{N}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p)$, provided $(\omega(\epsilon))_\epsilon$ belongs to ${\mathcal E}_M$. Therefore, ${\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p)\subseteq\widetilde{\mathcal{S}}^m_{\rho,\delta}(\Omega'\times\mathbb{R}^p)$. One easily proves that the following mapping properties hold: \[ \begin{split} \partial^\alpha_\xi\partial^\beta_x &:{\widetilde{\mathcal{S}}}^{m,\mu} _{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p)\to{\widetilde{\mathcal{S}}} ^{m-\rho|\alpha|+\delta|\beta|,\mu-|\beta|}_{\rho,\delta,\omega} (\Omega'\times\mathbb{R}^p),\\ + &:{\widetilde{\mathcal{S}}}^{m_1,\mu}_{\rho,\delta,\omega} (\Omega'\times\mathbb{R}^p)\times{\widetilde{\mathcal{S}}} ^{m_2,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p) \to{\widetilde{\mathcal{S}}}^{\max(m_1,m_2),\mu}_{\rho,\delta,\omega} (\Omega'\times\mathbb{R}^p). \end{split} \] As mentioned in the Introduction, an important notion is that of a {\em slow scale net}. Recall that a net $(r_\epsilon)\in\mathbb{C}^{(0,1]}$ is a slow scale net if for every $q\ge 0$ there exist $c_q>0$ such that for all $\epsilon\in(0,1]$ \begin{equation} \label{slow} |r_\epsilon|^q\le c_q\epsilon^{-1}. \end{equation} \begin{remark} \label{remnew} \rm If in addition to the usual assumptions on $(\omega(\epsilon))_\epsilon$, we assume that $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net, we can uniformly bound the contributions of the derivatives in \eqref{mod} by a single negative power of $\epsilon$, obtaining for a suitable constant $c$, for $x\in K$ and for all $\epsilon$ small enough \[ |\partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi)|\le c\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}\epsilon^{-N-1}. \] This means that if $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net then ${\mathcal{S}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p)\subseteq{\mathcal{S}}^{m}_{\rho,\delta,{\rm{rg}}}(\Omega'\times\mathbb{R}^p)$. If in addition $(\omega(\epsilon))_\epsilon \in {\mathcal E}_M$, then also ${\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega' \times\mathbb{R}^p) \subseteq {\widetilde{\mathcal{S}}}^{m}_{\rho,\delta,{\rm{rg}}}(\Omega'\times \mathbb{R}^p)$. Further, if $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net and $\sup_{\epsilon\in(0,1]}\omega(\epsilon)<\infty$ then $$ {\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega' \times\mathbb{R}^p) = {\widetilde{\mathcal{S}}}^{m}_{\rho,\delta, {\rm{rg}}}(\Omega'\times\mathbb{R}^p) $$ since ${\mathcal{S}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p) = {\mathcal{S}}^m_{\rho,\delta,\rm{rg}}(\Omega\times\mathbb{R}^p)$ and ${\mathcal{N}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p) = \mathcal{N}^m_{\rho,\delta}(\Omega'\times\mathbb{R}^p)$. \end{remark} \begin{definition} \label{defsip} \rm We denote by $\mathcal{S}^{-\infty}_{\rm{rg}}(\Omega'\times\mathbb{R}^p)$ the set of all $(a_\epsilon)_\epsilon\in\mathcal{S}^{-\infty}[\Omega'\times\mathbb{R}^p]$ such that \begin{equation} \label{sim} \begin{gathered} \forall K\Subset\Omega',\ \exists N\in\mathbb{N}:\ \forall m\in\mathbb{R},\ \forall\alpha\in\mathbb{N}^p,\ \forall\beta\in\mathbb{N}^{n'}\hskip-4pt,\ \exists\eta\in(0,1],\ \exists c>0\ :\\ \forall x\in K,\ \forall\xi\in\mathbb{R}^p,\ \forall\epsilon\in(0,\eta],\ |\partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi)|\le c\langle\xi\rangle^{m-|\alpha|}\epsilon^{-N}. \end{gathered} \end{equation} We denote by $\mathcal{N}^{-\infty}(\Omega'\times\mathbb{R}^p)$ the set of all $(a_\epsilon)_\epsilon\in\mathcal{S}^{-\infty}[\Omega'\times\mathbb{R}^p]$ such that \begin{equation} \label{ni} \begin{gathered} \forall K\Subset\Omega',\ \forall m\in\mathbb{R},\ \forall q\in\mathbb{N},\ \forall\alpha\in\mathbb{N}^p,\ \forall\beta\in\mathbb{N}^{n'},\ \exists\eta\in(0,1],\ \exists c>0\ :\\ \forall x\in K,\ \forall\xi\in\mathbb{R}^p,\ \forall\epsilon\in(0,\eta],\ |\partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi)|\le c\langle \xi\rangle^{m-|\alpha|}\epsilon^{q}. \end{gathered} \end{equation} The factor space $\mathcal{S}^{-\infty}_{\rm{rg}}(\Omega'\times\mathbb{R}^p)/ \mathcal{N}^{-\infty}(\Omega'\times\mathbb{R}^p)$ will be denoted by ${\widetilde{\mathcal{S}}}^{-\infty}_{\rm{rg}}(\Omega'\times\mathbb{R}^p)$. \end{definition} \begin{definition} \label{defsmu} \rm Let $\Omega$ be an open subset of $\mathbb{R}^n$. The elements of the factor spaces ${\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$ and ${\widetilde{\mathcal{S}}} ^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\Omega\times\mathbb{R}^n)$ will be called symbols and amplitudes of order $m$ and type $(\rho,\delta,\mu,\omega)$, respectively. The elements of the factor spaces $\widetilde{\mathcal{S}}^{-\infty}_{\rm{rg}}(\Omega\times\mathbb{R}^n)$ and $\widetilde{\mathcal{S}}^{-\infty}_{\rm{rg}}(\Omega\times\Omega\times \mathbb{R}^n)$ will be called smoothing symbols and smoothing amplitudes, respectively. \end{definition} \begin{example} \label{exZyg} \rm Let $(\omega(\epsilon))_\epsilon$ be a net as in Definition \ref{defsMmup} tending to 0 as $\epsilon$ goes to 0. Given $\mu\in\mathbb{R}\setminus\mathbb{N}$, we denote by $\mathcal{G}^\mu_{\ast,loc,\omega}(\Omega)$ the space of all generalized functions $a\in\mathcal{G}(\Omega)$ having a representative $(a_\epsilon)_\epsilon$ satisfying the condition: \[ \forall K\Subset\Omega,\ \forall\alpha\in\mathbb{N}^n,\quad \Vert\partial^\alpha a_\epsilon\Vert_{L^\infty(K)} =\begin{cases} O(1), &0\le|\alpha|\le\mu,\\ O(\omega(\epsilon)^{\mu-|\alpha|}), & |\alpha|>\mu \end{cases}\quad (\epsilon\to 0). \] This notion is a modified version (with scales) of the generalized Zygmund regularity introduced in \cite{GH:02b}. It is clear that for any $b\in{\widetilde{\mathcal{S}}}^{m}_{\rho,\delta,{\rm{rg}}} (\Omega\times\mathbb{R}^n)$ and $a \in \mathcal{G}^\mu_{\ast,loc,\omega}(\Omega)$, the product $a(x)b(x,\xi)$ defines a symbol in ${\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$. It follows from the results in \cite{GH:02b} that if $f$ is a function belonging to the Zygmund class $\mathcal{C}^\mu_\ast(\mathbb{R}^n)$ and $\varphi\in{\mathscr{S}}(\mathbb{R}^n)$ is a radial mollifier with $\int_{\mathbb{R}^n}\varphi(x)dx=1$ and $\int_{\mathbb{R}^n}x^\alpha\varphi(x)dx=0$ for all $\alpha\neq 0$ then $f\ast\varphi_{\omega(\epsilon)}$ satisfies the generalized Zygmund property defining $\mathcal{G}^\mu_{\ast,loc,\omega}(\Omega)$. Thus, for any $b$ as above, $((f\ast\varphi_{\omega(\epsilon)})_{\vert_\Omega}(x)b(x,\xi))_\epsilon$ may serve as a representative for a symbol in $\widetilde{\mathcal S}_{\rho,\delta,\omega}^{m,\mu}(\Omega \times \mathbb{R}^n)$. \end{example} In the sequel we assume $\rho>0$, $\delta<1$ and let $d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi=(2\pi)^{-n}d\xi$. \begin{proposition} \label{propseudo} Let $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$. The oscillatory integral \begin{equation} \label{pdo} Au:=\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a(x,y,\xi)u(y)\, dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi = \big(A_\epsilon u_\epsilon(x)\big)_\epsilon + \mathcal{N}(\Omega) \end{equation} where \[ A_\epsilon u_\epsilon(x) =\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a_\epsilon(x,y,\xi)u_\epsilon(y) \, dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi \] defines a linear map from $\mathcal{G}_c(\Omega)$ into $\mathcal{G}(\Omega)$. If $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net then the oscillatory integral \eqref{pdo} defines a linear map from $\mathcal{G}^\infty_c(\Omega)$ to $\mathcal{G}^\infty(\Omega)$. Finally, if $a\in\widetilde{\mathcal{S}}^{-\infty}_{\rm{rg}}(\Omega\times\Omega \times\mathbb{R}^n)$ then \eqref{pdo} defines a linear map from $\mathcal{G}_c(\Omega)$ to $\mathcal{G}^\infty(\Omega)$. \end{proposition} \begin{proof} In \eqref{pdo}, $\phi(x,y,\xi)=(x-y)\xi$ satisfies the assumptions of Remark \ref{remoipam}. It is immediate to prove that the map ${\widetilde{\mathcal{S}}}^{m,\mu} _{\rho,\delta,\omega}(\Omega\times\Omega\times\mathbb{R}^n)\times\mathcal{G} _c(\Omega)\to\widetilde{\mathcal{S}}^m_{\rho,\delta} (\Omega\times\Omega\times\mathbb{R}^n):(a,u)\to a(x,y,\xi)u(y) :=(a_\epsilon(x,y,\xi)u_\epsilon(y))_\epsilon+ \mathcal{N}^m_{\rho,\delta}(\Omega\times\Omega\times\mathbb{R}^n)$ is well-defined. From Proposition \ref{propw} and Remark \ref{remrep}, assertion $i)$, we obtain that $A$ is a linear map from $\mathcal{G}_c(\Omega)$ into $\mathcal{G}(\Omega)$. If $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net and $u\in\mathcal{G}^\infty_c(\Omega)$ then $au\in{{\widetilde{\mathcal{S}}}}^m_{\rho,\delta,{\rm{rg}}} (\Omega\times\Omega\times\mathbb{R}^n)$ and as a consequence of Proposition \ref{propw}, assertion $ii)$, $A$ maps $\mathcal{G}^\infty_c(\Omega)$ into $\mathcal{G}^\infty(\Omega)$. Finally, assuming that $a\in\widetilde{\mathcal{S}}^{-\infty}_{\rm{rg}}(\Omega\times \Omega\times\mathbb{R}^n)$, the integral \[ A_\epsilon u_\epsilon(x)=\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a_\epsilon(x,y,\xi)u_\epsilon(y)\, dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi \] is absolutely convergent. Differentiating we obtain that \[ \begin{tabular}{cccc} $(a_\epsilon)_\epsilon\in{\mathcal{S}}^{-\infty}_{\rm{rg}} (\Omega\times\Omega\times\mathbb{R}^n) $, & $(u_\epsilon)_\epsilon\in\mathcal{E}_{c,M}(\Omega)$ & $\Rightarrow$ & $(A_\epsilon u_\epsilon)_\epsilon\in\mathcal{E}^\infty_M(\Omega)$,\\[0.2cm] $(a_\epsilon)_\epsilon\in{\mathcal{N}}^{-\infty}(\Omega\times\Omega \times\mathbb{R}^n) $, & $(u_\epsilon)_\epsilon\in\mathcal{E}_{c,M}(\Omega)$ & $\Rightarrow$ & $(A_\epsilon u_\epsilon)_\epsilon\in\mathcal{N}(\Omega)$,\\[0.2cm] $(a_\epsilon)_\epsilon\in{\mathcal{S}}^{-\infty}_{\rm{rg}} (\Omega\times\Omega\times\mathbb{R}^n) $, & $(u_\epsilon)_\epsilon\in\mathcal{N}_c(\Omega)$ & $\Rightarrow$ & $(A_\epsilon u_\epsilon)_\epsilon\in\mathcal{N}(\Omega)$. \end{tabular} \] This completes the proof. \end{proof} \begin{definition} \label{defpseudo} \rm Let $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$. The linear map defined by \[ A:\mathcal{G}_c(\Omega)\to\mathcal{G}(\Omega):u\to Au(x):=\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a(x,y,\xi)u(y) \, dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi \] will be called a (generalized) pseudodifferential operator with amplitude $a$. \end{definition} The formal transpose of $A$ is the pseudodifferential operator ${\ }^tA:\mathcal{G}_c(\Omega)\to\mathcal{G}(\Omega)$ defined by \begin{equation} \label{trans} u\to \int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a(x,y,\xi)u(x)\, dx\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi=\int_{\Omega\times\mathbb{R}^n} \hskip-18pt e^{i(x-y)\xi}a(y,x,-\xi)u(y)\, dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi. \end{equation} The first integral in \eqref{trans} is an oscillatory integral in $x$ and $\xi$ depending on the parameter $y\in\Omega$. Renaming variables, we see that ${\ }^tA$ can be written in the usual pseudodifferential form and thus satisfies the mapping properties of Proposition \ref{propseudo} as well. \begin{definition} \label{defkera} \rm Let $A$ be a pseudodifferential operator. The map $k_A\in L(\mathcal{G}_c(\Omega\times\Omega),\widetilde{\mathbb{C}})$ defined by \begin{equation} \label{ken} k_A(u)=\int_\Omega A(u(x,\cdot))(x)dx. \end{equation} is called the kernel of $A$. \end{definition} We have to prove that the integral in \eqref{ken} makes sense. Let $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$ be an amplitude defining the operator $A$. Let $u \in {\mathcal{G}}_c(\Omega\times\Omega)$. From Definition \ref{defpseudo}, \[ A(u(x,\cdot))(x)=\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a(x,y,\xi)u(x,y)\, dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi, \] where $a(x,y,\xi)u(x,y)\in\widetilde{\mathcal{S}}^m_{\rho,\delta} (\Omega\times\Omega\times\mathbb{R}^n)$. From Proposition \ref{propw} and Remark \ref{remrep} this oscillatory integral defines a generalized function in $\mathcal{G}(\Omega)$ and $A(u(x,\cdot))(x)\in\mathcal{G}_c(\Omega)$. Consequently, $\int_\Omega A(u(x,\cdot))(x)dx$ is an element of $L(\mathcal{G}_c(\Omega\times\Omega),\widetilde{\mathbb{C}})$ and from Remark \ref{remrep}, assertion $ii)$ \begin{align*} k_A(u)&=\int_\Omega\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a(x,y,\xi)u(x,y)\, dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi\,dx\\ &=\int_{\Omega\times\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a(x,y,\xi)u(x,y)\, dx\,dy\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi. \end{align*} \begin{proposition} \label{propkera} Let $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$ and $A$ be the corresponding pseudodifferential operator. \begin{itemize} \item[i)] For all $u\in\mathcal{G}_c(\Omega)$ and $v\in\mathcal{G}_c(\Omega)$ \begin{equation} \label{k1} k_A(v\otimes u)=\int_\Omega Au(x)v(x)dx=\int_\Omega u(y){\ }^tAv(y)dy, \end{equation} where $v\otimes u:=(v_\epsilon(x)u_\epsilon(y))_\epsilon+\mathcal{N}_c (\Omega\times\Omega)$; \item[ii)] $k_A\in\mathcal{G}(\Omega\times\Omega\setminus\Delta)$, where $\Delta$ is the diagonal of $\Omega\times\Omega$. Moreover, for open subsets $W$ and $W'$ of $\Omega$ with $W\times W'\subseteq\Omega\times\Omega\setminus\Delta$, and for all $u\in\mathcal{G}_c(W')$ \begin{equation} \label{k2} Au_{\left|_W\right.}(x)=\int_{\Omega}k_A(x,y)u(y)dy; \end{equation} \item[iii)] if $\mathop{\rm supp}_{x,y}\,a\subseteq\Omega\times \Omega\setminus\Omega'$, where $\Omega'$ is an open neighborhood of $\Delta$, then $k_A\in\mathcal{G}(\Omega\times\Omega)$; \item[iv)] if $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net then $ii)$ and $iii)$ are valid with $\mathcal{G}^\infty(\Omega\times\Omega\setminus \Delta)$ and $\mathcal{G}^\infty(\Omega\times\Omega)$ in place of $\mathcal{G}(\Omega\times\Omega\setminus\Delta)$ and $\mathcal{G}(\Omega\times\Omega)$ respectively; \item[v)] if $a\in\widetilde{\mathcal{S}}^{-\infty}_{\rm{rg}}(\Omega\times \Omega\times\mathbb{R}^n)$ then $k_A\in\mathcal{G}^\infty(\Omega\times\Omega)$. \end{itemize} \end{proposition} \begin{proof} For the first point of the assertion it is sufficient, as in the classical theory of pseudodifferential operators, to write down the three oscillatory integrals in \eqref{k1} and to change order in integration. We observe that for $\phi(x,y,\xi)=(x-y)\xi$, $C_\phi\equiv\Delta\times\mathbb{R}^n\setminus\{0\}$ and $R_\phi\equiv\Omega\times\Omega\setminus\Delta$. Recalling the first statement of Remark \ref{remcphi}, the oscillatory integral $\int_{\mathbb{R}^n} e^{i(x-y)\xi}a(x,y,\xi)d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi$ defines a generalized function in $\mathcal{G}(\Omega\times\Omega\setminus\Delta)$. Now for all $u\in\mathcal{G}_c(\Omega\times\Omega\setminus\Delta)$ \begin{equation} \label{k3} k_A(u)=\int_{\Omega\times\Omega}\hskip-15pt u(x,y)\int_{\mathbb{R}^n}\hskip-5pt e^{i(x-y)\xi}a(x,y,\xi)\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi\, dx\,dy \end{equation} and, since as a consequence of Proposition \ref{propmo}, $\mathcal{G}(\Omega\times\Omega\setminus\Delta)$ is included in $L(\mathcal{G}_c(\Omega\times\Omega\setminus\Delta),\widetilde{\mathbb{C}})$, \eqref{k3} shows that $k_A\in\mathcal{G}(\Omega\times\Omega\setminus\Delta)$. In particular if $u\in\mathcal{G}_c(W')$ and $W\times W'\subseteq\Omega\times\Omega\setminus\Delta$, \eqref{k2} follows from \eqref{k3} and the inclusion $\mathcal{G}(W)\subseteq L(\mathcal{G}_c(W),\tilde{\mathbb{C}})$. Under the hypothesis that$(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net, $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$ can also be considered as an element of ${{\widetilde{\mathcal{S}}}}^m_{\rho,\delta,{\rm{rg}}} (\Omega\times\Omega\times\mathbb{R}^n)$. Thus the assertions $iii)$ and $iv)$ are obtained from the analogous statements $ii)$ and $iii)$ in Remark \ref{remcphi}. Finally, if $a\in\widetilde{\mathcal{S}}^{-\infty}_{\rm{rg}} (\Omega\times\Omega\times\mathbb{R}^n)$, $\int_{\mathbb{R}^n} e^{i(x-y)\xi}a(x,y,\xi)\, d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}} \xi\in\mathcal{G}^\infty(\Omega\times\Omega)$ and \eqref{k3} holds for all $u\in\mathcal{G}_c(\Omega\times\Omega)$. Hence $k_A(x,y)=\int_{\mathbb{R}^n}e^{i(x-y)\xi}a(x,y,\xi)\, d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi\in\mathcal{G}^\infty (\Omega\times\Omega)$. \end{proof} We see from \eqref{k1} and Proposition \ref{propmo} that two pseudodifferential operators having the same kernel coincide. The definition of the kernel $k_A$ is very useful in proving the following result of pseudolocality. \begin{proposition} \label{propseduoloc} Let $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$ and $(\omega^{-1}(\epsilon)) _\epsilon$ a slow scale net; let $A$ be the corresponding pseudodifferential operator. Then for all $u\in\mathcal{G}_c(\Omega)$ \[ {\rm{ sing\,supp}_g}\, Au\subseteq {\rm{ sing\,supp}_g}\,u. \] \end{proposition} \begin{proof} For $u\in\mathcal{G}_c(\Omega)$, we consider an arbitrary open neighborhood $V$ of $\mathop{\rm sing\,supp}_g\,u$ contained in $\Omega$ and a function $\psi\in\mathcal{C}^\infty_c(V)$ identically equal to 1 in a neighborhood of ${\rm{{sing\,supp}_g}}\,u$. Then we write $u=\psi u+(1-\psi)u$ where $\psi u\in\mathcal{G}_c(\Omega)$ and $(1-\psi)u\in\mathcal{G}^\infty_c(\Omega)$. From Proposition \ref{propseudo}, $A((1-\psi)u)\in\mathcal{G}^\infty(\Omega)$ and then our assertion becomes \begin{equation} \label{th} {\rm{ sing\,supp}_g}\, A(\psi u)\subseteq {\rm{ sing\,supp}_g}\, u. \end{equation} To prove \eqref{th}, we show that for all $u\in\mathcal{G}_c(\Omega)$ \begin{equation} \label{th2} {\rm{ sing\,supp}_g}\, Au\subseteq\mathop{\rm supp}u. \end{equation} Let $K\equiv\mathop{\rm supp}\,u$ and $x_0\in\Omega\setminus K$ so that $x_0\times K\subseteq \Omega\times\Omega\setminus\Delta$. Since $\Omega\times\Omega\setminus\Delta$ is open, there exist an open neighborhoods $W$ and $W'$ of $x_0$ and $K$, respectively, such that $W\times W'\subseteq\Omega\times\Omega\setminus\Delta$. We want to demonstrate that $x_0\in\Omega\setminus{\mathop{\rm supp}_g}\, Au$, i.e. $Au_{|_{W}}\in\mathcal{G}^\infty(W)$. It is sufficient to recall Pro\-po\-si\-tion \ref{propkera}, point $iv)$, and the equality \eqref{k2} where $k_A\in\mathcal{G}^{\infty}(W\times W')$. Writing $\psi u$ in place of $u$ in \eqref{th2}, we conclude that \[ {\rm{ sing\,supp}_g}\, A(\psi u)\subseteq\mathop{\rm supp}\psi u\subseteq V. \] Since $V$ is arbitrary, the proof is complete. \end{proof} Let us now consider a linear operator $A:\mathcal{G}_c(\Omega)\to\mathcal{G} (\Omega)$ of the form \begin{equation} \label{ka} Au(x)=\int_\Omega k(x,y)u(y) dy, \end{equation} where $k\in\mathcal{G}^\infty(\Omega\times\Omega)$. As noted in Remark \ref{remkernel}, $k$ is uniquely determined by \eqref{ka} as an element of $\mathcal{G}(\Omega\times\Omega)$. For this reason, we may call it {\em the} kernel of $A$, adopt the notation $k_A$, and we may call $A$ an operator with regular generalized kernel. Obviously, every operator with regular generalized kernel is regularizing, i.e. it maps $\mathcal{G}_c(\Omega)$ into $\mathcal{G}^\infty(\Omega)$. \begin{proposition} \label{propcarsmooth} $A$ is an operator with regular generalized kernel if and only if it is a pseudodifferential operator with smoothing amplitude in $\widetilde{\mathcal{S}}^{-\infty}_{\rm{rg}}(\Omega\times\Omega\times \mathbb{R}^n)$. \end{proposition} \begin{proof} Every pseudodifferential operator with smoothing amplitude has a regular generalized kernel by Proposition \ref{propkera}. To prove the converse, let $k_A\in\mathcal{G}^\infty(\Omega\times\Omega)$. Then for all $u\in\mathcal{G}_c(\Omega)$, $Au$ has as a representative \[ A_\epsilon u_\epsilon=\int_\Omega k_{A,\epsilon}(x,y)u_\epsilon(y) dy =\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi} e^{-i(x-y)\xi}k_{A,\epsilon}(x,y)\chi(\xi)u_\epsilon(y)\, dy d\llap{\raisebox{.9ex}{$\scriptstyle-\!$}}\xi , \] where $\chi(\xi)\in\mathcal{C}^\infty_c(\mathbb{R}^n)$ with $\int\chi(\xi)d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi=1$. Now if we define \[ a_\epsilon(x,y,\xi):=e^{-i(x-y)\xi}k_{A,\epsilon}(x,y)\chi(\xi) \] then each $a_\epsilon$ belongs to $S^{-\infty}(\Omega\times\Omega\times\mathbb{R}^n)$. Further, $(k_{A,\epsilon})_\epsilon\in\mathcal{E}^\infty_M(\Omega\times\Omega)$ implies $(a_\epsilon)_\epsilon\in\mathcal{S}^{-\infty}_{\rm{rg}}(\Omega\times \Omega\times\mathbb{R}^n)$ and $(k_{A,\epsilon})_\epsilon\in\mathcal{N}(\Omega\times\Omega)$ implies $(a_\epsilon)_\epsilon\in\mathcal{N}^{-\infty}(\Omega\times\Omega\times \mathbb{R}^n)$. In conclusion, \[ Au(x)=\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a(x,y,\xi)u(y)\, dyd\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi \] for $a:=(a_\epsilon)_\epsilon+\mathcal{N}^{-\infty}(\Omega\times\Omega \times\mathbb{R}^n)$. \end{proof} We introduce properly supported pseudodifferential operators using their kernels in $L(\mathcal{G}_c(\Omega\times\Omega),\widetilde{\mathbb{C}})$. \begin{definition} \label{defpropsuppop} \rm A pseudodifferential operator $A$ is properly supported if and only if $\mathop{\rm supp}k_A$ is a proper set. An amplitude $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$ is called properly supported if and only if $\mathop{\rm supp}_{x,y}a$ is a proper set. \end{definition} We note that if $A$ is properly supported then ${\ }^tA$ is properly supported. \begin{proposition} \label{propropam} Let $A$ be a pseudodifferential operator with amplitude $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$. If $(\omega(\epsilon))_\epsilon$ is bounded then A is properly supported if and only if it can be written with a properly supported amplitude in ${\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times \Omega\times\mathbb{R}^n)$. If $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net then A is properly supported if and only if it can be written with a properly supported amplitude in ${{\widetilde{\mathcal{S}}}}^m_{\rho,\delta,{\rm{rg}}} (\Omega\times\Omega\times\mathbb{R}^n)$. \end{proposition} \begin{proof} Let us consider the first case when $(\omega(\epsilon))_\epsilon$ is bounded. If $A$ is properly supported then choosing a proper function $\chi\in\mathcal{C}^\infty(\Omega\times\Omega)$ identically equal to $1$ in a neighborhood of supp\,$k_A$ we have that $\chi a:=(\chi a_\epsilon)_\epsilon+{\mathcal{N}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ belongs to ${\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\Omega\times\mathbb{R}^n)$. This uses the boundedness of $\omega(\epsilon)$. Clearly, $\chi a$ is properly supported. Moreover, since for all $u\in\mathcal{G}_c(\Omega\times\Omega)$ \[ k_A((1-\chi)u)=\int_{\Omega\times\mathbb{R}^n}\hskip-18pt e^{i(x-y)\xi}a(x,y,\xi)(1-\chi(x,y))u(x,y)\, dx\,dy\, d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi =0\quad \text{in}\ \widetilde{\mathbb{C}}, \] the operators with amplitudes $a$ and $\chi a$ have the same kernel and hence they coincide. To prove the converse, assume that $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times \Omega\times\mathbb{R}^n)$ is a properly supported amplitude. Since supp\,$k_A\subseteq{\rm{supp}}_{x,y}a$ we see that $A$ is a properly supported pseudodifferential operator. If $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net we can repeat the same arguments, substituting ${{\widetilde{\mathcal{S}}}}^m_{\rho,\delta,{\rm{rg}}} (\Omega\times\Omega\times\mathbb{R}^n)$ for ${\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$. \end{proof} \begin{remark} \label{rempropam} Proposition \ref{propropam} implies that any properly supported pseudodifferential operator $A$ can be written with a properly supported amplitude $a$ such that there exist a representative $(a_\epsilon)_\epsilon$ of a and a proper closed subset of $\Omega\times\Omega$ containing ${\rm{supp}}_{x,y}\,a_\epsilon$ for all $\epsilon$. \end{remark} \begin{proposition} \label{propropsuppop} If $A$ is a properly supported pseudodifferential operator then for all $K\Subset\Omega$ there exist $K',K''\Subset\Omega$ such that for all $u\in\mathcal{G}_c(\Omega)$ the following statements hold: \begin{itemize} \item[i)] ${\rm{supp}}\,u\subset K$ implies ${\rm{supp}}\,Au\subset K'$, \item[ii)]${\rm{supp}}\,u\subset \Omega\setminus K''$ implies ${\rm{supp}}\,Au\subset\Omega\setminus K$. \end{itemize} \end{proposition} \begin{proof} From \eqref{k1} we obtain for all $u\in\mathcal{G}_c(\Omega)$ \begin{equation} \label{properly} \mathop{\rm supp}Au\subseteq \pi_1(\pi_2^{-1}(\mathop{\rm supp}u) \cap\mathop{\rm supp}k_A). \end{equation} The first assertion follows from \eqref{properly} putting $K'=\pi_1(\pi_2^{-1}(K)\cap\mathop{\rm supp}k_A)$. Defining $K''$ as $\pi_2(\pi_1^{-1}(K)\cap\mathop{\rm supp}k_A)$, \eqref{properly} leads to assertion $ii)$. \end{proof} Proposition \ref{propropsuppop} is the well-known topological characterization of a properly supported linear operator. \begin{proposition} \label{propext} If $A$ is a properly supported pseudodifferential operator with amplitude $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$ then \begin{itemize} \item[i)] $A$ maps $\mathcal{G}_c(\Omega)$ into $\mathcal{G}_c(\Omega)$, \item[ii)] $A$ can be uniquely extended to a linear map from $\mathcal{G}(\Omega)$ into $\mathcal{G}(\Omega)$ such that for all $u\in\mathcal{G}(\Omega)$ and $v\in\mathcal{G}_c(\Omega)$, \begin{equation} \label{inttrans} \int_\Omega Au(x)v(x)dx=\int_\Omega u(y){\ }^tAv(y)dy. \end{equation} \end{itemize} In the particular case when $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net \begin{itemize} \item[iii)] $A$ maps $\mathcal{G}^\infty_c(\Omega)$ into $\mathcal{G}^\infty_c(\Omega)$, \item[iv)] the extension defined above maps $\mathcal{G}^\infty(\Omega)$ into $\mathcal{G}^\infty(\Omega)$. \end{itemize} The same results hold with ${\ }^tA$ in place of $A$. \end{proposition} \begin{proof} The first assertion is clear from $i)$ in Proposition \ref{propropsuppop}. To prove the second, we use the well-known sheaf-theoretic argument. We only need to define $A$ locally. Let $V_1\subset V_2\subset\dots$ be an exhausting sequence of relatively compact open sets, let $K_j=\overline{V_j}$ and $K_j''$ as in $ii)$ of Proposition \ref{propropsuppop}, where we assume that $\{ K_j''\}_{j\in\mathbb{N}}$ is increasing. Given $u\in\mathcal{G}(\Omega)$, we define $A_ju\in\mathcal{G}(V_j)$ by $(A(\psi_ju))_{\vert_{V_j}}$ where $\psi_j\in\mathcal{C}^\infty_c(\Omega)$, $\psi_j\equiv 1$ in an open neighborhood of $K_j''$. As in the proof of Proposition \ref{propkernel}, it is clear that the family $\{A_ju\}_{j\in\mathbb{N}}$ is coherent. In this way we obtain a linear extension of the original pseudodifferential operator on $\mathcal{G}_c(\Omega)$, which satisfies \eqref{inttrans}. In fact, choosing $u\in\mathcal{G}(\Omega)$ and $v\in\mathcal{G}_c(\Omega)$, we have that supp\,$v\subseteq V_j$ for some $j$. Since the function $\psi_j$ is identically 1 in an open neighborhood of $\pi_2(\pi_1^{-1}(\overline{V_j})\cap\mathop{\rm supp}k_A)$ and supp\,${\ }^tAv\subseteq\pi_2(\pi_1^{-1}(\overline{V_j})\cap\mathop{\rm supp}k_A)$, we conclude that \[ \int_\Omega Au(x)v(x)dx=\int_\Omega A(\psi_ju)v(x)dx=\int_\Omega \psi_ju(y){\ }^tAv(y)dy=\int_\Omega u(y){\ }^tAv(y)dy. \] The uniqueness is proved as in (\ref{prop3}). Assume now that $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net. From Proposition \ref{propseudo} we already know that $A$ maps $\mathcal{G}^\infty_c(\Omega)$ into $\mathcal{G}^\infty(\Omega)$. This mapping property combined with assertion $i)$ implies that $A:\mathcal{G}^\infty_c(\Omega)\to\mathcal{G}^\infty_c(\Omega)$. Finally using the sheaf property of $\mathcal{G}^\infty(\Omega)$, the extension to $\mathcal{G}(\Omega)$ defined above maps $\mathcal{G}^\infty(\Omega)$ into $\mathcal{G}^\infty(\Omega)$. \end{proof} It is clear that if $A$ is properly supported and $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net then the pseudolocality property $\mathop{\rm sing\,supp}_g Au\subseteq \mathop{\rm sing\,supp}_g u$ holds for every $u\in\mathcal{G}(\Omega)$. In fact, it suffices to recall that the restrictions of $Au$ to the open subsets $V_j$ are expressed by $(A(\psi_j u))_{\vert_{V_j}}$, where $\psi_ju$ has compact support. \begin{proposition} \label{propsum} Let $a\in{\widetilde{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega\times\Omega\times\mathbb{R}^n)$ with $(\omega^{-1}(\epsilon))_\epsilon$ a slow scale net. The corresponding pseudodifferential operator $A$ can be written as the sum $A_0+A_1$ where $A_0$ is a properly supported pseudodifferential operator and $A_1$ has regular generalized kernel. \end{proposition} \begin{proof} Take a proper function $\chi\in\mathcal{C}^\infty(\Omega\times\Omega)$ identically equal to 1 in a neighborhood of the diagonal $\Delta\subset\Omega\times\Omega$. Given $u\in\mathcal{G}_c(\Omega)$ we can write $Au=A_0u+A_1u$ where $A_0$ is the properly supported pseudodifferential operator with amplitude $ a_0=\chi a\in{{\widetilde{\mathcal{S}}}}^m_{\rho,\delta,{\rm{rg}}} (\Omega\times\Omega\times\mathbb{R}^n)$ and $A_1$ is the pseudodifferential operator with amplitude $a_1=a(1-\chi)\in{{\widetilde{\mathcal{S}}}}^m_{\rho,\delta,{\rm{rg}}} (\Omega\times\Omega\times\mathbb{R}^n)$. Since $\mathop{\rm supp}_{x,y}\,a_1$ is included in the complement of an open neighborhood of $\Delta$ and $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net, the arguments in the proof of Proposition \ref{propkera} show that the kernel of $A_1$ belongs to $\mathcal{G}^\infty(\Omega\times\Omega)$. \end{proof} \section{Formal series and generalized symbols} In this section we develop a pseudodifferential calculus: formal series, symbols, transposition, and composition. Formal series and symbols play a basic role in the classical theory of pseudodifferential operators. The aim of this section is to generalize these concepts to our context. As we shall see in detail in Theorem \ref{theofsM}, we will have to consider the subspaces $$ {\widetilde{\underline{\mathcal{S}}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega'\times\mathbb{R}^p) :={\underline{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega} (\Omega'\times\mathbb{R}^p) /{\underline{\mathcal{N}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p) $$ of ${\widetilde{\mathcal{S}}}^{m,\mu} _{\rho,\delta,\omega}(\Omega'\times\mathbb{R}^p)$, $$ {\underline{\widetilde{\mathcal{S}}}}^m_{\rho,\delta,{\rm{rg}}} (\Omega'\times\mathbb{R}^p) :={\underline{{\mathcal{S}}}}^m_{\rho,\delta,{\rm{rg}}} (\Omega'\times\mathbb{R}^p)/ {\underline{{\mathcal{N}}}}^m_{\rho,\delta}(\Omega'\times\mathbb{R}^p) $$ of ${\widetilde{\mathcal{S}}}^m_{\rho,\delta,{\rm{rg}}} (\Omega'\times\mathbb{R}^p)$ and $$ {\widetilde{\underline{\mathcal{S}}}}^{-\infty}_{\rm{rg}}(\Omega'\times\mathbb{R}^p) := \underline{\mathcal{S}}^{-\infty}_{\rm{rg}}(\Omega'\times\mathbb{R}^p) /\underline{\mathcal{N}}^{-\infty}(\Omega'\times\mathbb{R}^p) $$ of ${\widetilde{\mathcal{S}}}^{-\infty}_{\,\rm{rg}}(\Omega'\times\mathbb{R}^p)$, obtained by fixing $\eta=1$ in \eqref{mod}, \eqref{neg}, in the definitions of ${\mathcal{S}}^{m}_{\rho,\delta,{\rm{rg}}}(\Omega'\times\mathbb{R}^p)$ and ${\mathcal{N}}^{m}_{\rho,\delta}(\Omega'\times\mathbb{R}^p)$ and in \eqref{sim}, \eqref{ni}, respectively. This is needed in order to guarantee that the infinite number of terms in the formal series will be defined for $\epsilon$ in a common interval. What concerns smoothing symbols, we need a refined version of Definition \ref{defsip}. We denote by ${\underline{\mathcal{S}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n)$ the set of all $(a_\epsilon)_\epsilon\in\mathcal{E}[\Omega\times\mathbb{R}^n]$ such that \begin{gather*} \forall K\Subset\Omega,\ \exists N\in\mathbb{N}:\ \forall m\in\mathbb{R},\ \forall\alpha,\beta\in\mathbb{N}^n,\ \exists c>0:\ \forall x\in K,\ \forall\xi\in\mathbb{R}^n,\ \forall\epsilon\in(0,1],\\ |\partial^\alpha_\xi\partial^\beta_xa_\epsilon(x,\xi)|\le c\langle\xi\rangle^{m-|\alpha|}\epsilon^{-N}\omega(\epsilon)^{-(|\beta|-\mu)_+} \end{gather*} and by ${\underline{\mathcal{N}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n)$ the set of all $(a_\epsilon)_\epsilon\in\mathcal{E}[\Omega\times\mathbb{R}^n]$ such that \[ \begin{array}{cc} \forall K\Subset\Omega,\ \forall q\in\mathbb{N},\ \forall m\in\mathbb{R},\ \forall\alpha,\beta\in\mathbb{N}^n,\ \exists c>0:\ \forall x\in K,\ \forall\xi\in\mathbb{R}^n,\ \forall\epsilon\in(0,1],\\ |\partial^\alpha_\xi\partial^\beta_xa_\epsilon(x,\xi)|\le c\langle\xi\rangle^{m-|\alpha|}\epsilon^{q}\omega(\epsilon)^{-(|\beta|-\mu)_+}. \end{array} \] We introduce the notation ${\underline{\widetilde{\mathcal{S}}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n)$ for ${\underline{\mathcal{S}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n) / {\underline{\mathcal{N}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n)$. Obviously, if $(\omega(\epsilon))_\epsilon$ is bounded and $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net then ${\underline{\widetilde{\mathcal{S}}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n) = {\widetilde{\underline{\mathcal{S}}}}^{-\infty}_{\rm{rg}}(\Omega\times\mathbb{R}^n)$. Finally, let $(a_\epsilon)_\epsilon\in{\mathcal{S}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ and $K$ be a compact subset of $\Omega$. We say that $(a_\epsilon)_\epsilon$ is of {\em growth type} $N_K\in\mathbb{N}$ on $K$ if and only if \[ \begin{array}{cc} \forall\alpha,\beta\in\mathbb{N}^n,\ \exists c>0:\ \forall x\in K,\ \forall\xi\in\mathbb{R}^n,\ \forall\epsilon\in(0,1],\\[0.2cm] |\partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi)|\le c\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}\epsilon^{-N_K}\omega(\epsilon)^{-(|\beta|-\mu)_+}. \end{array} \] \begin{definition} \label{defsM} \rm Let $\{m_j\}_{j\in\mathbb{N}}$ and $\{\mu_j\}$ be sequences of real numbers with $m_j\searrow -\infty$, $m_0=m$ and $\mu_0=\mu$. Let $\{(a_{j,\epsilon})_\epsilon\}_{j\in\mathbb{N}}$ be a sequence of elements $(a_{j,\epsilon})_\epsilon\in {\underline{\mathcal{S}}}^{m_j,\mu_j}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$, satisfying the following condition: \begin{equation} \label{require} \forall K\Subset\Omega,\ \exists N_K\in\mathbb{N}:\ \forall j\in\mathbb{N}\quad (a_{j,\epsilon})_\epsilon\ \text{is of growth type $N_K$ on $K$}.\quad \end{equation} We say that the formal series $\sum_{j=0}^\infty(a_{j,\epsilon})_\epsilon$ is the asymptotic expansion of $(a_\epsilon)_\epsilon\in\mathcal{E}[\Omega\times\mathbb{R}^n]$, $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$ for short, if and only if for all $K\Subset\Omega$ there exists $M_K\in\mathbb{N}$ such that for all $r\ge 1$, $(a_\epsilon-\sum_{j=0}^{r-1}a_{j,\epsilon})_\epsilon$ is an element of ${\underline{\mathcal{S}}}^{m_r,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$ of growth type $M_K$ on $K$. \end{definition} \begin{remark}\label{uniqasym} \rm If the elements $(a_\epsilon)_\epsilon$ and $(a'_\epsilon)_\epsilon$ in $\in\mathcal{E}[\Omega\times\mathbb{R}^n]$ have the same asymptotic expansion $\sum_j(a_{j,\epsilon})_\epsilon$ then $(a_\epsilon-a'_\epsilon)_\epsilon \in{\underline{\mathcal{S}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n)$. Indeed, we can write \[ a_\epsilon-a'_\epsilon=a_{\epsilon}-\sum_{j=0}^{r}a_{j,\epsilon} +\sum_{j=0}^{r}a_{j,\epsilon}-a'_\epsilon. \] From the definition of an asymptotic expansion, we have that for all $K\Subset\Omega$ there exists a natural number $M_K$ such that $\big(a_{\epsilon}-\sum_{j=0}^{r}a_{j,\epsilon} \big)_\epsilon$ and $\big(\sum_{j=0}^{r}{a_{j,\epsilon}}-{a'_\epsilon}\big)_\epsilon$ are elements of ${\underline{\mathcal{S}}}^{m_r,\mu}_{\rho,\delta,\omega}(\Omega \times\mathbb{R}^n)$ of growth type $M_K$ on $K$. Since $M_K$ does not depend on $r$ and the sequence of $\{m_r\}_{r\in\mathbb{N}\setminus\{0\}}$ tends to $-\infty$, we conclude that $(a_\epsilon-a'_\epsilon)_\epsilon\in{\underline{\mathcal{S}}} ^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n)$, as desired. \end{remark} \begin{theorem} \label{theofsM} Let $\{m_j\}_j$, $\{\mu_j\}_j$ and $(a_{j,\epsilon})_\epsilon \in{\underline{\mathcal{S}}}^{m_j,\mu_j}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$ for all $j\in\mathbb{N}$ as in Definition \ref{defsM}. If in addition one of the following hypotheses \begin{equation} \label{h(1)} \mu_j\ge\mu\ \text{for\ all } j\in\mathbb{N} \text{ and } \sup_{\epsilon\in(0,1]}\omega(\epsilon)<\infty \end{equation} or \begin{equation} \label{h(2)} \mu_j\le\mu\ \text{for all } j \in\mathbb{N} \text{ and }(\omega^{-1}(\epsilon))_\epsilon \text{ is a slow scale net} \end{equation} holds then there exists $(a_\epsilon)_\epsilon\in{\underline{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ such that $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$. \end{theorem} \begin{proof} The proof follows the classical line of arguments, but we will have to keep track of the $\epsilon$-dependence carefully. We consider a sequence of relatively compact open sets $\{V_l\}$ contained in $\Omega$, such that for all $l\in\mathbb{N}$, $V_l\subset K_l=\overline{V_l}\subset V_{l+1}$ and $\bigcup_{l\in\mathbb{N}}V_l=\Omega$. Let $\psi\in\mathcal{C}^\infty(\mathbb{R}^n)$, $0\le\psi(\xi)\le 1$, such that $\psi(\xi)=0$ for $|\xi|\le 1$ and $\psi(\xi)=1$ for $|\xi|\ge 2$. We introduce \[ b_{j,\epsilon}(x,\xi)=\psi(\lambda_j\xi)a_{j,\epsilon}(x,\xi), \] where $\lambda_j$ will be positive constants, independent of $\epsilon$, with $\lambda_{j+1}<\lambda_j<1$, $\lambda_j\to 0$. We can define \begin{equation} \label{a} a_\epsilon(x,\xi)=\sum_{j\in\mathbb{N}}b_{j,\epsilon}(x,\xi). \end{equation} This sum is locally finite. We observe that $\partial^\alpha(\psi(\lambda_j\xi))=\partial^\alpha \psi(\lambda_j\xi)\lambda_j^{|\alpha|}$, that $\mathop{\rm supp}(\partial^\alpha\psi(\lambda_j\xi)) \subseteq\{\xi:\ 1/\lambda_j \le|\xi|\le 2/\lambda_j\}$, and that $1/\lambda_j\le|\xi|\le 2/\lambda_j$ implies $\lambda_j\le 2/|\xi|\le 4/(1+|\xi|)$. \smallskip \noindent\textbf{Case 1:} We assume now hypothesis \eqref{h(1)}. Fixing $K\Subset\Omega$ and $\alpha,\beta\in\mathbb{N}^n$, we obtain for $j\in\mathbb{N}$, $\epsilon\in(0,1]$, $x\in K$, $\xi\in\mathbb{R}^n$, \begin{equation} \label{comp} \begin{split} &|\partial^\alpha_\xi\partial^\beta_x b_{j,\epsilon}(x,\xi)|\\ &\le\sum_{\gamma\le\alpha}\binom{\alpha}{\gamma}\lambda_j^{|\alpha-\gamma|} |\partial^{\alpha-\gamma}\psi(\lambda_j\xi)|c_{j,\gamma,\beta,K} \langle\xi\rangle^{m_j-\rho|\gamma|+\delta|\beta|}\epsilon^{-N_{K}} \omega(\epsilon)^{-(|\beta|-\mu_j)_+}\\ &\le\sum_{\gamma\le\alpha}{c}_{j,\gamma,\beta,K}4^{|\alpha-\gamma|} \langle\xi\rangle^{-|\alpha-\gamma|}\langle\xi\rangle^{m_j-\rho|\gamma| +\delta|\beta|}\epsilon^{-N_{K}}\omega(\epsilon)^{-(|\beta|-\mu_j)_+}\\ &\le C_{j,\alpha,\beta,K}\langle\xi\rangle^{m_j-\rho|\alpha|+\delta|\beta|} \epsilon^{-N_K}\omega(\epsilon)^{-(|\beta|-\mu)_+}, \end{split} \end{equation} where in the last computations we use the inequality $(|\beta|-\mu_j)_+\le(|\beta|-\mu)_+$ for $\mu_j\ge\mu$ and the boundeness of $\omega$. At this point we choose $\lambda_j$ such that for $|\alpha+\beta|\le j$, $l\le j$ \begin{equation} \label{cj} C_{j,\alpha,\beta,K_l}\lambda_j\le 2^{-j}. \end{equation} Our aim is to prove that $a_\epsilon(x,\xi)$ defined in \eqref{a} belongs to ${\underline{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$. We already know that $(a_\epsilon)_\epsilon\in\mathcal{E}[\Omega\times\mathbb{R}^n]$. We observe that \begin{equation} \label{lo} \begin{gathered} \forall K\Subset\Omega,\ \exists l\in\mathbb{N}:\ K\subset V_l\subset K_l,\\ \forall\alpha_0,\beta_0\in\mathbb{N}^n,\ \exists j_0\in\mathbb{N}, \ j_0\ge l:\ |\alpha_0+\beta_0|\le j_0,\quad m_{j_0}+1\le m. \end{gathered} \end{equation} Now, $(a_\epsilon)_\epsilon$ as the sum of the following two terms: \[ a_\epsilon(x,\xi)=\sum_{j=0}^{j_0-1}b_{j,\epsilon}(x,\xi) +\sum_{j=j_0}^{+\infty}b_{j,\epsilon}(x,\xi)=f_\epsilon(x,\xi) +s_\epsilon(x,\xi). \] First we study $f_\epsilon(x,\xi)$. For $x\in K$, using hypothesis (\ref{h(1)}), we have that \begin{equation}\label{f} \begin{split} &|\partial^{\alpha_0}_\xi\partial^{\beta_0}_x f_\epsilon(x,\xi)|\\ &\le \sum_{j=0}^{j_0-1}c_{j,\alpha_0,\beta_0,K}\langle\xi \rangle^{m_j-\rho|\alpha_0|+\delta|\beta_0|}\epsilon^{-N_{K}} \omega(\epsilon)^{-(|\beta_0|-\mu_j)_+}\\ &\le c'_{\alpha_0,\beta_0,K}\langle\xi\rangle^{m-\rho|\alpha_0| +\delta|\beta_0|}\epsilon^{-N_K}\omega(\epsilon)^{-(|\beta_0|-\mu)_+}. \end{split} \end{equation} We now turn to $s_\epsilon(x,\xi)$. From \eqref{comp} and \eqref{cj}, we get for $x\in K$ and $\epsilon\in(0,1]$, \[ \begin{split} &|\partial^{\alpha_0}_\xi\partial^{\beta_0}_x s_\epsilon(x,\xi)|\\ &\le\sum_{j=j_0}^{+\infty}C_{j,\alpha_0,\beta_0,K_l}\langle\xi \rangle^{m_j-\rho|\alpha_0|+\delta|\beta_0|}\epsilon^{-N_{K_l}} \omega(\epsilon)^{-(|\beta_0|-\mu)_+}\\ &\le\sum_{j=j_0}^{+\infty}2^{-j}\lambda_j^{-1}\langle\xi\rangle^{-1} \langle\xi\rangle^{ m_j+1-\rho|\alpha_0|+\delta|\beta_0|}\epsilon^{-N_{K_l}} \omega(\epsilon)^{-(|\beta_0|-\mu)_+}. \end{split} \] Since $\psi(\xi)$ is identically equal to $0$ for $|\xi|\le 1$, we can assume in our estimates that $\langle\xi\rangle^{-1}\le\lambda_j$, and therefore from \eqref{lo}, we conclude that \begin{equation} \label{s} |\partial^{\alpha_0}_\xi\partial^{\beta_0}_x s_\epsilon(x,\xi)| \le \langle\xi\rangle^{m-\rho|\alpha_0|+\delta|\beta_0|}\epsilon^{-N_{K_l}} \omega(\epsilon)^{-(|\beta_0|-\mu)_+}. \end{equation} In conclusion, we obtain that $(a_\epsilon)_\epsilon\in{\mathcal{S}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$. In order to prove that $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$ we fix $r\ge 1$ and we write \begin{align*} a_\epsilon(x,\xi)-\sum_{j=0}^{r-1}a_{j,\epsilon}(x,\xi) &=\sum_{j=0}^{r-1}(\psi(\lambda_j\xi)-1)a_{j,\epsilon}(x,\xi) +\sum_{j=r}^{+\infty}\psi(\lambda_j\xi)a_{j,\epsilon}(x,\xi)\\ &=g_\epsilon(x,\xi)+t_\epsilon(x,\xi). \end{align*} Recall that $\psi\in\mathcal{C}^\infty(\mathbb{R}^n)$ was chosen such that $\psi-1\in\mathcal{C}^\infty_c(\mathbb{R}^n)$ and ${\rm{supp}}(\psi-1)\subseteq\{\xi:\ |\xi|\le 2\}$. Thus, for $0\le j\le r-1$, \[ \mathop{\rm supp}(\psi(\lambda_j\xi)-1)\subseteq\{\xi:\ |\lambda_j\xi|\le 2\}\subseteq \{\xi:\ |\xi|\le 2\lambda_{r-1}^{-1}\}. \] As a consequence, for fixed $K\Subset\Omega$ and for all $\epsilon\in(0,1]$, \[ |\partial^\alpha_\xi\partial^\beta_x g_\epsilon(x,\xi)| \le c_{\alpha,\beta,K}\langle\xi\rangle^{m_r-\rho|\alpha| +\delta|\beta|}\epsilon^{-N_K}\omega(\epsilon)^{-(|\beta|-\mu)_+}. \] In this way $(g_\epsilon)_\epsilon$ is an element of ${\underline{\mathcal{S}}}^{m_r,\mu}_{\rho,\delta,\omega}(\Omega\times \mathbb{R}^n)$ of growth type $N_K$ on the compact set $K$. Moreover, repeating the same arguments used in the construction of $(a_\epsilon)_\epsilon$ we have that $(t_\epsilon)_\epsilon$ belongs to ${\underline{\mathcal{S}}}^{m_r,\mu}_{\rho,\delta,\omega}(\Omega\times \mathbb{R}^n)$ and it is of growth type $N_{K_l}$ on $K$, where $K\subset V_l\subset K_l$. Summarizing, we have that for all $r\ge 1$, $({a_\epsilon})_\epsilon-\sum_{j=0}^{r-1}({a_{j,\epsilon}})_\epsilon\in \underline{{\mathcal{S}}}^{m_r,\mu}_{\rho,\delta,\omega}(\Omega\times \mathbb{R}^n)$ and it is of growth type $\max(N_K,N_{K_l})$. \smallskip \noindent\textbf{Case 2:} The proof can be easily adapted to cover hypothesis \eqref{h(2)}. The crucial point is to observe that if $\mu_j\le\mu$ then $(|\beta|-\mu_j)_+\ge(|\beta|-\mu)_+$, and that we get the inequality \begin{equation} \label{slowscalein} \omega(\epsilon)^{-(|\beta|-\mu_j)_+}= \omega(\epsilon)^{-(|\beta|-\mu)_+}\omega(\epsilon)^{(|\beta|-\mu)_+ -(|\beta|-\mu_j)_+}\le c_{j,\beta}\,\omega(\epsilon)^{-(|\beta|-\mu)_+}\epsilon^{-1}, \end{equation} which follows from the definition of a slow scale net. Using \eqref{slowscalein}, we may transform \eqref{comp}, \eqref{f}, \eqref{s}, respectively, into \begin{equation} |\partial^\alpha_x\partial^\beta_x b_{j,\epsilon}(x,\xi)| \le C_{j,\alpha,\beta,K}\langle\xi\rangle^{m_j+1-\rho|\alpha| +\delta|\beta|}\langle\xi\rangle^{-1}\epsilon^{-N_K-1}\omega(\epsilon) ^{-(|\beta|-\mu)_+}, \end{equation} \begin{equation} |\partial^{\alpha_0}_\xi\partial^{\beta_0}_x f_\epsilon(x,\xi)| \le c'_{\alpha_0,\beta_0,K}\langle\xi\rangle^{m-\rho|\alpha_0| +\delta|\beta_0|}\epsilon^{-N_K-1}\omega(\epsilon)^{-(|\beta_0|-\mu)_+}, \end{equation} \begin{equation} |\partial^{\alpha_0}_\xi\partial^{\beta_0}_x s_\epsilon(x,\xi)| \le c''_{\alpha_0,\beta_0,K}\langle\xi\rangle^{m-\rho|\alpha_0| +\delta|\beta_0|}\epsilon^{-N_{K_l}-1}\omega(\epsilon)^{-(|\beta_0| -\mu)_+}, \end{equation} where $x\in K$, $\xi\in\mathbb{R}^n$ and $\epsilon\in(0,1]$. In this way $(g_\epsilon)_\epsilon\in {\underline{\mathcal{S}}}^{m_r,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$ is of growth type $N_K+1$ on $K$ and $(t_\epsilon)_\epsilon\in {\underline{\mathcal{S}}}^{m_r,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$ is of growth type $N_{K_l}+1$ on $K$. Finally, $(a_\epsilon)_\epsilon -\sum_{j=0}^{r-1}({a_{j,\epsilon}})_\epsilon$ belongs to ${\underline{\mathcal{S}}}^{m_r,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$ and it is of growth type $\max(N_K, N_{K_l})+1$. \end{proof} The following theorem studies the special case of formal series of negligible elements. \begin{theorem} \label{theofsn} Let $\{m_j\}_j$, $\{\mu_j\}_j$ be sequences of real numbers as in Definition \ref{defsM} and let $(a_{j,\epsilon})_\epsilon\in\underline{\mathcal{N}}^{m_j,\mu_j} _{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$ for all $j\in\mathbb{N}$. If the hypothesis \eqref{h(1)} holds or if $\mu_j \leq \mu$ for all $j \in \mathbb{N}$ then there exists $(a_\epsilon)_\epsilon\in{\underline{\mathcal{N}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ such that for all $r\ge 1$ \begin{equation} \label{estasymneg} \big(a_\epsilon-\sum_{j=0}^{r-1}a_{j,\epsilon}\big)_\epsilon\in \underline{\mathcal{N}}^{m_r,\mu}_{\rho,\delta,\omega}(\Omega\times \mathbb{R}^n). \end{equation} If $(a'_\epsilon)_\epsilon\in\mathcal{E}[\Omega\times\mathbb{R}^n]$ satisfies \eqref{estasymneg} then $(a_\epsilon-a'_\epsilon)_\epsilon\in{\underline{\mathcal{N}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n)$. \end{theorem} \begin{proof} Repeating the arguments and constructions from the proof of Theorem \ref{theofsM}, we conclude that \begin{gather*} \forall K\Subset\Omega, \forall\alpha ,\beta\in\mathbb{N}^n,\ \forall q\in\mathbb{N},\ \forall j\in\mathbb{N},\ \exists C_{j,\alpha,\beta,K,q}>0:\ \forall x\in K,\ \forall\xi\in\mathbb{R}^n,\\ \forall\epsilon\in(0,1],\ |\partial^\alpha_\xi\partial^\beta_x b_{j,\epsilon}(x,\xi)| \le C_{j,\alpha,\beta,K,q}\langle\xi\rangle^{m_j-\rho|\alpha| +\delta|\beta|}\epsilon^q\omega(\epsilon)^{-(|\beta|-\mu)_+}. \end{gather*} holds under either of the two hypotheses stated in Theorem \ref{theofsn}. At this point we choose $\lambda_j$ such that for $|\alpha+\beta|\le j$, $l\le j$, $q\le j$, \begin{equation} \label{cjq} C_{j,\alpha,\beta,K_l,q}\lambda_j\le 2^{-j}. \end{equation} We observe that (\ref{lo}) still holds, where, given $q_0 \in \mathbb{N}$, we may take $j_0 \ge \max(l,q_0)$. As a consequence, writing as before $a_\epsilon(x,\xi)=f_\epsilon(x,\xi)+s_\epsilon(x,\xi)$, we have \[ |\partial^{\alpha_0}_\xi\partial^{\beta_0}_x f_\epsilon(x,\xi)| \le c_{\alpha_0,\beta_0,q_0,K}\langle\xi\rangle^{m-\rho|\alpha_0| +\delta|\beta_0|}\epsilon^{q_0}\omega(\epsilon)^{-(|\beta_0|-\mu)_+} \] and from \eqref{cjq} \[ \begin{split} |\partial^{\alpha_0}_\xi\partial^{\beta_0}_x s_\epsilon(x,\xi)| &\le\sum_{j=j_0}^{+\infty}C_{j,\alpha_0,\beta_0,K_l,q_0}\langle\xi \rangle^{m_j-\rho|\alpha_0|+\delta|\beta_0|}\epsilon^{q_0}\omega(\epsilon) ^{-(|\beta_0|-\mu)_+}\\ &\le\sum_{j=j_0}^{+\infty}2^{-j}\lambda_j^{-1}\langle\xi\rangle^{-1} \langle\xi\rangle^{ m_j+1-\rho|\alpha_0|+\delta|\beta_0|}\epsilon ^{q_0}\omega(\epsilon)^{-(|\beta_0|-\mu)_+}\\ &\le\langle\xi\rangle^{m-\rho|\alpha_0|+\delta|\beta_0|}\epsilon^{q_0} \omega(\epsilon)^{-(|\beta_0|-\mu)_+}. \end{split} \] These results lead us to the conclusion that $(a_\epsilon)_\epsilon\in{\underline{\mathcal{N}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$. We omit the rest of the proof since it is a simple adaptation of the proof of Theorem \ref{theofsM}, where in place of the growth type $N_K$ we consider an arbitrary exponent $q\in\mathbb{N}$. \end{proof} \begin{definition} \label{defs} \rm Let $\{m_j\}_{j\in\mathbb{N}}$ and $\{\mu_j\}$ be sequences of real numbers with $m_j\searrow -\infty$, $m_0=m$ and $\mu_0=\mu$. Let $\{a_j\}_{j\in\mathbb{N}}$ be a sequence of symbols $a_j\in{\underline{\widetilde{\mathcal{S}}}}^{\, m_j,\mu_j}_{\rho, \delta,\omega}(\Omega\times\mathbb{R}^n)$ whose representatives $(a_{j,\epsilon})_\epsilon$ satisfy \eqref{require}. We say that the formal series $\sum_j a_j$ is the asymptotic expansion of $a\in{\widetilde{\underline{\mathcal{S}}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$, $a\sim\sum_j a_j$ for short, if and only if there exist a representative $(a_\epsilon)_\epsilon$ of $a$ and, for every $j$, representatives $(a_{j,\epsilon})_\epsilon$ of $a_j$, such that $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$. \end{definition} Note that if some representative of $a_j$ satisfies \eqref{require} then every representative does. Theorems 5.1 and 5.2 lead us to the following characterization of $a\sim\sum_j a_j$. \begin{proposition} \label{propasym} Under the hypotheses of Theorem \ref{theofsn}, $a\sim\sum_j a_j$ if and only if for any choice of representatives $(a_{j,\epsilon})_\epsilon$ of $a_j$ there exists a representative $(a_\epsilon)_\epsilon$ of $a$ such that $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$. \end{proposition} \begin{proof} We assume that there exist $(a_\epsilon)_\epsilon$ and $(a_{j,\epsilon})_\epsilon$ such that $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$. Let $(a'_{j,\epsilon})_\epsilon$ be another choice of representatives of $a_j$. It is clear that $\sum_j(a_{j,\epsilon}-a'_{j,\epsilon})_\epsilon$ fulfills the requirements of Theorem \ref{theofsn} and therefore there exists $(a''_\epsilon)_\epsilon\in{\underline{\mathcal{N}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ such that for all $r\ge 1$, $(a''_\epsilon-\sum_{j=0}^{r-1}(a_{j,\epsilon}-a'_{j,\epsilon}))_\epsilon \in\underline{\mathcal{N}}^{m_r,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$. As a consequence, $(a_\epsilon-a''_\epsilon)_\epsilon\in{\underline{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ is another representative of $a$ and $(a_\epsilon-a''_\epsilon)_\epsilon\sim\sum_j(a'_{j,\epsilon})_\epsilon$. In fact, for all $r\ge 1$ we have that $(a_\epsilon-a''_\epsilon-\sum_{j=0}^{r-1}a'_{j,\epsilon})_\epsilon$ can be written as the difference of $(a_\epsilon-\sum_{j=0}^{r-1}a_{j,\epsilon})_\epsilon\in {\underline{\mathcal{S}}}^{m_r,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$ and $(a''_\epsilon-\sum_{j=0}^{r-1}(a_{j,\epsilon}-a'_{j,\epsilon}))_\epsilon\in {\underline{\mathcal{N}}}^{m_r,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$, where the growth order on every compact set is independent of $r$. \end{proof} \begin{theorem} \label{theofs} Let $\{m_j\}_j$, $\{\mu_j\}_j$ and $a_j\in{\underline{\widetilde {\mathcal{S}}}}^{\, m_j,\mu_j}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ for all $j\in\mathbb{N}$ as in Definition \ref{defs}. If in addition the hypothesis \eqref{h(1)} or \eqref{h(2)} holds then there exists $a\in{\widetilde{\underline{\mathcal{S}}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ such that $a\sim\sum_j a_j$. Moreover, if $b\in{\widetilde{\underline{\mathcal{S}}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ has asymptotic expansion $\sum_j a_j$ then there exists a representative $(a_\epsilon)_\epsilon$ of $a$ and a representative $(b_\epsilon)_\epsilon$ of $b$ such that $(a_\epsilon-b_\epsilon)_\epsilon\in{\underline{\mathcal{S}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n)$. \end{theorem} \begin{proof} The existence of $a$ is a direct consequence of Theorem \ref{theofsM}. In particular, there is a choice of representatives $(a_{j,\epsilon})_\epsilon$ of $a_j$ and a representative $(a_\epsilon)_\epsilon$ of $a$ such that $(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$. Now if $b\sim\sum_ja_j$, the previous proposition guarantees the existence of a representative $(b_\epsilon)_\epsilon$ such that $(b_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$. Therefore, from Remark \ref{uniqasym}, $(a_\epsilon-b_\epsilon)_\epsilon\in{\underline{\mathcal{S}}}^{-\infty,\mu}_{\,\omega}(\Omega\times\mathbb{R}^n)$. \end{proof} Combining Theorem \ref{theofsn} with Remark \ref{uniqasym} we have that if $a\in{\widetilde{\underline{\mathcal{S}}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ has asymptotic expansion $\sum_j a_j$ where each term $a_j=0$ in ${\underline{\widetilde{\mathcal{S}}}}^{\, m_j,\mu_j}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$ then it has a representative of the form $a_\epsilon=a'_\epsilon+a''_\epsilon$, where $(a'_\epsilon)_\epsilon\in{\underline{\mathcal{N}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ and $(a''_\epsilon)_\epsilon\in\underline{\mathcal{S}}^{-\infty,\mu}_{\,\omega} (\Omega\times\mathbb{R}^n)$. In the sequel we always assume $0\le\delta<\rho\le 1$. \begin{theorem} \label{theosymbol} Let $A$ be a properly supported pseudodifferential operator with amplitude $a\in{\widetilde{\underline{\mathcal{S}}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\Omega\times\mathbb{R}^n)$ where $(\omega^{-1}(\epsilon))_\epsilon$ is a slow scale net. Then there exists $\sigma\in{\underline{\widetilde{\mathcal{S}}}}^{\,m}_{\rho,\delta,{\rm{rg}}}(\Omega\times\mathbb{R}^n)$ such that for all $u\in\mathcal{G}_\mathscr{S}(\mathbb{R}^n)$ \begin{equation} \label{sigma} A(u_{\vert_{\Omega}})(x)=\int_{\mathbb{R}^n}e^{ix\xi} \sigma(x,\xi)\widehat{u}(\xi)\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}}\xi. \end{equation} Moreover, $\sigma\sim\sum_j\frac{1}{\gamma !}\partial^\gamma_\xi D^\gamma_y a(x,y,\xi)_{\vert_{x=y}}$ where $D^\gamma=(-i)^{|\gamma|}\partial^\gamma$ and the asymptotic expansion is understood in the sense of ${\underline{\widetilde{\mathcal{S}}}}^{\,m}_{\rho,\delta,{\rm{rg}}}(\Omega\times\mathbb{R}^n)$. \end{theorem} \begin{proof} As shown in Proposition \ref{propropam}, given a proper function $\chi\in\mathcal{C}^\infty(\Omega\times\Omega)$ identically equal to 1 in a neighborhood of $\mathop{\rm supp}k_A$$\cup\Delta$, the pseudodifferential operator $A$ can be written with the properly supported amplitude $\chi a:=(\chi a_\epsilon)_\epsilon +{\underline{\mathcal{N}}}^{m}_{\rho,\delta}(\Omega\times\Omega \times\mathbb{R}^n)$ and can be viewed as an element of ${\widetilde{\underline{\mathcal{S}}}}^{\,m}_{\rho,\delta,{\rm{rg}}} (\Omega\times\Omega\times\mathbb{R}^n)$. Now we define \begin{gather*} \sigma_\epsilon(x,\xi)=\int_{\mathbb{R}^n} \widehat{b_\epsilon}(x,\eta,\xi+\eta)\,d\eta\,, \\ \widehat{b_\epsilon}(x,\eta,\xi)=\int_\Omega e^{i(x-y)\eta} \chi(x,y)a_\epsilon(x,y,\xi)\,d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}} y. \end{gather*} The net $(\widehat{b_\epsilon})_\epsilon$ belongs to $\mathcal{E}[\Omega\times\Omega\times\mathbb{R}^n]$. Using integration by parts and the assumptions on $(\omega^{-1}(\epsilon))_\epsilon$, we obtain for $x\in K\Subset\Omega$ and $\epsilon\in(0,1]$, \[ \begin{split} &|(-i\eta)^\gamma\partial^\alpha_\xi\partial^\beta_x \widehat{b_\epsilon}(x,\eta,\xi)|\\ &=\Bigl|\sum_{\beta'\le\beta} \binom{\beta}{\beta'}\int_\Omega e^{i(x-y)\eta}\partial^\alpha_\xi \partial^{\beta-\beta'}_x\partial^{\gamma+\beta'}_y(\chi(x,y) a_\epsilon(x,y,\xi))d\llap {\raisebox{.9ex}{$\scriptstyle-\!$}} y\Bigr|\\ &\le c_{\alpha,\beta,\gamma}\langle\xi\rangle^{m-\rho|\alpha| +\delta|\beta+\gamma|}\epsilon^{-N_K-1}. \end{split} \] Consequently, we have for any $M\in\mathbb{N}$ that \begin{equation} \label{b} |\partial^\alpha_\xi\partial^\beta_x\widehat{b_\epsilon}(x,\eta,\xi)| \le c_{\alpha,\beta,M}\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta| +\delta M}\langle\eta\rangle^{-M}\epsilon^{-N_K-1}. \end{equation} From \eqref{b} we obtain for $x\in K$ and $\epsilon\in(0,1]$ \[ \begin{split} |\partial^\alpha_\xi\partial^\beta_x\sigma_{\epsilon}(x,\xi)| &\le\hskip-2pt c_{\alpha,\beta,M}\epsilon^{-N_K-1}\int_{\mathbb{R}^n} \hskip-7pt \langle\xi+\eta\rangle^{m-\rho|\alpha| +\delta|\beta|+\delta M}\langle\eta\rangle^{-M}d\eta\\ &\le c'_{\alpha,\beta,M}\epsilon^{-N_K-1}\langle\xi\rangle^{p +\delta M}, \end{split} \] where $p=\max(m-\rho|\alpha|+\delta|\beta|,0)$ and $M$ is large enough. Next we estimate \[ \sigma_{\epsilon}(x,\xi)-\sum_{|\gamma|=0}^{h-1} \frac{1}{\gamma !}\partial^\gamma_\xi D^\gamma_y a_\epsilon(x,y,\xi)\vert_{x=y} \] for $h\ge 1$. Recalling that $\partial^\gamma_\xi D^\gamma_y a_\epsilon(x,y,\xi)\vert_{x=y}=\partial^\gamma_\xi D^\gamma_y(\chi(x,y) a_\epsilon(x,y,\xi))\vert_{x=y}$, a power series expansion of $\widehat{b_\epsilon}(x,\eta,\xi+\eta)$ in the last argument about $\xi$ and the same reasoning as in \cite[p.24-25]{Shubin:87} leads to the following estimates: \begin{gather*} |\sigma_{\epsilon}(x,\xi)-\hskip-5pt\sum_{|\gamma|From the previous computations, $(\sigma_\epsilon-\sum_{|\alpha|0$ for all $\epsilon\in(0,1]$ and $\inf_\epsilon r_\epsilon \neq 0$. \begin{definition} \label{defhyp} \rm Let $m,l,\mu,\rho,\delta$ be real numbers with $l\le m$ and $0\le\delta <\rho\le 1$. We say that $(a_\epsilon)_\epsilon\in{\underline{\mathcal{S}}}^{m,\mu} _{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ is an element of $ H{\underline{\mathcal{S}}}^{m,l,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$ if and only if for all $K\Subset\Omega$ there exists a strongly positive slow scale net $(r_{K,\epsilon})_\epsilon$, a net $(\omega_{1,K,\epsilon})_\epsilon$, $\omega_{1,K,\epsilon}\ge C_K\epsilon^{s_K}$ on the interval $(0,1]$ for certain constants $C_K>0$, $s_K\in\mathbb{R}$, and slow scale nets $(\omega_{2,K,\alpha,\beta,\epsilon})_\epsilon$, such that for all $x\in K$, for $|\xi|\ge r_{K,\epsilon}$, for all $\epsilon\in(0,1]$, \begin{equation} \label{hyp1} |a_\epsilon(x,\xi)|\ge \omega_{1,K,\epsilon}\langle\xi\rangle^l \end{equation} and \begin{equation} \label{hyp2} |\partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi)| \le \omega_{2,K,\alpha,\beta,\epsilon}|a_\epsilon(x,\xi)| \langle\xi\rangle^{-\rho|\alpha|+\delta|\beta|}. \end{equation} for all $(\alpha,\beta)\neq(0,0)$. When $\omega$ is independent of $\epsilon$ we use the notation $H{\underline{{\mathcal{S}}}}^{m,l}_{\rho,\delta,{\rm{rg}}} (\Omega\times\mathbb{R}^n)$. \end{definition} Before proceeding, we indicate a simple example of an element of $ H{\underline{\mathcal{S}}}^{m,l,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$. \begin{example} Let $(\omega^{-1}(\epsilon))_\epsilon$ be a slow scale net with $\sup_\epsilon\omega(\epsilon)<\infty$. Given $\mu\in\mathbb{R}\setminus\mathbb{N}$, let $(a_\epsilon)_\epsilon$ be a representative of a generalized function in $\mathcal{G}^\mu_{\ast,loc,\omega}(\Omega)$ (see Example \ref{exZyg}) such that \begin{equation} \label{exest} \forall K\Subset\Omega,\ \forall\alpha\in\mathbb{N}^n,\ \exists c>0:\ \forall x\in K,\ \forall\epsilon\in(0,1], \quad |\partial^\alpha a_\epsilon(x)|\le c\,\omega(\epsilon)^{-(|\alpha|-\mu)_+} \end{equation} and \begin{equation} \label{exest1} \forall K\Subset\Omega,\ \exists (\omega_{1,K,\epsilon})_\epsilon :\ \forall\epsilon\in(0,1],\quad \inf_{x\in K}|a_\epsilon(x)|\ge \omega_{1,K,\epsilon}, \end{equation} where $(\omega_{1,K,\epsilon}^{-1})_\epsilon$ is a slow scale net. Now, for any classical hypoelliptic symbol $b(x,\xi)\in HS^{m}_{\rho,\delta}(\Omega\times\mathbb{R}^n)$, the product $(a_\epsilon(x)b(x,\xi))_\epsilon$ belongs to $ H{\underline{\mathcal{S}}}^{m,l,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$. Since, as already proved in Section 4, $(a_\epsilon(x)b(x,\xi))_\epsilon\in{\underline{\mathcal{S}}}^{m,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$, we simply have to check the estimates \eqref{hyp1} and \eqref{hyp2}. Combining \eqref{exest} and \eqref{exest1} with the properties of $b$, we obtain that for all compact sets $K$ there exists a radius $R_K$ such that for $x\in K$, $|\xi|\ge R_K$, $\epsilon\in(0,1]$, \[ |a_\epsilon(x)b(x,\xi)|\ge c\omega_{1,K,\epsilon}\langle\xi\rangle^l, \] and \[ \begin{split} |\partial^\alpha_\xi\partial^\beta_x(a_\epsilon(x)b(x,\xi))| &\le \sum_{\beta'\le\beta}\binom{\beta}{\beta'}|\partial^{\beta'} _xa_\epsilon(x)||\partial^\alpha_\xi\partial^{\beta-\beta'}_xb(x,\xi)|\\ &\le c\omega(\epsilon)^{-(|\beta|-\mu)_+}\omega^{-1}_{1,K,\epsilon} \omega_{1,K,\epsilon}\langle\xi\rangle^{-\rho|\alpha|+\delta|\beta|}\\ &\le c'\omega(\epsilon)^{-(|\beta|-\mu)_+}\omega^{-1}_{1,K,\epsilon}| a_\epsilon(x)b(x,\xi)|\langle\xi\rangle^{-\rho|\alpha|+\delta|\beta|}, \end{split} \] where $(\omega(\epsilon)^{-(|\beta|-\mu)_+}\omega^{-1}_{1,K,\epsilon}) _\epsilon$ is a slow scale net. \end{example} Returning to $(a_\epsilon)_\epsilon$ in Definition \ref{defhyp}, it is evident from \eqref{hyp1} that $a_\epsilon(x,\xi)\neq 0$ for $x\in K$ and $|\xi|\ge r_{K,\epsilon}$. Let us choose a locally finite open covering $(\Omega_j)_{j\in\mathbb{N}}$ of $\Omega$ such that $\Omega_j\subset{\overline{\Omega}}_j\Subset\Omega_{j+1}$ for all $j$. Let $(\psi_j)_{j\in\mathbb{N}}$ be a partition of unity subordinate to $(\Omega_j)_j$ and let $(r_{j,\epsilon})_\epsilon:=(r_{{\overline{\Omega}}_j,\epsilon})_\epsilon$ be an increasing sequence of strongly positive slow scale nets satisfying \eqref{hyp1} and \eqref{hyp2} with $K={\overline{\Omega}}_j$. We take a function $\varphi\in\mathcal{C}^\infty(\mathbb{R}^n)$ such that $\varphi(\xi)=0$ for $|\xi|\le 1$ and $\varphi(\xi)=1$ for $|\xi|\ge 2$. At this point we can define \begin{equation} \label{po} p_{0,\epsilon}(x,\xi)=\sum_ja_\epsilon^{-1}(x,\xi) \varphi\big(\frac{\xi}{r_{j,\epsilon}}\big)\psi_j(x), \end{equation} where by construction $(p_{0,\epsilon})_\epsilon\in \mathcal{E}[\Omega\times\mathbb{R}^n]$ since every $(a_\epsilon^{-1}(x,\xi)\varphi(\frac{\xi}{r_{j,\epsilon}})) _\epsilon\in\mathcal{E}[\Omega\times\mathbb{R}^n]$ and the sum is locally finite. Before proceeding with the study of $(p_{0,\epsilon})_\epsilon$ we need a technical lemma. \begin{lemma} \label{lema-1} Let $(a_\epsilon)_\epsilon\in H{\underline{\mathcal{S}}}^{m,l,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$. For all $K\Subset\Omega$ and $\alpha,\beta\in\mathbb{N}^n$, there exists a slow scale net $(d_{K,\alpha,\beta,\epsilon})_\epsilon$ such that \begin{equation} \label{esta-1} |\partial^\alpha_\xi\partial^\beta_xa_\epsilon^{-1}(x,\xi)|\le d_{K,\alpha,\beta,\epsilon}\langle\xi\rangle^{-\rho|\alpha| +\delta|\beta|}|a^{-1}_\epsilon(x,\xi)|, \quad x\in K,\ |\xi|\ge r_{K,\epsilon},\ \epsilon\in(0,1]. \end{equation} \end{lemma} \begin{proof} Obviously \eqref{esta-1} is true for $\alpha ,\beta =0$. Differentiating $a_\epsilon^{-1}a_\epsilon(x,\xi)\equiv 1$ on $K\times\{|\xi|\ge r_{K,\epsilon}\}$ we obtain \[ \frac{\partial^\alpha_\xi\partial^\beta_x a_\epsilon^{-1}(x,\xi)}{a_\epsilon^{-1}(x,\xi)} =-\sum_{\substack{0<\alpha'\le\alpha\\ 0<\beta'\le\beta}} \binom{\alpha}{\alpha'}\binom{\beta}{\beta'}\frac{\partial ^{\alpha'}_\xi\partial^{\beta'}_x a_\epsilon(x,\xi)}{a_\epsilon(x,\xi)} \frac{\partial^{\alpha-\alpha'}_\xi\partial^{\beta-\beta'}_x a_\epsilon^{-1}(x,\xi)}{a_\epsilon^{-1}(x,\xi)}. \] Using induction, we conclude that \[ \begin{split} \frac{|\partial^\alpha_\xi\partial^\beta_x a_\epsilon^{-1}(x,\xi)|}{|a_\epsilon^{-1}(x,\xi)|} &\le\sum_{\substack{\alpha'\le\alpha\\ \beta'\le\beta}} d_{K,\alpha',\beta',\epsilon}\langle\xi\rangle^{-\rho|\alpha'| +\delta|\beta'|}\omega_{2,K,\alpha-\alpha',\beta-\beta',\epsilon} \langle\xi\rangle^{-\rho|\alpha-\alpha'|+\delta|\beta-\beta'|}\\ &\le d_{K,\alpha,\beta,\epsilon}\langle\xi\rangle^{-\rho|\alpha| +\delta|\beta|}, \end{split} \] where $(d_{K,\alpha,\beta,\epsilon})_\epsilon$ is a slow scale net since it is a finite sum of products of slow scale nets. \end{proof} \begin{proposition} \label{propo} Let $(a_\epsilon)_\epsilon\in H{\underline{\mathcal{S}}}^{m,l,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$. Then $(p_{0,\epsilon})_\epsilon$ is an element of $H{\underline{\mathcal{S}}}^{-l,-m}_{\rho,\delta,{\rm{rg}}} (\Omega\times\mathbb{R}^n)$ and for all $K\Subset\Omega$ there exists a strongly positive slow scale net $(r'_{K,\epsilon})_\epsilon$ such that for all $x\in K$, $|\xi|\ge r'_{K,\epsilon}$ and for all $\epsilon\in(0,1]$, \begin{equation} \label{estpo} p_{0,\epsilon}(x,\xi)a_\epsilon(x,\xi)=1. \end{equation} \end{proposition} \begin{proof} We begin by observing that for all $K\Subset\Omega$ there exists $j_0\in\mathbb{N}$ such that for all $j\ge j_0$, $\mathop{\rm supp}\psi_j\cap K=\emptyset$ and then \[ p_{0,\epsilon}(x,\xi)_{\vert_{K\times\mathbb{R}^n}} =\sum_{j=0}^{j_0}\Bigl(a_\epsilon^{-1}(x,\xi)\varphi \big(\frac{\xi}{r_{j,\epsilon}}\big)\psi_j(x)\Bigr)_{\hskip-3pt \left\vert_{K\times\mathbb{R}^n}\right.} \] For $x\in K$ and $|\xi|\ge 2r_{{j_0},\epsilon}$ we have $p_{0,\epsilon}(x,\xi)=\sum_{j=0}^{j_0}a^{-1}_\epsilon(x,\xi)\psi_j(x) =a^{-1}_\epsilon(x,\xi)$. This result proves \eqref{estpo}. Using \eqref{hyp1} and the definition of $(p_{0,\epsilon})_\epsilon$ we conclude \[ \begin{split} |p_{0,\epsilon}(x,\xi)|&=|a_\epsilon^{-1}(x,\xi)| \ge c_{K}\langle\xi\rangle^{-m}\epsilon^{N_K}\omega(\epsilon)^{(-\mu)_+}, \quad x\in K,\ |\xi|\ge 2r_{j_0,\epsilon},\ \epsilon\in(0,1],\\ |p_{0,\epsilon}(x,\xi)|&\le c_K\max_{0\le j \le j_0}(\omega^{-1}_{1,\overline{\Omega}_j,\epsilon})\langle \xi\rangle^{-l}\le c'_K\epsilon^{-M_K}\langle\xi\rangle^{-l}, \quad x\in K,\ \xi\in\mathbb{R}^n,\ \epsilon\in(0,1], \end{split} \] where $M_K=\max_{0\le j\le j_0}(s_{\overline{\Omega}_j})_+$. Let us now consider $\partial^\alpha_\xi\partial^\beta_xp_{0,\epsilon}(x,\xi)$ for $(\alpha,\beta)\neq(0,0)$. Since $p_{0,\epsilon}$ coincides with $a^{-1}_\epsilon(x,\xi)$ on $K\times\{|\xi|\ge 2r_{j_0,\epsilon}\}$, Lemma \ref{lema-1} guarantees the estimate \begin{equation} \label{hyp3} |\partial^\alpha_\xi\partial^\beta_x p_{0,\epsilon}(x,\xi)|\le d_{K,\alpha,\beta,\epsilon}|p_{0,\epsilon}(x,\xi)|\langle\xi\rangle^{-\rho|\alpha|+\delta|\beta|} \end{equation} on this set. In order to prove that $(p_{0,\epsilon})_\epsilon$ is an element of $H{\underline{\mathcal{S}}}^{-l,-m}_{\rho,\delta,{\rm{rg}}} (\Omega\times\mathbb{R}^n)$, it remains to estimate every term $\partial^\alpha_\xi\partial^\beta_x(a_\epsilon^{-1}(x,\xi)\varphi \big(\frac{\xi}{r_{j,\epsilon}}\big)\psi_j(x))$, $j\le j_0$, on $K\times\{r_{j,\epsilon}\le|\xi|\le 2r_{j_0,\epsilon}\}$. From \eqref{esta-1} and \eqref{hyp2}, recalling the assumptions on the nets involved in our formulas, we obtain \begin{equation} \label{estpod} \begin{split} &|\partial^\alpha_\xi\partial^\beta_x(a_\epsilon^{-1} (x,\xi)\varphi\big(\frac{\xi}{r_{j,\epsilon}}\big)\psi_j(x))|\\ &\le\sum_{\alpha'\le\alpha,\beta'\le\beta} d_{\overline{\Omega}_j, \alpha',\beta',\epsilon}1_{j,j_0}(|\xi|)\langle \xi\rangle^{-\rho|\alpha'|+\delta|\beta'|}|a_\epsilon^{-1} (x,\xi)| \\ &\quad\times \sup_{1\le|\xi|\le 2}|\partial^{\alpha-\alpha'}\varphi(\xi)| \sup_x|\partial^{\beta-\beta'}\psi_j(x)|\\ &\le\sum_{\alpha'\le\alpha,\beta'\le\beta} d'_{\overline{\Omega}_j, \alpha',\beta',\epsilon}\langle\xi\rangle^{-l-\rho|\alpha| +\delta|\beta|}\omega^{-1}_{1,\overline{\Omega}_j,\epsilon} \langle 2r_{j_0,\epsilon}\rangle^{\rho|\alpha-\alpha'|}\\ &\le g_{\overline{\Omega}_j,\alpha,\beta,\epsilon}\epsilon^{-M_K} \langle\xi\rangle^{-l-\rho|\alpha|+\delta|\beta|}, \quad x\in K,\ |\xi|\le 2r_{j_0,\epsilon}, \end{split} \end{equation} where $1_{j,j_0}$ is the characteristic function of the interval $[r_{j,\epsilon},2r_{j_0,\epsilon}]$. As a consequence there exist certain slow scale nets $(g_{K,\alpha,\beta,\epsilon})_\epsilon$ such that the following estimate holds on $K\times\mathbb{R}^n$: \begin{equation} \label{estpod2} |\partial^\alpha_\xi\partial^\beta_x p_{0,\epsilon}(x,\xi)| \le g_{K,\alpha,\beta,\epsilon}\epsilon^{-M_K}\langle\xi \rangle^{-l-\rho|\alpha|+\delta|\beta|}. \end{equation} In conclusion, combining the estimate from below with \eqref{hyp3} and \eqref{estpod2}, we have that $(p_{0,\epsilon})_\epsilon$ belongs to $H{\underline{\mathcal{S}}}^{-l,-m}_{\rho,\delta,{\rm{rg}}} (\Omega\times\mathbb{R}^n)$ and it is of growth type $M_K+1$ on the compact set $K$. \end{proof} \begin{proposition} \label{propoa} Let $(a_\epsilon)_\epsilon$ be in $H{\underline{\mathcal{S}}}^{m,l,\mu}_{\rho,\delta,\omega} (\Omega\times\mathbb{R}^n)$. Then for all $\alpha,\beta$ in $\mathbb{N}^n$, we have $(p_{0,\epsilon}(x,\xi)\partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi))_\epsilon$ in ${\underline{\mathcal{S}}}^{-\rho|\alpha|+\delta|\beta|} _{\rho,\delta,{\rm{rg}}}$. More precisely, for every $K\Subset\Omega$, for all $x\in K$, $\xi\in\mathbb{R}^n$ and $\epsilon\in(0,1]$, \begin{equation} \label{estpoad} |\partial^\gamma_\xi\partial^\sigma_x(p_{0,\epsilon}(x,\xi) \partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi))| \le s_{K,\alpha,\beta,\gamma,\sigma,\epsilon}\langle\xi \rangle^{-\rho|\alpha+\gamma|+\delta|\beta+\sigma|}, \end{equation} where $(s_{K,\alpha,\beta,\gamma,\sigma,\epsilon})_\epsilon$ is a slow scale net. \end{proposition} \begin{proof} We fix $K\Subset\Omega$. From \eqref{hyp2} and \eqref{hyp3} we easily see that there exists a slow scale net satisfying \eqref{estpoad} on $K\times\{|\xi|\ge 2r_{j_0,\epsilon}\}$. Let us assume now $|\xi|\le 2r_{j_0,\epsilon}$. From \eqref{hyp2} and the same arguments as used in \eqref{estpod} we obtain \[ \begin{split} &|\partial^\gamma_\xi\partial^\sigma_x(p_{0,\epsilon}(x,\xi) \partial^\alpha_\xi\partial^\beta_x a_\epsilon(x,\xi))|\\ &\le\sum_{\gamma'\le\gamma, \sigma'\le\sigma} \binom{\gamma}{\gamma'}\binom{\sigma}{\sigma'}|\partial^{\gamma'} _\xi\partial^{\sigma'}_xp_{0,\epsilon}(x,\xi)||\partial^{\alpha +\gamma-\gamma'}_\xi\partial^{\beta+\sigma-\sigma'}_x a_\epsilon(x,\xi)|\\ &\le\sum_{\gamma'\le\gamma, \sigma'\le\sigma, j\le j_0} \hskip -12pt l_{\overline{\Omega}_j,\gamma',\sigma',\epsilon}\langle\xi \rangle^{-\rho|\gamma'|+\delta|\sigma'|}\omega_{2,\overline{\Omega}_j, \alpha+\gamma-\gamma',\beta+\sigma-\sigma',\epsilon}\langle\xi \rangle^{-\rho|\alpha+\gamma-\gamma'|+\delta|\beta+\sigma-\sigma'|}\\ &\le s_{K,\alpha,\beta,\gamma,\sigma,\epsilon}\langle\xi \rangle^{-\rho|\alpha+\gamma|+\delta|\beta+\sigma|}, \end{split} \] where $(l_{\overline{\Omega}_j,\gamma',\sigma',\epsilon})_\epsilon$ and $(s_{K,\alpha,\beta,\gamma,\sigma,\epsilon})_\epsilon$ are slow scale nets. \end{proof} \begin{proposition} \label{proph} Let $(a_\epsilon)_\epsilon\in H{\underline{\mathcal{S}}}^{m,l,\mu}_{\rho,\delta,\omega}(\Omega\times\mathbb{R}^n)$. We define for $h\ge 1$ \[ p_{h,\epsilon}(x,\xi)=-\Bigl\{\sum_{\substack{|\gamma|+j=h\\ j