\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 17, pp. 1--8.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/17\hfil Weighted function spaces] {Weighted function spaces of fractional derivatives for vector fields} \author[A. Domokos\hfil EJDE-2007/17\hfilneg] {Andr\'{a}s Domokos} \address{Andr\'{a}s Domokos \newline Department of Mathematics and Statistics, California State University Sacramento, Sacramento, CA 95819, USA} \email{domokos@csus.edu} \thanks{Submitted May 26, 2006. Published January 25, 2007.} \subjclass[2000]{26A33, 35H20} \keywords{Fractional derivatives; weak solutions; subelliptic PDE} \begin{abstract} We introduce and study weighted function spaces for vector fields from the point of view of the regularity theory for quasilinear subelliptic PDEs. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}{section} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \section{Results} We consider a bounded domain $\Omega \subset {\mathbb R}^n$ and a system of smooth vector fields $X = (X_1 , \dots , X_m )$, $m \leq n$, defined on $\Omega$. Denote by $Xf = (X_1 f, \dots , X_m f )$ the $X$-gradient of a function $f$ and use the notation $|Xf|^2 = \sum_{i=1}^m (X_i f)^2$. In terms of the vector fields $X_1,\dots ,X_m$, in the theory of second order PDE, usually we have one of the following two cases: \begin{enumerate} \item[(1)] $X_i = \frac{\partial}{\partial x_i}$, $1 \leq i \leq n$ and we refer to it as the (classical) elliptic case. \item[(2)] There are points in $\Omega$ where the linear subspace of the tangent space spanned by the vector fields $X_1, \dots , X_m$ has dimension strictly less then $n$, but at the same time H\"{o}rmander's condition is satisfied, which means that there exists a positive integer $\nu \geq 2$ such that the vector fields $X_i$ and their commutators $$[X_{i_1} , [X_{i_2} , \dots , X_{i_k}]\dots ] \,, \quad 2 \leq k \leq \nu$$ of length at most $\nu \in {\mathbb N}$ span the tangent space at every point of $\Omega$. We refer to this case as the subelliptic case and the vector fields $X_i$ are called horizontal vector fields. \end{enumerate} Let $2 \leq p<\infty$ and $K \subset \Omega$ be a compact subset of $\Omega$. Consider the Sobolev space $$XW^{1,p} (\Omega ) = \Bigl\{ f \in L^p(\Omega ) : X_i f \in L^p (\Omega ) \mbox{ for all } \; i \in \{1,\dots ,m\} \Bigr\} \, .$$ In the elliptic case we use the usual $W^{1,p} (\Omega )$ notation. If $Z$ is a smooth vector field then we define its flow as the mapping $F(x,s) = e^{sZ}x$ which solves the initial value problem $$\label{e1} \begin{split} \frac{\partial F}{\partial s} (x,s)& = Z F (x,s)\\ F(x,0) & =x \; . \end{split}$$ For $f \in XW^{1,p} (\Omega )$, we define the weight $$w(Xf,s,x) = \Bigl( 1 + |Xf(x)|^2 + |Xf(e^{sZ} x)|^2 \Bigr)^{1/2}$$ and the following first and second order differences: \begin{gather*} \Delta_{Z,s} f(x) = f(e^{sZ}x) - f(x) \, ,\\ \Delta_{Z,-s} f(x) = f(x) - f( e^{-sZ}x)\, ,\\ \Delta_{Z,s}^2 f(x) = f(e^{sZ}x) + f(e^{-sZ}x) -2f(x) \, . \end{gather*} Notice that $$\Delta_{Z,s}^2 f(x) = \Delta_{Z,-s} \Delta_{Z,s} f(x) = \Delta_{Z,s} \Delta_{Z,-s} f(x)\, .$$ Let $0 < \theta <2$, $0 \leq \alpha \leq p-2$ and $2 \leq q \leq p - \alpha$. Consider $s_K > 0$ sufficiently small such that $$e^{sZ}x \in \Omega \, , \quad \mbox{for all } 0 < |s| < s_K \text{ and } x \in K \, ,$$ and the Jacobian of the transformation $x \mapsto e^{s Z} x$ to be bounded in the following way: $$0 < a^q \leq \left| J \bigl( e^{s Z} x \bigr) \right| \leq b^q\, , \quad \mbox{for all } 0 < |s| < s_K \text{ and } x \in K \, ,$$ where $0< a \leq 1 \leq b$. Consider the following two pseudo-norms: \begin{gather*} \|f\|_{Z,\alpha,p,q}^{\theta,1} = \|f\|_{L^P(\Omega)} + \sup_{0<|s|0$depending on the$XW^{1,p}$norm of$f, we have \label{e3} \begin{aligned} &\Big( \int_{\Omega} \bigl( 1 + |Xf(x)|^2 \bigr)^{\alpha/2} \left| \triangle_{Z,\frac{s}{2^n}} f(x) \right|^q dx \Big)^{1/q}\\ &\leq \frac{1}{2^n} \Big( \int_{\Omega} \bigl( 1 + |Xf(x)|^2 \bigr)^{\alpha/2} \left| \triangle_{Z,s} f (x) \right|^q dx \Big)^{1/q}+ c \frac{M}{a 2^{\theta}} |s|^{\theta} 2^{-\theta n}\\ &\leq C \Bigl( \frac{1}{2^n} + |s|^{\theta} 2^{-\theta n} \Bigr) \,. \end{aligned} For allh$with$0< |h| < s_K/2$there exist$n \in {\mathbb N}$and$s \in {\mathbb R}$such that$|s| \in [s_K/2 , s_K ]$and$h = s/2^n$. In this way we get \begin{equation*} \frac{1}{|h|^{\theta}} \Big(\int_{\Omega} \bigl( 1 + |Xf(x)|^2 \bigr)^{\alpha/2} \; \left| \triangle_{Z,h} f(x) \right|^q dx \Big)^{1/q} \leq C \Big( \frac{|h|^{1-\theta}}{s_K} + 1 \Big) \, . \end{equation*} Also, for$s_K / 2 \leq |h| \leq s_K$we have $$\frac{1}{|h|^{\theta}} \Big(\int_{\Omega} \bigl( 1 + |Xf(x)|^2 \bigr)^{\alpha/2} \left| \triangle_{Z,h} f(x) \right|^q dx \Big)^{1/q} \leq C\, ,$$ and therefore, $$\sup_{0<|h|2, by Theorem \ref{thm1}, the inequality \frac{1}{2}+\frac{1}{p}<1 implies that$$ \eta^2 u \in B_{T,0,p,p}^{\frac{1}{2}+\frac{1}{p},1} (\mathop{\rm supp} \eta , \Omega) \,, $$and hence we can restart the whole process again with \frac{1}{2}+\frac{1}{p} instead of \frac{1}{2} and a new cut-off function \eta with a conveniently chosen support to get$$ \eta^2 u \in B_{T,0,p,p}^{\frac{1}{2}+\frac{1}{p}+ \frac{2}{p^2},1} (\mathop{\rm supp} \eta , \Omega) \,. $$In general, after k iterations we get \eta^2 u \in B_{T,0,p,p}^{\gamma_k,2} (\mathop{\rm supp} \eta , \Omega), with$$ \gamma_k = \frac{1}{2} + \frac{1}{p} \Big( 1 + \frac{2}{p}+ \dots + \frac{2^{k-1}}{p^{k-1}} \Big) \, . $$If 2 \leq p < 4 then for a sufficiently large k we have \gamma_k >1 and then$$ \eta^2 u \in B_{T,0,p,p}^{1,1} (\mathop{\rm supp} \eta , \Omega) $$which implies that Tu \in L^p_{\rm loc} (\Omega ). Of course, there is the question of what is happening if, for a k \in {\mathbb N}, we get \gamma_k =1. In this case, we can choose a \gamma_{k+1} <1 sufficiently close to 1 such that after repeating the iteration to get \gamma_{k+2} > 1. \end{example} \begin{remark} \label{rmk2} \rm We study the case p\geq 2 in order to be able to give a uniform approach to our function spaces in various cases of horizontal vector fields. In \cite{dom} it is also proved that Tu \in L^p_{\rm loc} (\Omega) for 1 < p <2. The proof of this result is connected to Heisenberg group and does not work for other Carnot groups of step 3 or higher. However, let us give the sequence of spaces in which we include \eta^2 u. So, we start with B_{T,0,p,p}^{\frac{1}{2},1} (\mathop{\rm supp} \eta , \Omega) and continue with \begin{gather*} XB_{T,p-2,p,2}^{\frac{1}{4},1} (\mathop{\rm supp} \eta , \Omega) \,, \quad XB_{T,0,p,p}^{\frac{1}{4},1} (\mathop{\rm supp} \eta , \Omega), \\ B_{T,0,p,p}^{\frac{3}{4},2} (\mathop{\rm supp} \eta , \Omega) \,, \quad B_{T,0,p,p}^{\frac{3}{4},1} (\mathop{\rm supp} \eta , \Omega) \,, \dots \,, \\ B_{T,0,p,p}^{\frac{2^{k+1} -1}{2^{k+1}},1} (\mathop{\rm supp} \eta , \Omega) \,, \quad B_{T,0,p,p}^{\frac{1}{2}+\gamma_k,2} (\mathop{\rm supp} \eta , \Omega), \end{gather*} where \gamma_k = \frac{2^k -1}{2^{k+2}}(p-1) + \frac{2^{k+1} -1}{2^{k+2}}> 1/2 for k sufficiently large. \end{remark} \begin{example} \label{exa2} \rm We consider now an example involving commutators of length higher than 2. Our preference goes with Grushin type vector fields, but we could use T from the center of any nilpotent Lie Algebra generated by a system of horizontal vector fields. Consider \Omega \subset {\mathbb R}^2 intersecting the line x_1 =0 and the vector fields X_1 = \frac{\partial}{\partial x_1} and X_2 = x_1^3 \frac{\partial}{\partial x_2}. At the points (0,x_2) \in \Omega the vector fields X_1 and X_2 span a 1 dimensional subspace, so we need their commutator of length 4$$ T = [X_1 , [X_1 , [X_1 , X_2]]] = 6 \frac{\partial}{\partial x_2} $$to span the whole tangent space. \end{example} According to \cite{ho} we have$$ \eta^2 u \in B^{\frac{1}{4},1}_{T,0,p,p} (\Omega ) $$for every \eta \in C_0^{\infty} (\Omega ) and we can start the iteration process with the test function$$ \varphi = \frac{\triangle_{T,-s}}{s^{1/4}} \Big( \frac{\triangle_{T,s} (\eta^2 u)}{s^{1/4}} \Big) \,. $$In a similar to way to Example \ref{exa1} we get the series of inclusions \begin{gather*} \eta^2 u \in XB_{T,p-2,p,2}^{\frac{1}{4},1} (\mathop{\rm supp} \eta , \Omega) ,\\ \eta^2 u \in XB_{T,0,p,p}^{\frac{1}{2p},1} (\mathop{\rm supp} \eta , \Omega) \, ,\\ \eta^2 u \in B_{T,0,p,p}^{\frac{1}{4}+\frac{1}{2p},2} (\mathop{\rm supp} \eta , \Omega) \,. \end{gather*} By Theorem \ref{thm1}, the inequality \frac{1}{4}+\frac{1}{2p}<1 implies that$$ \eta^2 u \in B_{T,0,p,p}^{\frac{1}{4}+\frac{1}{2p},1} (\mathop{\rm supp} \eta , \Omega) \,, $$and hence we can restart the whole process again with \frac{1}{4}+\frac{1}{2p} instead of \frac{1}{4} and get$$ \eta^2 u \in B_{T,0,p,p}^{\frac{1}{4}+\frac{1}{2p}+ \frac{1}{p^2},1} (\mathop{\rm supp} \eta , \Omega) \,. $$Therefore, after k iterations we get$$ \eta^2 u \in B_{T,0,p,p}^{\gamma_k,2} (\mathop{\rm supp} \eta , \Omega), $$with$$ \gamma_k = \frac{1}{4} + \frac{1}{2p} \Big( 1 + \frac{2}{p}+ \dots + \frac{2^{k-1}}{p^{k-1}} \Big) \, . $$If 2 \leq p < 8/3 then for a sufficiently large k we have \gamma_k >1 and then$$ \eta^2 u \in B_{T,0,p,p}^{1,1} (\mathop{\rm supp} \eta , \Omega)$$which implies that$Tu \in L^p_{\rm loc} (\Omega )$. \begin{thebibliography}{99} \bibitem{cap} L. 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