\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 170, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/170\hfil Generalized Leray-alpha equations]
{Local and global low-regularity solutions to  generalized Leray-alpha equations}

\author[N. Pennington \hfil EJDE-2015/170\hfilneg]
{Nathan Pennington}

\address{Nathan Pennington \newline
Department of Mathematics, Creighton University,
2500 California Plaza,
Omaha, NE 68178, USA}
\email{nathanpennington@creighton.edu}

\thanks{Submitted  May 15, 2015. Published June 18, 2015.}
\subjclass[2010]{76D05, 35A02, 35K58}
\keywords{Leray-alpha model; Besov space; fractional Laplacian}

\begin{abstract}
 It has recently become common to study approximating equations
 for the Navier-Stokes equation. One of these is the Leray-$\alpha$ equation,
 which regularizes the Navier-Stokes  equation by replacing (in most locations)
 the solution $u$   with $(1-\alpha^2\Delta)u$.  Another is the generalized
 Navier-Stokes equation,  which replaces the Laplacian with a Fourier multiplier
 with symbol of the   form $-|\xi|^\gamma$ 
 ($\gamma=2$ is the standard Navier-Stokes  equation),  
 and recently in \cite{taolog} Tao also considered multipliers of
 the form   $-|\xi|^\gamma/g(|\xi|)$, where $g$ is (essentially) a logarithm.
 The generalized Leray-$\alpha$ equation combines these two modifications by
 incorporating the regularizing term and replacing the Laplacians with more
 general Fourier multipliers, including allowing for $g$ terms similar to those
 used in \cite{taolog}.  Our goal in this paper is to obtain existence and
 uniqueness results with low regularity and/or non-$L^2$ initial data.
 We will also use energy estimates to extend some of these local existence
 results to global existence results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}


The incompressible form of the Navier-Stokes equation is given by
\begin{equation}\label{NS}
\begin{gathered}
 \partial_t u + (u\cdot \nabla)u=\nu\Delta u-\nabla p,
\\ u(0,x)=u_0(x), \quad \operatorname{div}(u)=0
\end{gathered}
\end{equation}
where $u:I\times \mathbb{R}^n\to \mathbb{R}^n$ for some time
strip $I=[0,T)$, $\nu>0$ is a constant due to the viscosity of the fluid,
$p:I\times\mathbb{R}^n\to \mathbb{R}^n$ denotes the fluid pressure,
and $u_0:\mathbb{R}^n\to\mathbb{R}^n$.  The requisite differential
operators are defined by $\Delta=\sum_{i=1}^n \frac{\partial^2}{\partial_{x_i}^2}$
and $\nabla=(\frac{\partial} {\partial_{x_i}},\dots ,\frac{\partial}{\partial_{x_n}})$.

In dimension $n=2$, local and global existence of solutions to the Navier-Stokes
equation are well known (see \cite{lady356}; for a more modern reference,
see \cite[Chapter 17]{T3}).  For dimension $n\geq 3$, the problem is significantly
 more complicated.  There is a robust collection of local existence results,
including \cite{Kato}, in which Kato proves the existence of local solutions
to the Navier-Stokes equation with initial data in $L^n(\mathbb{R}^n)$; \cite{KP},
 where Kato and Ponce solve the equation with initial data in the Sobolev space
$H^{n/p-1,p}(\mathbb{R}^n)$; and \cite{KT}, where Koch and Tataru establish
local existence with initial data in the space $BMO^{-1}(\mathbb{R}^n)$
(for a more complete accounting of local existence theory for the Navier-Stokes
equation, see \cite{nsbook}).  In all of these local results, if the initial
datum is assumed to be sufficiently small, then the local solution can be
extended to a global solution.   However, the issue of global existence of
solutions to the Navier-Stokes equation in dimension $n\geq 3$ for arbitrary
initial data is one of the most challenging open problems remaining in analysis.

Because of the intractability of the Navier-Stokes equation, many approximating
equations have been studied.  One of these is the Leray-$\alpha$ model, which is
%\label{leray1}
\begin{gather*}
\partial_t(1-\alpha^2\Delta )u+\nabla_u (1-\alpha^2 \Delta) u-\nu\Delta (1-\alpha^2 \Delta )u=-\nabla p,
\\ u(0,x)=u_0(x), ~~ \operatorname{div}u_0=\operatorname{div}u=0,
\end{gather*}
where we recall that $\nabla_u v=(u\cdot \nabla)v$.  Note that setting
$\alpha=0$ returns the standard Navier-Stokes equation.
Like the Lagrangian Averaged Navier-Stokes (LANS) equation
(which differs from the Leray-$\alpha$ in the presence of an additional
nonlinear term), the system \eqref{leray1} compares favorably with
numerical data; see \cite{holmleray}, in which the authors compared the
 Reynolds numbers for the Leray-$\alpha$ equation and the LANS equation
with the Navier-Stokes equation.

Another commonly studied equation is the generalized Navier-Stokes equation,
given by
\begin{gather*}
\partial_t u + (u\cdot \nabla)u=\nu\mathcal{L} u-\nabla p,\\
u(0,x)=u_0(x), \quad \operatorname{div}(u)=0
\end{gather*}
where $\mathcal{L}$ is a Fourier multiplier with symbol $m(\xi)=-|\xi|^\gamma$
for $\gamma>0$.  Choosing $\gamma=2$ returns the standard Navier-Stokes equation.
 In \cite{Wu}, Wu proved (among other results) the existence of unique local
solutions for this equation provided the data is in the Besov space
$B^s_{p,q}(\mathbb{R}^n)$ with $s=1+n/p-\gamma$ and $1<\gamma\leq 2$.
If the norm of the initial data is sufficiently small, these local solutions
can be extended to global solutions.

It is well known that if $\gamma\geq \frac{n+2}{2}$, then this equation
has a unique global solution.  In \cite{taolog}, Tao strengthened this result,
 proving global existence with the symbol $m(\xi)=-|\xi|^\gamma/g(|\xi|)$,
with $\gamma\geq \frac{n+2}{2}$ and $g$ a non-decreasing, positive function
that satisfies
\[
\int_1^\infty \frac{ds}{sg(s)^2}=+\infty.
\]
Note that $g(|x|)=\log^{1/2}(2+|x|^2)$ satisfies the condition.
Similar types of results involving $g$ terms that are, essentially,
logarithms have been proven for the nonlinear wave equation;
see \cite{taolog} for a more detailed description.

Here we consider a combination of these two models, called the
generalized Leray-$\alpha$ equation, which is
\begin{equation} \label{leray1}
\begin{gathered}
\partial_t(1-\alpha^2 \mathcal{L}_2 )u+\nabla_u (1-\alpha^2  \mathcal{L}_2) u
 -\nu\mathcal{L}_1 (1-\alpha^2 \mathcal{L}_2 )u=-\nabla p, \\
u(0,x)=u_0(x), \quad \operatorname{div}u_0=\operatorname{div}u=0,
\end{gathered}
\end{equation}
with the operators $\mathcal{L}_i$ defined by
\[
\mathcal{L}_iu(x)=\int -\frac{|\xi|^{\gamma_i}}{g_i(\xi)}\hat{u}(\xi)
e^{ix\cdot \xi}d\xi,
\]
where $g_i$ are radially symmetric, nondecreasing, and bounded below by $1$.
Note that if $g_2=1$ and $\gamma_2=0$, then $\mathcal{L}_2 u(x)=-u(x)$, so
choosing $g_1=g_2=1$, $\gamma_1=2$, and $\gamma_2=0$ returns the Navier-Stokes
equation (after absorbing $(1+\alpha^2)^{-1}$ into the pressure function $p$).
Choosing $g_1=g_2=1$ and $\gamma_1=\gamma_2=2$ gives the Leray-$\alpha$ equation,
and choosing $g_2=\gamma_2=1$ returns the generalized Navier-Stokes equation.

In \cite{barbato}, the authors proved the existence of a smooth global solution
to the generalized Leary-$\alpha$ equation with smooth initial data provided
$\gamma_1+\gamma_2\geq n/2+1$, $g_2=1$, and $g_1$ is in a category similar to,
though inclusive of, the type of $g$ required in Tao's argument in \cite{taolog}.

In \cite{kazuo}, Yamazaki obtains a unique global solution to equation
\eqref{leray1} in dimension three provided  $(1-\alpha^2\mathcal{L}_2)u_0$
is in the Sobolev space $H^{m,2}(\mathbb{R}^3)$, where $u_0$ is the initial data,
$m>\max\{5/2, 1+2\gamma_1\}$, and provided $\gamma_1$ and $\gamma_2$ satisfy
the inequality $2\gamma_1+\gamma_2\geq 5$ and that $g_1$ and $g_2$ satisfy
\begin{equation}\label{g restriction}
\int_1^\infty \frac{ds}{s g_1^2(s)g_2(s)} =\infty.
\end{equation}

The goal of this article is to obtain a much wider array of existence results,
specifically existence results for initial data with low regularity and for
initial data outside the $L^2$ setting.  We will also, where applicable,
use the energy bound from \cite{kazuo} to extend these local solutions to
global solutions.  Our plan is to follow the general contraction-mapping based
procedure outlined by Kato and Ponce in \cite{KP} for the Navier-Stokes equation,
with two key modifications.

First, the approach used in \cite{KP} relies heavily on operator estimates
for the heat kernel $e^{t\Delta}$.  We will require similar estimates for our
solution operator $e^{t\mathcal{L}_1}$ and some operator estimates for
$(1-\alpha^2\mathcal{L}_2)$, and establishing these estimates is the purpose
of Section \ref{L operator estimates} and Section \ref{operator est}.
This will require some technical restrictions on the choices of $g_1$ and $g_2$
that will be more fully addressed below.  We also note that these estimates should
allow the application of this general technique to other similar equations,
like the MHD equation in \cite{kazuomhd} and \cite{kazuo}, the generalized MHD
equation found in \cite{globalmhd} (see \cite{genmdh} for a general study of
the generalized MHD equation), the logarithmically super critical Boussinesq
system in \cite{hmidi}, and the Navier-Stokes like equation studied
in \cite{olsontitialphalike}.

The second modification is in how we will deal with the nonlinear term.
 For the first set of results, we will use the standard Leibnitz-rule estimate
to handle the nonlinear terms.  Our second set of results rely on a product
estimate (due to Chemin in \cite{chemin}) which will allow us to obtain lower
regularity existence but will (among other costs) require us to work in Besov spaces.  The advantages and disadvantages of each approach will be detailed later in this introduction. 
 The product estimates themselves are stated as Proposition \ref{product est 1}
 and Proposition $\ref{product est 2}$ in Section \ref{Besov space}.  
We will also need bounds on the terms $(1-\mathcal{L}_2)$ and 
$(1-\mathcal{L}_2)^{-1}$, and establishing these bounds is the subject 
of Section \ref{L operator estimates}.

The rest of this paper is organized as follows.  The remainder of this introduction
is devoted to stating and contextualizing the main results of the paper.
Section \ref{Besov space} reviews the basic construction of Besov spaces and
states some foundational results, including our two product estimates.
In Section \ref{type 1} we carry out the existence argument using the standard
product estimate, and in Section \ref{type 2} we obtain existence results
using the other product estimate.  As stated above, Sections
\ref{L operator estimates} and \ref{operator est} contain the proofs of the
operator estimates that are central to the arguments used in Sections \ref{type 1}
and \ref{type 2}.

Our last task before stating the main results is to establish some notation.
First, we denote Besov spaces by $B^s_{p,q}(\mathbb{R}^n)$, with norm denoted
by $\|\cdot\|_{B^s_{p,q}}=\|\cdot\|_{s,p,q}$ (a complete definition of these
spaces can be found in Section \ref{Besov space}).  We define the space
\[
C^T_{a;s,p,q}=\{f\in C((0,T):B^s_{p,q}(\mathbb{R}^n)):\|f\|_{a;s,p,q}<\infty\},
\]
where
\[
\|f\|_{a;s,p,q}=\sup\{t^a\|f(t)\|_{s,p,q}:t\in (0,T)\},
\]
$T>0$, $a\geq 0$, and $C(A:B)$ is the space of continuous functions from $A$ to $B$.
We let $\dot{C}^T_{a;s,p,q}$ denote the subspace of $C^T_{a;s,p,q}$ consisting
of $f$ such that
\[
\lim_{t\to 0^+}t^a f(t)=0 \quad \text{(in } B^s_{p,q}(\mathbb{R}^n)).
\]
Note that while the norm $\|\cdot\|_{a;s,p,q}$ lacks an explicit reference to
$T$, there is an implicit $T$ dependence. We also say $u\in BC(A:B)$ if
$u\in C(A:B)$ and $\sup_{a\in A}\|u(a)\|_{B}<\infty$.

Now we are ready to state the existence results.  As expected in these types
of arguments, the full result gives unique local solutions provided the
parameters satisfy a large collection of inequalities.
Here we state special cases of the full results.  Our first Theorem uses
the standard product estimate (Proposition \ref{product est 1} in
Section \ref{Besov space}).

\begin{theorem}\label{old style short}
Let  $\gamma_1>1$, $\gamma_2>0$, $q\geq 1$, $p\geq 2$, and let $s_1$, $s_2$ 
be real numbers such that $s_2>\gamma_2$, $0<s_2-s_1<\min\{\gamma_1/2, 1\}$ and 
$\gamma_1\geq s_2-s_1+1+n/p$.   We also assume that $g_1$ and $g_2$ are
 Mikhlin multipliers (see inequality \eqref{mikhlin condition don}).  
Then for any divergence free $u_0\in B^{s_1}_{p,q}(\mathbb{R}^n)$, there exists 
a unique local solution $u$ to the generalized Leray-alpha equation \eqref{leray1}, 
with
\[
u\in BC([0,T):B^{s_1}_{p,q}(\mathbb{R}^n))\cap \dot{C}^T_{a;s_2,p,q},
\]
where $a=(s_2-s_1)/\gamma_1$.  $T$ can be chosen to be a non-increasing function
 of $\|u_0\|_{B^{s_1}_{p, q}}$ with $T=\infty$ if $\|u_0\|_{B^{s_1}_{p, q}}$ 
is sufficiently small.
\end{theorem}

Before stating our second theorem, we remark that this result also holds if 
the Besov spaces are replaced by Sobolev spaces.  This is not true of the next 
theorem, which is a special case of the more general 
Theorem \ref{full generality}, and relies on our second product estimate 
(Proposition \ref{product est 2} in Section \ref{Besov space}).
\begin{theorem}\label{special case 1}
Let $\gamma_1>1$, $\gamma_2>0$, $q\geq 1$, $p\geq 2$, $s_1$, and $s_2$ satisfy
\begin{gather*} 
0<s_2-s_1<\gamma_1/2, \\ 
s_1>\gamma_2-n/p-1, \\ 
\gamma_1\geq 2s_2-s_1-\gamma_2+n/p+1,\\ 
n/p>\gamma_2/2, \\ 
s_2\geq \gamma_2/2.
\end{gather*}
We also assume that $g_1$ and $g_2$ are Mikhlin multipliers 
(see inequality \eqref{mikhlin condition don}).  Then for any divergence 
free $u_0\in B^{s_1}_{p,q}(\mathbb{R}^n)$, there exists a unique local solution 
$u$ to the generalized Leray-alpha equation \eqref{leray1}, with
\[
u\in BC([0,T):B^{s_1}_{p,q}(\mathbb{R}^n))\cap \dot{C}^T_{a;s_2,p,q},
\]
where $a=(s_2-s_1)/\gamma_1$.  $T$ can be chosen to be a non-increasing 
function of $\|u_0\|_{B^{s_1}_{p, q}}$ with $T=\infty$ if 
$\|u_0\|_{B^{s_1}_{p, q}}$ is sufficiently small.
\end{theorem}

We remark that in the first theorem, $\gamma_2$ can be arbitrarily large, 
but $s_1>-1$, while in the second theorem $\gamma_2<2n/p$, but for sufficiently 
large $\gamma_1$ and sufficiently small $\gamma_2$, $s_1>\gamma_2-n/p-1$ 
can be less than $-1$.  Thus the non-standard product estimate allows us 
to obtain existence results for initial data with lower regularity, 
but requires $\gamma_2$ to be small and requires the use of Besov spaces.

We also note that if we set $\gamma_2=0$ and $g_2(|\xi|)=1$ (and thus are
 back in the case of the generalized Navier-Stokes equation), then these 
techniques would recover the results of Wu in \cite{Wu} for the generalized 
Navier-Stokes equation.

As was stated above, these results will hold if the $g_i$ are Mikhlin multipliers.  
However, there are interesting choices of $g_i$ (specifically $g_i$ 
being, essentially, a logarithm) which are not Mikhlin multipliers.  
For this case we have analogous, but slightly weaker, results.  
In what follows, we let $r^-$ indicate a number arbitrarily close to, 
but strictly less than, $r$ (and similarly let $r^+$ be a number arbitrarily 
close to, but strictly greater than, $r$).

\begin{theorem}\label{old style short log}
Let $\gamma_1>1$, $\gamma_2>0$, $q\geq 1$, $p\geq 2$, and $s_1$, $s_2$ 
be real numbers such that $s_2>\gamma_2$, $0<s_2-s_1<\min\{\gamma_1^-/2, 1\}$ and 
$\gamma_1^-\geq s_2-s_1+1+n/p$.   We also assume that, for $i=1,2$, 
$|g_i(r)|\leq Cr^\delta$ for any $\delta>0$ and $|g_i^{(k)}(r)|\leq Cr^{-k}$ 
for $1\leq k\leq n/2+1$.  Then for any divergence free 
$u_0\in B^{s_1}_{p,q}(\mathbb{R}^n)$, there exists a unique local solution 
$u$ to the generalized Leray-alpha equation \eqref{leray1}, with
\[
u\in BC([0,T):B^{s_1}_{p,q}(\mathbb{R}^n))\cap \dot{C}^T_{a;s_2,p,q},
\]
where $a=(s_2-s_1)/\gamma_1^-$ for arbitrarily small $\varepsilon>0$.
 $T$ can be chosen to be a non-increasing function of
$\|u_0\|_{B^{s_1}_{p, q}}$ with $T=\infty$ if $\|u_0\|_{B^{s_1}_{p, q}}$
is sufficiently small.
\end{theorem}


\begin{theorem}\label{special case 1 log}
Let $\gamma_1>1$, $\gamma_2>0$, $q\geq 1$, $p\geq 2$,  $s_1$ and $s_2$ satisfy
\begin{gather*} 
0<s_2-s_1<\gamma^-_1/2,\\ 
s_1>\gamma^-_2-n/p-1, \\ 
\gamma^-_1\geq 2s_2-s_1-\gamma^-_2+n/p+1,\\ 
n/p>\gamma^-_2/2,\\ 
s_2\geq \gamma^-_2/2.
\end{gather*}
We also assume that, for $i=1,2$, $|g_i(r)|\leq Cr^\delta$ for any 
$\delta>0$ and $|g_i^{(k)}(r)|\leq Cr^{-k}$ for $1\leq k\leq n/2+1$.  
Then for any divergence free $u_0\in B^{s_1}_{p,q}(\mathbb{R}^n)$, 
there exists a unique local solution $u$ to the generalized Leray-alpha 
equation \eqref{leray1}, with
\[
u\in BC([0,T):B^{s_1}_{p,q}(\mathbb{R}^n))\cap \dot{C}^T_{a;s_2,p,q},
\]
where $a=(s_2-s_1)/\gamma^-_1$.  $T$ can be chosen to be a non-increasing 
function of $\|u_0\|_{B^{s_1}_{p, q}}$ with $T=\infty$ if
 $\|u_0\|_{B^{s_1}_{p, q}}$ is sufficiently small.
\end{theorem}

Incorporating the additional constraints from the energy bound in \cite{kazuo}, 
we can now state the global existence result.

\begin{corollary} \label{coro1.1}
Let $p=2$ and let $n=3$.  Then, for any of our local existence results, 
if we additionally assume that
\begin{gather*} 
2\gamma_1+\gamma_2\geq 5, \\  
\int_1^\infty \frac{ds}{s g_1^2(s)g_2(s)} =\infty,
\end{gather*}
then the local solutions can be extended to global solutions.
\end{corollary}

Note that if $g_1$ and $g_2$ are Mikhlin multipliers, then all of the 
constraints on $g_1$ and $g_2$ are satisfied.  Also,   
if $g_1$ and $g_2$ are logarithms, then the corollary extends the appropriate 
local solutions from Theorem \ref{old style short log} and 
\ref{special case 1 log} to global solutions.

The corollary follows directly from the smoothing effect of the operator 
$e^{t\mathcal{L}_1}$, which ensures that, for any $t>0$, our local solution 
$u(t,\cdot)\in B^r_{2,q}(\mathbb{R}^3)$ for any $r\in \mathbb{R}$.  
This provides the smoothness necessary to use the energy bound from 
\cite{kazuo} to obtain a uniform-in-time bound on the $B^{s_1}_{2,q}(\mathbb{R}^3)$
 norm of the solution, and then a standard bootstrapping argument completes 
the proof of global existence.  
In Section \ref{Higher regularity for the local existence result}, we include 
an argument detailing this smoothing effect for the solution to 
Theorem \ref{old style short}.

Finally, we remark that extending the local solutions to global solutions for 
$p\neq 2$ and $n>3$ will be the subject of future work.  Handling $n>3$ should 
follow by tweaking the argument used in \cite{kazuo}.  Obtaining global 
solutions for $p\neq 2$ is significantly more complicated, and the argument
 will follow the interpolation based argument used by 
Gallagher and Planchon \cite{galplan} for the two dimensional Navier-Stokes equation.

\section{Besov spaces}\label{Besov space}

We begin by defining the Besov spaces $B^s_{p,q}(\mathbb{R}^n)$.  
Let $\psi_0$ be an even, radial, Schwartz function with Fourier transform 
$\hat{\psi_0}$ that has the following properties:
\begin{gather*} 
\hat{\psi_0}(x)\geq 0,\\  
\operatorname{support}\hat{\psi_0}\subset A_0
 :=\{\xi\in \mathbb{R}^n:2^{-1}<|\xi|<2\},\\ 
\sum_{j\in\mathbb{Z}} \hat{\psi_0}(2^{-j}\xi)=1, \quad \text{for all } \xi\neq 0.
\end{gather*}

We then define
$\hat{\psi_j}(\xi)=\hat{\psi}_0(2^{-j}\xi)$ (from Fourier inversion, 
this also means $\psi_j(x)=2^{jn}\psi_0(2^jx)$), and remark that 
$\hat{\psi_j}$ is supported in $A_j:=\{\xi\in\mathbb{R}^n:2^{j-1}<|\xi|<2^{j+1}\}$. 
 We also define $\Psi$ by
\begin{equation}\label{low freq part}
\hat{\Psi}(\xi)=1-\sum_{k=0}^\infty \hat{\psi}_k(\xi).
\end{equation}

We define the Littlewood Paley operators $\Delta_j$ and $S_j$ by
\[
\Delta_j f=\psi_j\ast f, \quad
S_jf=\sum_{k=-\infty}^{j}\Delta_k f,
\]
and record some properties of these operators.  
Applying the Fourier Transform and
recalling that $\hat{\psi}_j$ is supported on 
$2^{j-1}\leq |\xi|\leq2^{j+1}$, it follows that
\begin{equation}\label{besovlemma1}
\begin{gathered}
\Delta_j\Delta_k f= 0, \quad |j-k|\geq 2, \\ 
\Delta_j (S_{k-3}f\Delta_{k}g)= 0 \quad |j-k|\geq 4,
\end{gathered}
\end{equation}
and, if $|i-k|\leq 2$, then
\begin{equation}\label{besovpieces67}
\Delta_j(\Delta_kf\Delta_i g)=0 \quad j>k+4.
\end{equation}

For $s\in\mathbb{R}$ and $1\leq p,q\leq \infty$ we define
the space $\tilde{B}^s_{p,q}(\mathbb{R}^n)$ to be the set of 
distributions such that
\[
\|u\|_{\tilde{B}^s_{p,q}}=\Big(\sum_{j=0}^\infty (2^{js}\|\Delta_j
u\|_{L^p})^q\Big)^{1/q}<\infty,
\]
with the usual modification when $q=\infty$.  Finally, we define the Besov 
spaces $B^s_{p,q}(\mathbb{R}^n)$ by the norm
\[
\|f\|_{B^s_{p,q}}=\|\Psi*f\|_p+\|f\|_{\tilde{B}^s_{p,q}},
\]
for $s>0$.  For $s>0$, we define $B^{-s}_{p',q'}$ to be the dual
of the space $B^s_{p,q}$, where $p',q'$ are the Holder-conjugates to
$p,q$.

These Littlewood-Paley operators are also used to define Bony's paraproduct. 
 We have
\begin{equation}\label{lp start}
fg=\sum_{k} S_{k-3}f\Delta_k g + \sum_{k}S_{k-3}g\Delta_k f
+ \sum_{k}\Delta_k f\sum_{l=-2}^2 \Delta_{k+l} g.
\end{equation}

The estimates \eqref{besovlemma1} and \eqref{besovpieces67} imply that
\begin{equation} \label{bony256}
\begin{aligned}
\Delta_j (fg)&= \sum_{k=-3}^3 \Delta_j (S_{j+k-3}f\Delta_{j+k} g)
 + \sum_{k=-3}^3 \Delta_j (S_{j+k-3}g\Delta_{j+k} f)\\ 
&\quad +\sum_{k>j-4}\Delta_j \Big(\Delta_k f\sum_{l=-2}^2 \Delta_{k+l}g\Big).
\end{aligned}
\end{equation}

Now we turn our attention to establishing some basic Besov space estimates.  
First, we let $1\leq q_1\leq q_2\leq \infty$, $\beta_1\leq \beta_2$, 
$1\leq p_1\leq p_2\leq\infty$, $\alpha>0$, and set $\tilde{p}=np/(n-\alpha p)$
 with $\alpha<n/p$.  Then we have the following Besov embedding results:
\begin{subequations}\label{besov embedding}
\begin{gather}\label{besov embedding1} 
\|f\|_{B^{\beta_1}_{p,q_2}}\leq C\|f\|_{B^{\beta_2}_{p,q_1}},\\ 
\label{besov embedding3} 
\|f\|_{B^{\beta_1}_{\tilde{p},q_1}}\leq C\|f\|_{B^{\beta_1+\alpha}_{p_1,q_1}},\\ 
\label{besov embedding4} 
\|f\|_{H^{\beta_1,2}}=\|f\|_{B^{\beta_1}_{2,2}}.
\end{gather}
\end{subequations}

The following result is straightforward, but will be used often.
\begin{proposition}\label{sob embedding}Let $1\leq p<\infty$, $0<\alpha<n/p$
 and set $\tilde{p}=np/(n-\alpha p)$.  Then
\begin{equation}
\|f\|_{L^{\tilde{p}}}\leq C\|f\|_{B^{\alpha^+}_{p,q}},
\end{equation}
for any $1\leq q \leq \infty$.
\end{proposition}
For any $\varepsilon>0$, we have
\begin{equation}
\|f\|_{L^{\tilde{p}}}\leq \|f\|_{B^\varepsilon_{\tilde{p},q}}
\leq \|f\|_{B^{\alpha+\varepsilon}_{p,q}},
\end{equation}
where we used the definition of Besov spaces for the first inequality 
and \eqref{besov embedding3} for the second.

Next we record our two  Leibnitz-rule type estimate.  
The first is the standard estimate, which can be found in (among many other places) 
\cite[Lemma 2.2]{chae}.  See also \cite[Proposition 1.1]{TT}.

\begin{proposition}\label{product est 1}
Let $s>0$ and $q\in [1,\infty]$.  Then
\[
\|fg\|_{B^s_{p,q}}\leq C\big(\|f\|_{L^{p_1}}\|g\|_{B^s_{p_2,q}}
+\|f\|_{B^s_{q_1,q}}\|g\|_{L^{q_2}}\big),
\]
where $1/p=1/p_1+1/p_2=1/q_1+1/q_2$ and $p_i, q_i \in [1,\infty]$ for $i=1,2$.
\end{proposition}

Our second product estimate is less common.  
The estimate originated in \cite{chemin}; another proof can be found in
 \cite{besovpaper2}.

\begin{proposition}\label{product est 2} 
Let $f\in B^{s_1}_{p_1,q}(\mathbb{R}^n)$ and let 
$g\in B^{s_2}_{p_2,q}(\mathbb{R}^n)$.  Then, for any $p$ such that 
$1/p\leq 1/p_1+1/p_2$ and with $s=s_1+s_2-n(1/p_1+1/p_2-1/p)$,  we have
\[
\|fg\|_{B^s_{p,q}}\leq C\|f\|_{B^{s_1}_{p_1,q}}\|g\|_{B^{s_2}_{p_2,q}},
\]
provided $s_1<n/p_1$, $s_2<n/p_2$, and $s_1+s_2>0$.
\end{proposition}

\section{Local existence by Proposition \ref{product est 1}}\label{type 1}

Theorem \ref{old style short} and Theorem \ref{old style short log} are both 
proven using the standard product estimate (Proposition \ref{product est 1}).  
These theorems are both special cases of more general theorems, and the primary 
task of this section is to prove the theorem which implies 
Theorem \ref{old style short log}.  There is a similar result associated with 
Theorem \ref{old style short}, and it will be discussed at the end of the section.


\begin{theorem}\label{full version 2 log}
Let $\gamma_1>1$, $\gamma_2>0$, $q\geq 1$, and $p\geq 2$.  Assume $g_1$ and 
$g_2$ satisfy $|g_i(r)|\leq C(1+r)^\delta$ for any $\delta>0$ and 
$|g_i^{(k)}(r)|\leq Cr^{-k}$ for $1\leq k\leq n/2+1$.  
Let $u_0\in B^{s_1}_{p,q}(\mathbb{R}^n)$ be divergence-free.  
Then there exists a unique local solution $u$ to the generalized 
Leray-alpha equation \eqref{leray1}, with
\[
u\in BC([0,T):B^{s_1}_{p,q}(\mathbb{R}^n))\cap \dot{C}^T_{a;s_2,p,q},
\]
where $a=(s_2-s_1)/(\gamma_1-\varepsilon)$ for any sufficiently small 
$\varepsilon>0$ if there exists $k>0$ such that the parameters satisfy 
\eqref{full 2 list}.  $T$ can be chosen to be a non-increasing function of 
$\|u_0\|_{B^{s_1}_{p, q}}$ with $T=\infty$ if $\|u_0\|_{B^{s_1}_{p, q}}$ 
is sufficiently small.
\end{theorem}

We begin by re-writing equation \eqref{leray1} as
\begin{equation} \label{leray2}
\begin{gathered}
\partial_t u+P(1-\alpha^2\mathcal{L}_2)^{-1}\operatorname{div}(u\otimes (1-\alpha^2\mathcal{L}_2)u)-\nu\mathcal{L}_1 u=0,
\\ 
u(0,x)=u_0(x), \quad \operatorname{div}u_0=\operatorname{div}u=0,
\end{gathered}
\end{equation}
where $P$ is the Hodge projection onto divergence free vector fields and an 
application of the divergence free condition shows 
$\nabla_u (1-\alpha^2  \mathcal{L}_2) u=\operatorname{div}(u\otimes (1-\alpha^2\mathcal{L}_2)u)$, 
where $v\otimes w$ is the matrix with $ij$ entry equal to the product of the 
$i^{th}$ coordinate of $v$ and the $j^{th}$ coordinate of $w$.

Setting $\alpha=1$ and $\nu=1$ for notational simplicity and applying Duhamel's 
principle, we obtain that $u$ is a solution to the equation if and only if $u$
is a fixed point of the map $\Phi$ given by
\[
\Phi(u)=e^{t\mathcal{L}_1}u_0+\int_0^t e^{(t-s)\mathcal{L}_1}(W(u(s)))ds,
\]
where $W(u,v)=-P(1-\mathcal{L}_2)^{-1}\operatorname{div}(u(s)\otimes (1-\mathcal{L}_2)v(s))$.  
To simplify notation, we will also set $W(u,u)=W(u)$.  Our goal is to show that
 $\Phi$ is a contraction in the space
\begin{align*} 
X_{T,M}=\Big\{&f\in BC([0,T):B^{s_1}_{p,q}(\mathbb{R}^n))\cap \dot{C}_{a;s_2,p,q} 
 \text{and}
\\ &\sup_t \|f(t)-e^{t\mathcal{L}_1}u_0\|_{B^{s_1}_{p,q}}
 +\sup_t t^{a}\|u(t)\|_{B^{s_2}_{p,q}}<M\Big\},
\end{align*}
where $a=(s_2-s_1)/(\gamma_1^-)$, for $0<T\leq 1$ and $M>0$ to be chosen later.

Following the arguments outlined in \cite{KP} and \cite{sobpaper}, 
$\Phi$ will be a contraction if we can show that
\begin{equation}\label{main parts 1}
\begin{gathered} 
\sup_{t}t^{a}\|e^{t\mathcal{L}_1}u_0\|_{B^{s_2}_{p,q}}<M/3,\\ 
\sup_{t}\|\int_0^t e^{(t-s)\mathcal{L}_1} W(u(s))ds\|_{B^{s_1}_{p,q}}<M/3,\\ 
\sup_{t}t^a\|\int_0^t e^{(t-s)\mathcal{L}_1} W(u(s))ds\|_{B^{s_2}_{p,q}}<M/3,
\end{gathered}
\end{equation}
for $u\in X_{T,M}$.

For the first of these terms, we let $\varphi$ be in the Schwartz space.  
Then using Proposition \ref{lpsemi same} and 
Proposition \ref{heat kernel bound log} we have
\begin{align*}
&\sup_{t}t^{a}\|e^{t\mathcal{L}_1}(u_0-\varphi+\varphi)\|_{B^{s_2}_{p,q}}\\
&\leq \sup_{t} t^a\|e^{t\mathcal{L}_1}(u_0-\varphi)\|_{B^{s_2}_{p,q}}
 +\sup_t t^a \|e^{t\mathcal{L}_1}\varphi\|_{B^{s_2}_{p,q}}\\ 
&\leq \sup_{t} t^at^{-a}\|u_0-\varphi\|_{B^{s_1}_{p,q}}
 +\sup_t t^a \|\varphi\|_{B^{s_2}_{p,q}}\\
&\leq \|u_0-\varphi\|_{B^{s_1}_{p,q}}+T^a\|\varphi\|_{B^{s_2}_{p,q}}.
\end{align*}
Since the Schwartz space is dense in $B^{s_1}_{p,q}(\mathbb{R}^n)$, 
we can choose $\varphi$ so that the first term is arbitrarily small.  
Then we choose $T$ to be small enough so that the sum is bounded by $M/3$.

Turning to the second inequality, applying Minkowski's inequality and 
Proposition \ref{heat kernel bound log}, we have
\begin{equation}\label{23-1}
\begin{aligned} 
&\sup_{t}\|\int_0^t e^{(t-s){\mathcal{L}_1}} W(u(s))ds\|_{B^{s_1}_{p,q}}\\ 
&\leq \sup_{t}\int_0^t \|e^{(t-s){\mathcal{L}_1}} W(u(s))\|_{B^{s_1}_{p,q}}ds \\ 
&\leq \sup_{t}\int_0^t |t-s|^{-(s_1-r+n/p^*-n/p)/(\gamma_1^-)} 
 \|W(u(s))\|_{B^{r}_{p^*,q}}ds,
\end{aligned}
\end{equation}
where $p^*\leq p$ will be specified later.  Using Proposition \ref{annie2}, 
then Proposition \ref{product est 1}, and finally Propositions \ref{sob embedding}
and \ref{annie}, we have
\begin{equation}\label{2mal}
\begin{aligned} 
\|W(u(s))\|_{B^{r}_{p^*,q}}
&\leq C\|u\otimes (1-\mathcal{L}_2)u\|_{B^{r+1-\gamma_2^-}_{p^*,q}}\\ 
&\leq C\|u\|_{L^{p_1}}\|(1-\mathcal{L}_2)u\|_{B^{r+1-\gamma_2^-}_{p_2,q}}
 +C\|u\|_{B^{r+1-\gamma_2^-}_{q_1,q}}\|(1-\mathcal{L}_2)u\|_{L^{q_2}}\\ 
&\leq C\|u\|_{L^{p_1}}\|u\|_{B^{r+1+\gamma_2-\gamma_2^-}_{p_2,q}} 
 +C\|u\|_{B^{r+1-\gamma_2^-}_{q_1,q}}\|(1-\mathcal{L}_2)u\|_{B^{0^+}_{q_2,q}}\\ 
&\leq C\|u\|_{L^{p_1}}\|u\|_{B^{r+1+\gamma_2-\gamma_2^-}_{p_2,q}} 
 +C\|u\|_{B^{r+1-\gamma_2^-}_{q_1,q}}\|u\|_{B^{\gamma_2^+}_{q_2,q}},
\end{aligned}
\end{equation}
where $1/p^*=1/p_1+1/p_2=1/q_1+1/q_2$, provided $r+1-\gamma_2^->0$.  
To complete the argument, we need to bound this by $\|u\|_{B^{s_2}_{p,q}}^2$.  
To facilitate this, we define $\varepsilon$ by setting
 $\gamma_2-\gamma_2^-=\varepsilon$, choose $r+1+\varepsilon=s_2$ 
(which forces $s_2>\gamma_2$), and  $q_2=p_2=p$ (which forces $p_1=q_1$). 
 Applying these choices and using the Besov embedding \eqref{besov embedding1}, 
inequality \eqref{2mal} becomes
\begin{equation}\label{2mali}
\begin{aligned} 
\|W(u(s))\|_{B^{r}_{p^*,q}}
&\leq C\|u\|_{L^{p_1}}\|u\|_{B^{s_2}_{p_2,q}}
 +C\|u\|_{B^{s_2-\gamma_2}_{q_1,q}}\|u\|_{B^{\gamma_2^+}_{q_2,q}}\\ 
&\leq C\|u\|_{B^{s_2}_{p,q}}\big(\|u\|_{L^{p_1}}+\|u\|_{B^{s_2-\gamma_2}_{p_1,q}}\big).
\end{aligned}
\end{equation}
Finally, choosing $p_1=np/(n-kp)$ for some $k<n/p$ and $k\leq \gamma_2<s_2$, 
we use Proposition \ref{sob embedding} to obtain
\begin{equation}\label{lorde}
\begin{aligned} 
\|W(u(s))\|_{B^{r}_{p^*,q}}
&\leq C\|u\|_{B^{s_2}_{p,q}}
 \big(\|u\|_{L^{p_1}} +\|u\|_{B^{s_2-\gamma_2}_{p_1,q}}\big)\\ \
&leq C\|u\|_{B^{s_2}_{p,q}}\big(\|u\|_{B^{k^+}_{p,q}}
 +\|u\|_{B^{s_2-\gamma_2+k}_{p,q}} \big)
\leq C\|u\|_{B^{s_2}_{p,q}}^2,
\end{aligned}
\end{equation}
provided
\begin{equation}\label{product 1 term}
\begin{gathered} 1/p^*-1/p=1/p_1=(n-k p)/np,\\
 s_2>\gamma_2, \quad 
s_2=r+1+\varepsilon, \\ 
k\leq \gamma_2, \quad 
kp<n.
\end{gathered}
\end{equation}

Returning to the estimate begun in \eqref{23-1}, using \eqref{lorde}, 
we have
\begin{align*}
 &\sup_{t}\|\int_0^t e^{(t-s){\mathcal{L}_1}} W(u(s))ds\|_{B^{s_1}_{p,q}}\\ 
&\leq C\sup_t \int_0^t |t-s|^{-(s_1-r+n/p^*-n/p)/(\gamma_1^-)}
 s^{-2a}s^{2a}\|u(s)\|^2_{B^{s_2}_{p,q}}ds \\ 
&\leq C\sup_t \|u\|^2_{a;s_2,p,q}t^{-(s_1-r+n/p^*-n/p)
 /(\gamma_1^-)-2(s_2-s_1)/(\gamma_1^-) +1}\\ 
&\leq CM^2T^{-(s_1-r+n/p^*-n/p)/(\gamma_1^-)-2(s_2-s_1)/(\gamma_1^-) +1}<M/3,
\end{align*}
provided
\begin{equation}\label{2ty1}
\begin{gathered} 
0\leq (s_1-r+n/p^*-n/p)/(\gamma_1^-)<1\\ 
1>2(s_2-s_1)/(\gamma_1^-)\\
0\leq -(s_1-r+n/p^*-n/p)/(\gamma_1^-)-2(s_2-s_1)/(\gamma_1^-)+1.
\end{gathered}
\end{equation}
The first inequality in this list ensures that the $|t-s|$ term is integrable
 as $s$ goes to $t$, the second inequality does the same for the $s^{-2a}$ 
term as $s$ goes to $0$, and the last inequality makes the power on the 
post-integration $t$ positive.  The last inequality follows by recalling 
that $T\leq 1$ and by choosing a sufficiently small $M$.

For the last term in \eqref{main parts 1}, a similar argument gives
\begin{align*}
&\sup_{t}t^{a}\|\int_0^t e^{(t-s){\mathcal{L}_1}} W(u(s))ds\|_{B^{s_2}_{p,q}}\\ 
&\leq \sup_{t}t^{a}\int_0^t |t-s|^{-(s_2-r+n/p^*-n/p)/(\gamma_1^-)} 
 \|W(u(s))\|_{B^{r}_{p^*,q}}ds \\ 
&\leq \sup_{t}t^{a}\int_0^t |t-s|^{-(s_2-r+n/p^*-n/p)/(\gamma_1^-)}s^{-2a}
 s^{2a}\|u(s)\|^2_{B^{s_2}_{p,q}}ds \\ 
&\leq C\|u\|^2_{a;s_2, p, q}\sup_t t^{a}t^{-(s_2-r+n/p^*-n/p)
 /(\gamma_1^-)-2(s_2-s_1)/(\gamma_1^-)+1} \\ 
&\leq CM^2T^{-(s_2-r)/(\gamma_1^-)-(s_2-s_1)/(\gamma_1^-)+1}<M/3,
\end{align*}
provided
\begin{equation}\label{2ty2}
\begin{gathered}
 0\leq (s_2-r+n/p^*-n/p)/(\gamma_1^-)<1,\\ 
1>2(s_2-s_1)/(\gamma_1^-), \\ 
0\leq -(s_2-r+n/p^*-n/p)/(\gamma_1^-)-(s_2-s_1)/(\gamma_1^-)+1.
\end{gathered}
\end{equation}

Combining \eqref{2ty1}, and \eqref{2ty2} (and removing redundancies) gives
\begin{equation}\label{pac 1}
\begin{gathered} 
s_1>r, \\ 
(\gamma_1^-)/2>s_2-s_1,\\
0\leq s_2-r+n/p^*-n/p<(\gamma_1^-),\\ 
(\gamma_1^-)\geq 2s_2-r+n/p^*-n/p-s_1.
\end{gathered}
\end{equation}
Incorporating \eqref{product 1 term}, and observing that, since $s_2>s_1$, 
the last inequality in \eqref{pac 1} implies  the third inequality, we obtain
\begin{equation}\label{full 2 list}
\begin{gathered}
 s_2>\gamma_2\geq k,\\ 
kp <n, \\ 
s_2-s_1<\min\{(\gamma_1^-)/2, 1\},\\ 
\gamma_1^-\geq s_2-s_1+1+n/p+\varepsilon-k.
\end{gathered}
\end{equation}

This completes Theorem \ref{full version 2 log}.  
Replacing $\gamma_1^-$ with $\gamma_1$ and setting $\varepsilon=0$ recovers 
the result for the case where the $g_i$ are Mikhlin multipliers.  
In that case, note that for $\gamma_1=2$ and $\gamma_2=0$, this recovers, 
up to a slight modification in the argument, the result from \cite{KP}  
for the Navier-Stokes equation.

In comparison with the existence result for the next section, this existence 
result requires a larger initial regularity, but imposes no restrictions on 
the value of $\gamma_2$ (beyond the requirement that $\gamma_2>0$).   
To get Theorem \ref{old style short} or Theorem \ref{old style short log}, 
choose $k$ to be an arbitrarily small positive number, which removes $k$ 
from the last inequality and forces $\gamma_2>0$.

\section{Local existence using Proposition \ref{product est 2}}\label{type 2}

In this section we prove the following local existence result, 
which implies Theorem \ref{special case 1}.  
We address Theorem \ref{special case 1 log} at the end of the section.

\begin{theorem}\label{full generality}
Let $\gamma_1>1$, $\gamma_2>0$, $q\geq 1$, $p\geq 2$, and assume $g_1$ and 
$g_2$ satisfy the Mikhlin condition 
(see inequality \eqref{mikhlin condition don}).  
Let $u_0\in B^{s_1}_{p,q}(\mathbb{R}^n)$ be divergence-free.  
Then there exists a unique local solution $u$ to the generalized Leray-alpha 
equation \eqref{leray1}, with
\[
u\in BC([0,T):B^{s_1}_{p,q}(\mathbb{R}^n))\cap \dot{C}^T_{a;s_2,p,q},
\]
where $a=(s_2-s_1)/\gamma_1$, if there exists $r$, $r_1$ and $r_2$ such 
that all the parameters satisfy \eqref{final list 2}.  
$T$ can be chosen to be a non-increasing function of $\|u_0\|_{B^{s_1}_{p, q}}$
 with $T=\infty$ if $\|u_0\|_{B^{s_1}_{p, q}}$ is sufficiently small.
\end{theorem}

With the same set-up as the previous section, our goal is to show that
\begin{equation}\label{2main parts 1}
\begin{gathered} 
\sup_{t}t^{a}\|e^{t\mathcal{L}_1}u_0\|_{B^{s_2}_{p,q}}<M/3,\\ 
\sup_{t}\|\int_0^t e^{(t-s)\mathcal{L}_1} W(u(s))ds\|_{B^{s_1}_{p,q}}<M/3,\\ 
\sup_{t}t^a\|\int_0^t e^{(t-s)\mathcal{L}_1} W(u(s))ds\|_{B^{s_2}_{p,q}}<M/3.
\end{gathered}
\end{equation}
The first inequality follows exactly as it did in the previous section, 
and for the second, using Minkowski's inequality and Proposition 
\ref{heat kernel bound}, we have
\begin{equation}\label{3-1}
\begin{aligned} 
&\sup_{t}\|\int_0^t e^{(t-s){\mathcal{L}_1}} W(u(s))ds\|_{B^{s_1}_{p,q}}\\ 
&\leq \sup_{t}\int_0^t \|e^{(t-s){\mathcal{L}_1}} W(u(s))\|_{B^{s_1}_{p,q}}ds \\ 
&\leq \sup_{t}\int_0^t |t-s|^{-(s_1-r)/\gamma_1} \|W(u(s))\|_{B^{r}_{p,q}}ds,
\end{aligned}
\end{equation}
where $r\leq s_1$ and will be specified later.  
Using Proposition \ref{product est 2}, then Proposition \ref{annie2}, and 
finally Proposition \ref{annie}, we have
\begin{equation}\label{mal}
\begin{aligned} 
\|W(u(s))\|_{B^{r}_{p,q}}
&\leq \|u\otimes (1-\mathcal{L}_2)u\|_{B^{r+1-\gamma_2}_{p,q}}\\ 
&\leq \|u\|_{B^{r_1}_{p,q}}\|(1-\mathcal{L}_2)u\|_{B^{r_2}_{p,q}}\\
&\leq \|u\|_{B^{r_1}_{p,q}}\|u\|_{B^{r_2+\gamma_2}_{p,q}}
 \leq \|u\|^2_{B^{s_2}_{p,q}},
\end{aligned}
\end{equation}
provided
\begin{equation}\label{aldi}
\begin{gathered}
 r+1-\gamma_2\leq r_1+r_2-n/p,\\ 
r_1+r_2>0, \\ 
r_1, r_2<n/p, \\ 
s_2\geq \max\{r_1, r_2+\gamma_2\}.
\end{gathered}
\end{equation}


Returning to equation \eqref{3-1}, we have
\begin{equation}
\begin{aligned} 
 &\sup_{t}\|\int_0^t e^{(t-s){\mathcal{L}_1}} W(u(s))ds\|_{B^{s_1}_{p,q}}\\ 
&\leq \sup_t \int_0^t |t-s|^{-(s_1-r)/\gamma_1}s^{-2a}s^{2a}
 \|u(s)\|^2_{B^{s_2}_{p,q}}ds\\ 
&\leq C\sup_t \|u\|^2_{a;s_2,p,q}t^{-(s_1-r)/\gamma_1-2(s_2-s_1)/\gamma_1+1}\\
&\leq CM^2T^{-(s_1-r)/\gamma_1-2(s_2-s_1)/\gamma_1+1}<M/3,
\end{aligned}
\end{equation}
provided
\begin{equation}\label{ty1}
\begin{gathered} 
0\leq (s_1-r)/\gamma_1<1,\\ 
1>2(s_2-s_1)/\gamma_1,\\
0\leq -(s_1-r)/\gamma_1-2(s_2-s_1)/\gamma_1+1.
\end{gathered}
\end{equation}
Estimating the last term of \eqref{main parts 1} in a similar fashion, we have
\begin{equation}
\begin{aligned} 
&\sup_{t}t^{a}\|\int_0^t e^{(t-s){\mathcal{L}_1}} W(u(s))ds\|_{B^{s_2}_{p,q}}\\
&\leq \sup_{t}t^{a}\int_0^t |t-s|^{-(s_2-r)/\gamma_1} \|W(u(s))\|_{B^{r}_{p,q}}ds\\ 
&\leq \sup_{t}t^{a}\int_0^t |t-s|^{-(s_2-r)/
 \gamma_1}s^{-2a}s^{2a}\|u(s)\|^2_{B^{s_2}_{p,q}}ds\\ 
&\leq C\|u\|^2_{a;s_2,p,q}\sup_t t^{a}t^{-(s_2-r)/\gamma_1-2(s_2-s_1)/\gamma_1+1}\\
&\leq CM^2T^{-(s_2-r)/\gamma_1-(s_2-s_1)/\gamma_1+1}<M/3,
\end{aligned}
\end{equation}
provided
\begin{equation}\label{ty2}
\begin{gathered} 
0\leq (s_2-r)/\gamma_1<1,\\ 
1>2(s_2-s_1)/\gamma_1,\\ 
0\leq -(s_2-r)/\gamma_1-(s_2-s_1)/\gamma_1+1.
\end{gathered}
\end{equation}

Our final task is to unify the conditions on the parameters.  
The sets of inequalities from equations \eqref{ty1} and \eqref{ty2}
 can be simplified to
\begin{equation}
\begin{gathered} 
0<s_2-s_1<\gamma_1/2,\\ 
s_1\geq r >s_2-\gamma_1,\\ 
\gamma_1\geq (s_2-s_1)+(s_2-r).
\end{gathered}
\end{equation}
Incorporating the inequalities from \eqref{aldi}, we have
\begin{equation}\label{vague list}
\begin{gathered} 
0<s_2-s_1<\gamma_1/2,\\ 
s_1\geq r >s_2-\gamma_1,\\ 
\gamma_1\geq (s_2-s_1)+(s_2-r),\\ 
r+1-\gamma_2\leq r_1+r_2-n/p,\\ 
r_1+r_2>0,\\ 
r_1, r_2<n/p, \\ 
s_2\geq \max \{r_1,r_2+\gamma_2\},
\end{gathered}
\end{equation}
and this completes the proof of Theorem \ref{full generality}.  
To obtain the results in Theorem \ref{special case 1}, 
we fix the values of the parameters $r_1, r_2,$ and $r$ in the following way. 
 First, since our primary interest is in minimizing $s_1$ and $s_2$, we see 
from the last inequality that this is helped by minimizing 
$\max\{r_1, r_2+\gamma_2\}$, subject to the constraints $r_1+r_2>0$ and
 $r_1, r_2<n/p$.  This is accomplished by choosing $r_2=-\gamma_2/2$ and 
$r_1=\gamma_2/2+R$, where $R$ is some positive number.  
Choosing the fourth inequality in the list \eqref{vague list} to be an equality, 
the list \eqref{vague list} becomes
\begin{equation}\label{vague list 2}
\begin{gathered} 
0<s_2-s_1<\gamma_1/2,\\ 
s_1\geq r >s_2-\gamma_1,\\ 
\gamma_1\geq (s_2-s_1)+(s_2-r),\\
 r=-1+\gamma_2+R-n/p, \\ 
n/p>\gamma_2/2+R, \\ 
s_2\geq \gamma_2/2+R.
\end{gathered}
\end{equation}
Using the fourth line to eliminate $r$ from the other inequalities, 
and then removing extraneous inequalities, we finally get
\begin{equation}\label{final list 2}
\begin{gathered} 
0<s_2-s_1<\gamma_1/2,\\ 
s_1\geq \gamma_2+R-n/p-1,\\ 
\gamma_1\geq 2s_2-s_1-\gamma_2-R+n/p+1, \\
 n/p>\gamma_2/2+R, \\
 s_2\geq \gamma_2/2+R.
\end{gathered}
\end{equation}
Eliminating the free parameter $R$ weakens this to
\begin{equation}\label{final list 3}
\begin{gathered} 
0<s_2-s_1<\gamma_1/2,\\ 
s_1>\gamma_2-n/p-1,\\ 
\gamma_1\geq 2s_2-s_1-\gamma_2+n/p+1,\\ 
n/p>\gamma_2/2, \\
s_2\geq \gamma_2/2,
\end{gathered}
\end{equation}
which finishes Theorem \ref{special case 1}.  
The analog of Theorem \ref{special case 1} for logarithmic $g$ 
is obtained by replacing $\gamma_1$ and $\gamma_2$ with $\gamma_1^-$ 
and $\gamma_2^-$.

\section{Operator estimates for $\mathcal{L}^g_\gamma$}\label{L operator estimates}

In this section, we define the Fourier multiplier $\mathcal{L}^g_\gamma$ by
\[
\mathcal{L}^g_\gamma u(x)=\int -\frac{|\xi|^{\gamma}}{g(|\xi|)}
\hat{u}(\xi)e^{ix\cdot \xi}d\xi,
\]
where $\gamma\in \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ is radial,
nondecreasing, and bounded below by 1.  Note that if we define $G$ to be
 the Fourier multiplier with symbol $1/g$, then
 $L^g_\gamma=-G(-\Delta)^{\gamma/2}$. The goal here is to prove operator
estimates for $\mathcal{L}^g_\gamma$, and we begin by stating the Mikhlin
 multiplier theorem, which will be referenced often in this section.

\begin{theorem}[Mikhlin multiplier theorem] 
Let $M$ be an operator with symbol $m:\mathbb{R}^n\to \mathbb{R}^n$.  
If $|x|^k|\nabla^k m(x)|$ is bounded for all $0\leq k\leq n/2+1$, then 
$M$ is an $L^p(\mathbb{R}^n)$ multiplier for all $1<p<\infty$.
\end{theorem}

The multipliers we are working with will be radial.  
In this context, the Mikhlin conditions is
\begin{equation}\label{mikhlin condition don}|m^{(k)}(r)|\leq Cr^{-k},
\end{equation}
for $0\leq k\leq n/2+1$.  Now we are ready to prove our fist result.

\begin{proposition}\label{annie} 
Let $1<p<\infty$ and let $|g^{(k)}(r)|\leq Cr^{-k}$ for $1\leq k\leq n/2+1$.  
Then $\mathcal{L}^g_\gamma: B^{s_1+\gamma}_{p,q}(\mathbb{R}^n)
\to B^{s_1}_{p,q}(\mathbb{R}^n)$, with
\[
\|\mathcal{L}^g_\gamma f\|_{B^{s_1}_{p,q}}\leq C\|f\|_{B^{s_1+\gamma}_{p,q}}.
\]
\end{proposition}

\begin{proof}
Without loss of generality, we assume $s_1=0$. Then we have 
\begin{align*}
 \|(1-\mathcal{L}^g_\gamma)^{-1} f\|_{B^{0}_{p,q}}
&= \|(1-\mathcal{L}^g_\gamma)^{-1} (1-\Delta)^{\gamma/2}(1-\Delta)^{-\gamma/2} f 
  \|_{B^{0}_{p,q}} \\ 
&=\|(1-\Delta)^{\gamma/2}(1-\mathcal{L}^g_\gamma)^{-1} f \|_{B^{-\gamma}_{p,q}}.
\end{align*}
We finish the proof by showing that the operator 
$(1-\Delta)^{\gamma}(1-\mathcal{L}^g_\gamma)^{-1}$, 
with symbol $\frac{(1+r^2)^{\gamma/2}}{(1+r^\gamma/g(r))}$, 
is a Mikhlin multiplier.  Note that the symbol can be written 
as $\frac{g(r)(1+r^2)^{\gamma/2}}{(g(r)+r^\gamma)}$, and since $g$ 
is already known to be a Mikhlin multiplier, this is equivalent 
to showing that the symbol $\frac{(1+r^2)^{\gamma/2}}{(g(r)+r^\gamma)}$ 
satisfies the Mikhlin condition, which follows from a straightforward 
(though lengthy) computation.
\end{proof}

If $g$ is a logarithm, then $g$ is not a Mikhlin multiplier, since 
$g$ is not bounded.  However, $g$ would satisfy $g(r)\leq C(1+r)^\delta$ 
for any $\delta>0$ and $|g^{(k)}(r)|\leq Cr^{-k}$ for all $1\leq k$.  
These observations inform the next proposition.

\begin{proposition}\label{annie2}
Let $1<p<\infty$, let $g(r)\leq C(1+r)^\delta$ for any $\delta>0$, 
and assume $|g^{(k)}(r)|\leq Cr^{-k}$ for all $1\leq k\leq n/2+1$.  
Then $(1-\mathcal{L}^g_\gamma)^{-1}: B^{s_1-(\gamma-\varepsilon)}_{p,q}
(\mathbb{R}^n)\rightarrow B^{s_1}_{p,q}(\mathbb{R}^n)$ for any 
small $\varepsilon>0$, with 
\[
\|(1-\mathcal{L}^g_\gamma)^{-1} f\|_{B^{s_1}_{p,q}}
\leq C\|f\|_{B^{s_1-(\gamma-\varepsilon)}_{p,q}}.
\]
\end{proposition}
\begin{proof}

As in the previous proposition, this follows by showing that the symbol 
$\frac{(1+r^2)^{(\gamma-\varepsilon)/2}}{1+r^\gamma/g(r)}$ is a Mikhlin 
multiplier.  First, we re-write this as 
$\frac{g(r)}{(1+r^2)^{\varepsilon/2}} \frac{(1+r^2)^{\gamma/2}}{g(r)+r^\gamma}$.  
That each of these terms individually satisfies the Mikhlin condition follows 
directly from the assumptions on $g$.  
\end{proof}

We remark that the $\varepsilon$ loss between these two results is due to 
the necessity of controlling the growth of the $g(r)$ term.

\section{Operator estimates for $e^{t\mathcal{L}}$}\label{operator est}

As in the previous section, we define the Fourier multiplier 
$\mathcal{L}^g_\gamma$ by
\[
\mathcal{L}^g_\gamma u(x)
=\int -\frac{|\xi|^{\gamma}}{g(|\xi|)}\hat{u}(\xi)e^{ix\cdot \xi}d\xi,
\]
where $\gamma\in \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ is 
nondecreasing and bounded below by 1. We define the operator 
$e^{t\mathcal{L}^g_\gamma}$ to be the Fourier multiplier with symbol 
$e^{-t|\xi|^\gamma/g(|\xi|)}$.  The goal of this section is to establish 
operator bounds for $e^{t\mathcal{L}^g_\gamma}$ in the case where 
$\gamma>1$, and following the general outline of the same task for 
$e^{t\Delta}$, we need to first establish $L^p-L^q$ boundedness 
for the operator.  Also note that, throughout the section, we will 
assume $0<t<1$.

We start with the special case of $p=q$.
\begin{proposition}\label{lpsemi same}
Let $1< p< \infty$ and $\gamma>1$.  Assume 
$|g(r)|\leq C(1+r)^\delta$ for any $\delta>0$  and assume
\begin{equation}\label{mik condition 65} 
|g^{(k)}(r)|\leq Cr^{-k}
\end{equation}
holds for $1\leq k\leq n/2+1$.  Then
\[
e^{t\mathcal{L}^g_\gamma}:L^p(\mathbb{R}^n)\to L^p(\mathbb{R}^n),
\]
and we have the bound
\begin{equation}\label{mal2}
\|e^{t\mathcal{L}^g_\gamma}f\|_{L^p}\leq C\|f\|_{L^p}.
\end{equation}
\end{proposition}

\begin{proof}
We will show that $e^{-r^\gamma/g(r)}$ satisfies the Mikhlin condition, 
and then the result follows by the Mikhlin multiplier theorem.  
First, we observe that the multiplier is clearly bounded.  Then
\[
\big|\frac{d}{dr}e^{-r^\gamma/g(r)} \big|
\leq C\Big(\frac{r^{\gamma-1}}{g(r)}+\frac{r^\gamma g'(r)}{g(r)^2}\Big) 
e^{-r^\gamma/g(r)}\leq Cr^{\gamma-1}e^{-Cr^{\gamma-\delta}}\leq Cr^{-1},
\]
where in the last inequality we used that $\gamma>1$ and we have chosen 
$\delta$ to be a small positive number.  Similar calculations hold for 
the remaining derivatives.
\end{proof}

Now we consider the case where $p\neq q$.

\begin{proposition}\label{lpsemi} 
Let $1\leq p< q\leq \infty$, and assume $\gamma>1$.  Then
\[
e^{t\mathcal{L}^g_\gamma}:L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n),
\]
and we have the bound
\begin{equation}\label{mal2b}
\|e^{t\mathcal{L}^g_\gamma}f\|_{L^q}\leq Ct^{-(n/p-n/q)/\gamma}\|f\|_{L^p},
\end{equation}
provided
\begin{equation}\label{mal condition}
\sup_{t\in (0, 1)}  \int_{0}^\infty r^{n-1} 
| \partial_r^{(n)}(e^{-r^\gamma/g(rt^{-1/\gamma})}) | dr<\infty.
\end{equation}
\end{proposition}

\begin{proof}
For notational convenience, we will suppress the subscript $\gamma$ and the 
superscript $g$ on the operator for the duration of the proof.  
Setting $e^{t\mathcal{L}}f=e^{t\mathcal{L}}\delta\ast f$, where the Fourier 
Transform of $e^{t\mathcal{L}}\delta(x)$ is equal to $e^{-t|\xi|^\gamma/g(|x|)}$, 
and applying  Young's inequality, we obtain that
\[
\|e^{t\mathcal{L}}f\|_{L^q}\leq \|e^{t\mathcal{L}}\delta\|_{L^r}\|f\|_{L^p},
\]
where $1+1/q=1/r+1/p$.  Formally, we have that
\[
e^{t\mathcal{L}}\delta(\xi)
=C\int_{\mathbb{R}^n}e^{-t|x|^\gamma/g(|x|)}e^{ix\cdot\xi}dx.
\]
Making the variable change $x\to t^{-1/\gamma}x$, we obtain
\[
e^{t\mathcal{L}}\delta(\xi)=Ct^{-n/\gamma}\int_{\mathbb{R}^n}
e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)} e^{it^{-1/\gamma}x\cdot\xi}dx.
\]
Taking the $L^r(\mathbb{R}^n)$ norm gives
\[
\|e^{t\mathcal{L}}\delta\|_{L^r}=Ct^{-n/\gamma}\Big(\int_{\mathbb{R}^n}
\Big| \int_{\mathbb{R}^n}e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)} 
e^{it^{-1/\gamma}x\cdot\xi}dx \Big|^rd\xi\Big)^{1/r}.
\]
Making the variable change $\xi\to t^{-1/\gamma}\xi$, this finally becomes
\[
\|e^{t\mathcal{L}}\delta\|_{L^r}=Ct^{-n/\gamma+n/(r\gamma)}
\Big(\int_{\mathbb{R}^n}\Big|\int_{\mathbb{R}^n}e^{-|x|^\gamma/
g(t^{-1/\gamma}|x|)} e^{ix\cdot\xi}dx \Big|^rd\xi\Big)^{1/r}.
\]

Since $1-1/r=1/p-1/q$, it only remains to obtain a $t$-independent bound for 
the integral.  First, we have
\begin{align*} 
&\int_{\mathbb{R}^n}\Big| \int_{\mathbb{R}^n}e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)}
 e^{ix\cdot\xi}dx \Big|^rd\xi \\ 
&\leq \int_{|\xi|<1}\Big|\int_{\mathbb{R}^n}e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)} 
 e^{ix\cdot\xi}dx \Big|^rd\xi \\
&\quad  + \int_{|\xi|>1}\Big|\int_{\mathbb{R}^n}e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)} 
 e^{ix\cdot\xi}dx \Big|^rd\xi \\ 
&\leq C\int_{|\xi|>1}\Big|\int_{\mathbb{R}^n}e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)} 
 dx \Big|^r \\
&\quad +\int_{|\xi|>1}\Big|\int_{\mathbb{R}^n}e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)} 
 \frac{\partial_{x_1}\partial_{x_2}\dots \partial_{x_n}}{\xi_1\xi_2\dots 
 \xi_n}e^{ix\cdot\xi}dx \Big|^rd\xi \\ 
&\leq C+\int_{|\xi|>1}\frac{1}{|\xi_1|^r|\xi_2|^r\dots |\xi_n|^r}
 \Big|\int_{\mathbb{R}^n}\partial_{x_1}\partial_{x_2}\dots 
 \partial_{x_n}(e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)}) e^{ix\cdot\xi}dx \Big|^rd\xi\\ 
&\leq C+C\Big(\int_{\mathbb{R}^n} \Big| \partial_{x_1}\partial_{x_2}
 \dots \partial_{x_n}(e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)}) \Big|  dx \Big)^r,
\end{align*}
where the last inequality follows from observing that since $p<q$, $r>1$, 
and therefore the integral in each $\xi$ coordinate is finite.

To compute the final integral, we convert to polar coordinates, and remark 
that since $e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)}$ is radial, we can 
disregard the radial portion of each coordinate derivative's expression 
in polar form, and we have
\[
\int_{\mathbb{R}^n}  \Big| \partial_{x_1}\partial_{x_2}
\dots \partial_{x_n}(e^{-|x|^\gamma/g(t^{-1/\gamma}|x|)})  
\Big|  dx
\leq C\int_{0}^\infty r^{n-1}\Big| \partial_r^{(n)}
 (e^{-r^\gamma/g(rt^{-1/\gamma})}) \Big|  dr,
\]
which is \eqref{mal condition}, and thus completes the proposition.
\end{proof}

Our next task is to establish sufficiency criteria for \eqref{mal condition}.

\begin{proposition}\label{rip city 5}
If
\begin{equation}\label{rip city}|g^{(k)}(r)|\leq Cr^{-k},
\end{equation}
for all $0\leq k\leq n$, then \eqref{mal condition} holds.
\end{proposition}

\begin{proof}
We begin with the special case where $g$ is a constant function, and without 
loss of generality assume $g(r)=1$.  Then \eqref{mal condition} becomes
\[
\int_{0}^\infty r^{n-1}\big| \partial_r^{(n)}(e^{-r^\gamma})\big|\,  dr.
\]
Computing the derivatives through repeated use of the product rule will yield 
the addition of $2^{n-1}$ terms of the form $r^\alpha e^{-r^\gamma}$.  
The decay provided by the exponential term makes each of these integrable 
for large $r$.  For the region where $r$ is small, the most singular term is 
of the form $r^{n-1}r^{n(\gamma-n)}e^{-r^\gamma}$, which is integrable 
for small $r$ because $\gamma>1$, and this finishes the argument for the 
special case where $g$ is constant.

If $\gamma>1$ and $g$ is not constant, the proof is similar, 
but more combinatorially intense.  To see this, we first consider the 
case where $n=1$.  Then we have
\begin{align*}
&\int_{0}^\infty \big| \partial_r(e^{-r^\gamma/g(rt^{-1/\gamma})})\big| \,  dr\\
&=\int_0^\infty \Big(\frac{Cr^{\gamma-1}}{g(rt^{-1/\gamma})}
+C\frac{r^\gamma g'(rt^{-1/\gamma})t^{-1/\gamma}}{(g(rt^{-1/\gamma}))^2}\Big) 
e^{-r^\gamma/g(rt^{-1/\gamma})} dr.
\end{align*}
Using our assumption on $g$, we have
\begin{align*} 
&\int_0^\infty \Big(\frac{Cr^{\gamma-1}}{g(rt^{-1/\gamma})}
+C\frac{r^\gamma g'(rt^{-1/\gamma})t^{-1/\gamma}}{(g(rt^{-1/\gamma}))^2}\Big) 
e^{-r^\gamma/g(rt^{-1/\gamma})} dr.\\ 
&\leq C\int_0^\infty \Big(\frac{Cr^{\gamma-1}}{g(rt^{-1/\gamma})}
+C\frac{r^\gamma t^{-1/\gamma}}{rt^{-1/\gamma} (g(rt^{-1/\gamma}))^2}\Big) 
e^{-r^\gamma/g(rt^{-1/\gamma})} dr.
\end{align*}
Finally, since $g$ is bounded below by one and is bounded above by assumption, 
we have
\[
\int_0^\infty \Big(\frac{Cr^{\gamma-1}}{g(rt^{-1/\gamma})}
+C\frac{r^\gamma t^{-1/\gamma}}{rt^{-1/\gamma} 
(g(rt^{-1/\gamma}))^2}\Big) e^{-r^\gamma/g(rt^{-1/\gamma})} dr
\leq C\int_0^\infty r^{\gamma-1}e^{-Cr^\gamma}dr,
\]
which reduces the problem to the constant $g$ case. 
 In general, when computing the derivatives, either the derivatives 
fall on the $r^\gamma$ term (and then since $g$ is bounded below, 
we are back in constant $g$ case) or the derivatives act on 
the $g(rt^{-1/\gamma})$ term.  In that case, because of \eqref{rip city}, 
the derivatives of $g$ introduce exactly as much decay in $r$ as 
occurs when differentiating $r^\gamma$, and the $t$ dependent terms cancel out,
 so this case also reduces to a term from the constant $g$ case.
\end{proof}

Recall from the introduction that we ultimately seek results for $g$ 
of the form $g(r)=\ln(a+r)$, where $\ln(a)>1$.  Since such a $g$ is unbounded, 
it does not satisfy the requirements of the previous proposition.  
The following result adapts that argument to this particular case.

\begin{proposition} 
Let $g(r)\leq C(1+r)^\delta$ for any $\delta>0$, and assume that
\[
|g^{(k)}(r)|\leq C|r|^{-k},
\]
for $1\leq k\leq n$.  Then
\begin{equation}\label{rip city 3}
\|e^{t\mathcal{L}^g_\gamma}f\|_{L^q}
\leq Ct^{-(n/p-n/q)/(\gamma-\varepsilon)}\|f\|_{L^p},
\end{equation}
for any small $\varepsilon>0$, provided $0<t<1$.
\end{proposition}

Before beginning the proof, we note that if $g(r)=\ln(a+r)$, and 
$\ln(a)\geq 1$, then $g$ satisfies the hypothesis of the proposition.

\begin{proof} First, we observe that when controlling the derivatives, 
we only required that $g$ be bounded below and that the derivatives of 
$g$ have sufficient decay.  Since that portion of the argument did not 
require $g$ to be bounded, we can apply that argument here, and we have that
\begin{equation}\label{rip city 11}
\|e^{t\mathcal{L}^g_\gamma}f\|_{L^q}
\leq Ct^{-(n/p-n/q)/\gamma}\|f\|_{L^p}
\int_{0}^\infty r^\alpha e^{-r^\gamma/g(rt^{-1/\gamma})}   dr,
\end{equation}
for some $\alpha>-1$.
Since $\alpha>-1$, we have
\begin{align*} 
\int_0^\infty r^{\alpha}e^{-r^\gamma/g(rt^{-1/\gamma})} dr
&\leq \int_0^1 r^{\alpha}e^{-r^\gamma/g(rt^{-1/\gamma})} dr
 +\int_1^\infty r^{\alpha}e^{-r^\gamma/g(rt^{-1/\gamma})} dr\\ 
&\leq C+\int_1^\infty r^{\alpha}e^{-r^\gamma/(1+rt^{-1/\gamma})^\delta} dr\\
&\leq C+\int_1^\infty r^{\alpha}e^{-Cr^{\gamma-\delta}t^{\delta/\gamma}} dr,
\end{align*}
where we used the assumption on $g$ and $\delta$ is small positive number 
to be specified later.  Making the variable change 
$r\to rt^{\delta/(\gamma(\gamma-\delta))}$, we finally obtain
\[
\int_1^\infty r^{\alpha}e^{-Cr^\gamma/g(rt^{-1/\gamma})} dr
\leq C+t^{-\delta\alpha/\gamma(\gamma-\delta)}
 \int_1^\infty r^\alpha e^{-Cr^{\gamma-\delta}} dr
\leq C(1+t^{-\delta\alpha/\gamma(\gamma-\delta)}),
\]
provided $\gamma-\delta>1$.  Plugging back into \eqref{rip city 11}, we have
\[
\|e^{t\mathcal{L}^g_\gamma}f\|_{L^q}
\leq Ct^{-(n/p-n/q)/\gamma-\delta\alpha/\gamma(\gamma-\delta)}\|f\|_{L^p}.
\]
Choosing a sufficiently small $\delta$ finishes the proposition.
\end{proof}

Before moving on, we remark that, as in the previous section, adapting 
to the logarithmic $g$ comes at a cost of additional singularity in the 
time variable.

Now that we have established $L^p-L^q$ boundedness for $e^{t\mathcal{L}^g_\gamma}$, 
we turn our attention to incorporating changes in regularity.  
We will rely heavily on the following, which is 
\cite[Proposition 7.2 Chapter 13]{T3}.

\begin{proposition}\label{taylor prop}
Let $e^{tA}$ be a holomorphic semigroup on a Banach space $X$.  Then, for $t>0$,
\[
\|Ae^{tA}f\|_X\leq \frac{C}{t}\|f\|_X,
\]
for $0<t\leq 1$.
\end{proposition}

For our purposes, $A=\mathcal{L}^g_\gamma$ and $X=L^p(\mathbb{R}^n)$.  
To use this proposition, we need to know that $e^{t\mathcal{L}^g_\gamma}$ 
is a holomorphic semigroup, and following the proof of
\cite[Proposition 7.1 Chapter 13]{T3},
 we see that we only need $e^{t\mathcal{L}^g_\gamma}$ to be uniformly bounded 
from $L^p(\mathbb{R}^n)$ into itself, which is the content of Proposition 
\ref{lpsemi same}.  Now we are ready to prove the following result.

\begin{proposition}\label{sobolev}
 Let $1<p<\infty$, $s_1\leq s_2$ and assume $g$ satisfies the Mikhlin 
condition (see inequality \eqref{mikhlin condition don}).   
Then $e^{t\mathcal{L}^g_\gamma}: B^{s_1}_{p,q}(\mathbb{R}^n)
\to B^{s_2}_{p,q}(\mathbb{R}^n)$ and
\begin{equation}
\|e^{t\mathcal{L}^g_\gamma} f\|_{B^{s_2}_{p,q}}
\leq t^{-(s_2-s_1)/\gamma}\|f\|_{B^{s_1}_{p,q}}.
\end{equation}
\end{proposition}

\begin{proof}
We first establish this result in the case $s_2=\gamma$ and $s_1=0$.  We have
\begin{align*} 
\|e^{t\mathcal{L}^g_\gamma} f\|_{B^{\gamma}_{ p,q}}
& =\|e^{t\mathcal{L}^g_\gamma}(\Psi\ast f)\|_{L^p}
+\Big(\sum_{j=0}^{\infty} 2^{jq\gamma} \|e^{t\mathcal{L}^g_\gamma}
 \Delta_j f\|_{L^p}^q\Big)^{1/q} \\ 
&\leq C\|\Psi\ast f\|_{L^p}+\Big(\sum_{j=0}^{\infty}
 \|(-\Delta)^{\gamma/2}\big(\mathcal{L}^g_\gamma\big)^{-1} 
 \mathcal{L}^g_\gamma e^{t\mathcal{L}^g_\gamma} \Delta_j f\|_{L^p}^q\Big)^{1/q}\\ 
&\leq C\|\Psi\ast f\|_{L^p}+Ct^{-1}
\Big(\sum_{j=0}^{\infty} \| \mathcal{L}^g_\gamma e^{t\mathcal{L}^g_\gamma} 
 \Delta_j f \|_{L^p}^q\Big)^{1/q}\\ 
&\leq Ct^{-1}\|f\|_{B^0_{p,q}},
\end{align*}
where we used (essentially) Proposition \ref{annie2} in the first inequality, 
Proposition \ref{taylor prop} in the second, and the fact that $t\leq 1$ 
for the last inequality.  Standard interpolation and duality arguments extend 
this result to the general case of $s_1\leq s_2$.
\end{proof}

We again state the parallel result for the special case where $g$ is, essentially, 
a logarithm.
\begin{proposition}\label{sobolev log}
Let $1<p<\infty$, $s_1\leq s_2$, let $g(r)\leq Cr^{\varepsilon}$ for any 
$\varepsilon>0$ and let $|g^{(k)}(r)|\leq C|r|^{-k}$ for all $1\leq k\leq n/2+1$. 
 Then $e^{t\mathcal{L}^g_\gamma}: B^{s_1}_{p,q}(\mathbb{R}^n)\to 
B^{s_2}_{p,q}(\mathbb{R}^n)$ and
\begin{equation}
\|e^{t\mathcal{L}^g_\gamma} f\|_{B^{s_2}_{p,q}}
\leq t^{-(s_2-s_1)/(\gamma-\varepsilon)}\|f\|_{B^{s_1}_{p,q}},
\end{equation}
for any $\varepsilon>0$.
\end{proposition}

\begin{proof}
As in the previous proposition, we let $s_2=\gamma-\varepsilon$ and set 
$s_1=0$, and the rest of the argument will follow as before provided we show 
that the Fourier multiplier with symbol
\[
m(\xi)=\frac{|\xi|^{\gamma-\varepsilon}g(\xi)}{|\xi|^\gamma},
\]
and with support in the annulus $|\xi|\geq 1/2$, is bounded on 
$L^p(\mathbb{R}^n)$.  As this follows from the assumptions on $g$, 
the proof is complete.
\end{proof}

The following is direct combination of Proposition \ref{rip city 5} and 
Proposition \ref{sobolev}.

\begin{proposition}\label{heat kernel bound} 
Let $1<p<\infty$, $s_1\leq s_2$, $p_1\leq p_2$ and let $g$ satisfy the
 Mikhlin condition.  Then 
$e^{t\mathcal{L}^g_\gamma}: B^{s_1}_{p_1,q}(\mathbb{R}^n)\to 
B^{s_2}_{p_2,q}(\mathbb{R}^n)$ and
\begin{equation}
\|e^{t\mathcal{L}^g_\gamma} f\|_{B^{s_2}_{p_2,q}} 
\leq t^{-(s_2-s_1+n/p_1-n/p_2)/\gamma}\|f\|_{B^{s_1}_{p_1,q}}.
\end{equation}
\end{proposition}

We also record the analogous result for our special case.
\begin{proposition}\label{heat kernel bound log}
Let $1<p<\infty$, $s_1\leq s_2$, $p_1\leq p_2$, $g(r)\leq Cr^{\varepsilon}$ 
for any $\varepsilon>0$, and let $|g^{(k)}(r)|\leq C|r|^{-k}$ for all 
$1\leq k\leq n/2+1$. 
  Then $e^{t\mathcal{L}^g_\gamma}: B^{s_1}_{p_1,q}(\mathbb{R}^n)\to
 B^{s_2}_{p_2,q}(\mathbb{R}^n)$ and
\begin{equation}
\|e^{t\mathcal{L}^g_\gamma} f\|_{B^{s_2}_{p_2,q}}
\leq t^{-(s_2-s_1+n/p_1-n/p_2)/(\gamma-\varepsilon)}\|f\|_{B^{s_1}_{p_1,q}},
\end{equation}
for any small $\varepsilon>0$.
\end{proposition}

We remark that these results also apply to Sobolev spaces 
(see \cite[Section 2]{besovpaper2} for an example of a similar process 
applied to the standard heat kernel).

\section{Higher regularity for the local existence result}
\label{Higher regularity for the local existence result}

As was mentioned in the introduction, the solutions to the generalized
 Leray-alpha equations constructed here are smooth for all $t>0$ at which 
the solution exists.  In this section we prove that the solutions to 
Theorem \ref{old style short} have this additional regularity and quantify 
the blow-up that occurs in these higher regularity norms as $t\to 0$.  
We use an induction argument inspired by the results in \cite{katoinduction} 
for the Navier-Stokes equation.  We remark that similar results can be proven 
for the other theorems in this paper, but require different (and in some cases
 much more involved) arguments.

\begin{proposition}\label{higher regularity theorem}
Let $u_0\in B^{s_1}_{p,q}(\mathbb{R}^n)$ be divergence-free.   
Let $u$ be a solution to the generalized Leray-alpha equation \eqref{leray1}  
given by Theorem \ref{old style short}.
Then for all $r\geq s_1$ we have that $u\in \dot{C}^T_{(r-s_1)/\gamma_1;r,p,q}$.
\end{proposition}

Before starting the proof, recall from Theorem \ref{old style short} that 
$s_1>0$ and that
\begin{equation}\label{durant}
\begin{gathered} 
\gamma_1>1, \quad  \gamma_2>0, \quad s_2>\gamma_2,\\
s_2-s_1<\min\{\gamma_1/2, 1\},\\
\gamma_1\geq s_2-s_1+n/p+1.
\end{gathered}
\end{equation}

\begin{proof}
We start with the solution $u$ given by Theorem \ref{old style short}.  
Then let $\delta>0$ be arbitrary and let $v=t^\delta u$.  We note that 
$v(0)=v_0=0$.  Then
\begin{align*} 
\partial_t v
&=\delta t^{\delta-1} u+t^\delta \partial_t u \\ 
&=\delta t^{-1} v+t^\delta P(\mathcal{L}_1 u-(1-\mathcal{L}_2)^{-1}\operatorname{div}(u\otimes (1-\mathcal{L}_2)u)) \\ 
&=\delta t^{-1} v+ \mathcal{L}_1 v-t^{-\delta}P(1-\mathcal{L}_2)^{-1}\operatorname{div}(v\otimes (1-\mathcal{L}_2)v)).
\end{align*}
Applying Duhamel's principle, we obtain
\begin{align*} 
v&=e^{t\mathcal{L}_1}v_0+\delta\int_0^t e^{(t-s)\mathcal{L}_1}s^{-1}v(s)ds
 +\int_0^t e^{(t-s)\mathcal{L}_1}s^{-\delta}W(v(s))ds\\ 
&=\delta\int_0^t e^{(t-s)\mathcal{L}_1}s^{-1}v(s)ds
 +\int_0^t e^{(t-s)\mathcal{L}_1}s^{-\delta}W(v(s))ds,
\end{align*}
where we recall 
$W(f,g)=-P(1-\mathcal{L}_2)^{-1}\operatorname{div}(f(s)\otimes (1-\mathcal{L}_2)g(s))$ 
(and for notational convenience set $W(f,f)=W(f)$) and in the last line 
used that $v_0=0$.  Using $v=t^\delta u$, we obtain
\[
u=\delta t^{-\delta}\int_0^t e^{(t-s)\mathcal{L}_1}s^{\delta-1}u(s)ds
 +t^{-\delta}\int_0^t e^{(t-s)\mathcal{L}_1}s^\delta W(u(s))ds.
\]
The key idea here is that we can choose $\delta$ to be large enough to cancel 
any singularities that occur at $s=0$.  Now we are ready to set up the induction.  
We have by Theorem \ref{old style short} that the local solution 
$u$ is in $\dot{C}^T_{(s_2-s_1)/\gamma_1;s_2,p,q}$, where $s_2>\gamma_2$. 
 For induction, we assume this solution $u$ is also in 
$\dot{C}^T_{(k-s_1)/\gamma_1;k,p,q}$ for some $k\geq s_2$, and seek to 
show that $u$ is in $\dot{C}^T_{a^*;k+h,p,q}$, where $a^*=(k+h-s_1)/\gamma_1$ 
and $h$ is a fixed number between $0$ and $1$ which will be chosen later.

An application of Proposition \ref{heat kernel bound} gives
\begin{equation}\label{hugo}
\begin{aligned} 
&\|u\|_{B^{k+h}_{p,q}}\\
&\leq t^{-\delta}\Big(\delta\int_0^t \|e^{(t-s)\mathcal{L}_1} 
s^{\delta-1}u(s)\|_{B^{k+h}_{p,q}}ds 
+ \int_0^t \|e^{(t-s)\mathcal{L}_1} s^{\delta}W(u(s))\|_{B^{k+h}_{p,q}}ds\Big)\\
&\leq Ct^{-\delta}\delta \int_0^t |t-s|^{-h/\gamma_1}s^{\delta-1}
 \|u(s)\|_{B^{k}_{p,q}} \\
&\quad + t^{-\delta}\int_0^t |t-s|^{-b_1/\gamma_1}s^{\delta}
 \|W(u(s))\|_{B^{k-1}_{\tilde{p},q}} ds,
\end{aligned}
\end{equation}
where $b_1=h+1+n/\tilde{p}-n/p$.

For the first term in the right hand side of \eqref{hugo}, we have
\begin{equation}\label{hugo2}
\begin{aligned} 
&t^{-\delta}\int_0^t |t-s|^{-h/\gamma_1}s^{\delta-1}\|u(s)\|_{B^{k}_{p,q}}ds\\ 
&\leq t^{-\delta}\|u\|_{(k-s_1)/\gamma_1;k,p,q}
 \int_0^t |t-s|^{-h/\gamma_1}s^{\delta-1-(k-s_1)/\gamma_1}ds\\ 
&\leq C\|u\|_{(k-s_1)/\gamma_1;k,p,q}t^{-\delta}t^{-h/\gamma_1}
 t^{\delta-1-(k-s_1)/\gamma_1+1}\\ 
&\leq Ct^{-(k+h-s_1)/\gamma_1}\|u\|_{(k-s_1)/\gamma_1;k,p,q}
\end{aligned}
\end{equation}
This calculation implicitly assumes that the exponents of $|t-s|$ and $s$ 
in the integral are both strictly greater than negative $1$.  
For $|t-s|$, this holds provided $h/\gamma_1<1$.  For $s$, it works for 
a sufficiently large choice of $\delta$.  We note that without modifying 
the PDE to include these $t^\delta$ terms, we would need $(k-s_1)/\gamma_1$ 
to be less than $1$, which does not hold for large $k$.

For the second piece, we start by bounding $\|W(u)\|_{B^{k-1}_{\tilde{p},q}}$.  
Using Proposition \ref{annie2}, Proposition \ref{product est 1}, and 
finally Proposition \ref{annie} and Proposition \ref{sob embedding}, we obtain
\begin{equation}\label{mal38}
\begin{aligned} 
\|W(u(s))\|_{B^{k-1}_{\tilde{p},q}}
&\leq \|u\otimes (1-\mathcal{L}_2)u\|_{B^{k-\gamma_2}_{\tilde{p},q}}\\
&\leq \|u\|_{L^{p_1}}\|(1-\mathcal{L}_2)u\|_{B^{k-\gamma_2}_{p,q}}
 +\|u\|_{B^{k-\gamma_2}_{p, q}}\|(1-\mathcal{L}_2)u\|_{L^{p_3}} \\ 
&\leq \|u\|_{B^{r_1^+}_{p,q}}\|u\|_{B^{k}_{p,q}}
 +\|u\|_{B^{k-\gamma_2}_{p,q}}\|u\|_{B^{\gamma_2+r_2^+}_{p,q}}
\end{aligned}
\end{equation}
where
\begin{equation}\label{aldi65}
\begin{gathered} 
1/\tilde{p}=1/p_1+1/p=1/p+1/p_2,\\
p_i=\frac{np}{n-r_ip}, \quad i=1,2
\end{gathered}
\end{equation}
and
\begin{equation}\label{aldi652}
\begin{gathered} r_i <n/p, \quad i=1,2,\\ 
r_1<s_2\leq k, \\ 
r_2<s_2-\gamma_2\leq k-\gamma_2.
\end{gathered}
\end{equation}
Using \eqref{mal38} in the last term in \eqref{hugo}, we obtain
\begin{equation}\label{hugo3}
t^{-\delta}\int_0^t |t-s|^{-b_1/\gamma_1}s^{\delta}
\|W(u)\|_{B^{k-1}_{\tilde{p},q}} ds\leq I_1+I_2,
\end{equation}
where
\begin{gather*} 
I_1=t^{-\delta}\int_0^t |t-s|^{-b_1/\gamma_1}s^{\delta} 
\|u(s)\|_{B^{r^+_1}_{p,q}}\|u(s)\|_{B^{ k}_{p,q}}ds,\\ 
I_2=t^{-\delta}\int_0^t |t-s|^{-b_1/\gamma_1}s^{\delta}
\|u(s)\|_{B^{k-\gamma_2}_{p,q}} \|u(s)\|_{B^{\gamma_2+r^+_2}_{p,q}} ds.
\end{gather*}
Working on $I_1$, and setting $a_1=(r^+_1-s_1)/\gamma_1$  
$a_2=( k-s_1)/\gamma_1$, and recalling that $b_1=h+1+n/\tilde{p}-n/p$, we have
\begin{align*} 
I_1&\leq Ct^{-\delta}\|u\|_{a_1;r^+_1,p,q}\|u\|_{a_2; k,p,q} 
 \int_0^t |t-s|^{-b_1/\gamma_1}s^{\delta-a_1-a_2}ds\\ 
&\leq Ct^{-\delta}\|u\|_{a_1;r^+_1p, q}\|u\|_{a_2; k,p, q} 
  t^{-b_1/\gamma_1+\delta-a_1-a_2+1}\\ 
&\leq Ct^{-(h+1+k+r^+_1+n/\tilde{p}-n/p-2s_1)/\gamma_1+1}
\end{align*}
provided $b_1<\gamma_1$.  For the last inequality, we recall that 
$\|u\|_{a_2; k, p, q}$ is bounded (by the induction hypothesis) 
and since $r_1<k$ (by \eqref{aldi652}), we have by interpolation that 
$\|u\|_{a_1; r_1^+,p,q}$ is also bounded.   
Incorporating the relevant constraints from \eqref{aldi65} and 
\eqref{aldi652} gives
\begin{equation}\label{induct2}
\begin{aligned} 
I_1&\leq Ct^{-(h+1+k+r^+_1+n/\tilde{p}-n/p-2s_1)/\gamma_1+1}\\ 
&\leq C t^{-(h+k-s_1)/\gamma_1-(1+n/p-s_1+\varepsilon)/\gamma_1+1},
\end{aligned}
\end{equation}
with the constraint $h+1+n/p-r^+_1<\gamma_1$ and where 
$\varepsilon=r^+-r_1$. By the last inequality in \eqref{durant}, we know 
that $\gamma_1-1-n/p>0$, so the constraint will be satisfied if we choose
 $h$ to be small enough so that $r^+_1>h.$  Note that $r_1<\min\{s_2, n/p\}$, 
so this choice only depends on $s_2, n$, and $p$.

Also from \eqref{durant}, we have
\[
1-(1+n/p-s_1+\varepsilon)/\gamma_1=1-(1+n/p-s_1+s_2)
/\gamma_1+(s_2-\varepsilon) /\gamma_1\geq s_2/\gamma.
\]
Applying this to \eqref{induct2}, we finally get
\begin{equation}\label{ward}
I_1\leq Ct^{-(h+k-s_1)/\gamma_1+(s_2-\varepsilon)/\gamma_1}.
\end{equation}

A similar calculation for $I_2$ yields
\begin{equation} \label{durant2}
\begin{aligned}
I_2&\leq C\|u\|_{a_3;k-\gamma_2,p,q}\|u\|_{a_4;\gamma_2+r^+_2,p,q}
 t^{-(h+k-s_1)/\gamma_1+s_2/\gamma_1}\\ 
&\leq C t^{-(h+k-s_1)/\gamma_1+(s_2-\varepsilon)/\gamma_1},
\end{aligned}
\end{equation}
where we recall that $r^+_2<s_2-\gamma_2<k-\gamma_2$  and we have set
\[ 
a_3=(k-\gamma_2+r^+_2-s_1)/\gamma_1, \quad
a_4=(\gamma_2+r_3-s_1)/\gamma_1.
\]
This also requires setting $r_2^+>h$, which now means the choice of $h$ 
also depends on $\gamma_2$.

Using \eqref{ward} and \eqref{durant2} in \eqref{hugo3}, we have
\begin{equation}\label{durant3}
t^{-\delta}\int_0^t |t-s|^{-b_1/\gamma_1}s^{\delta}
\|W(u)\|_{B^{k-1}_{\tilde{p},q}} ds\leq C t^{-(h+k-s_1)
/\gamma_1+(s_2-\varepsilon)/\gamma_1},
\end{equation}
and using \eqref{hugo2} and \eqref{durant3} in \eqref{hugo} gives
\[
\|u(t)\|_{B^{k+h}_{p,q}}\leq C\delta t^{-(k+h-s_1)/\gamma_1}
\|u\|_{(k-s_1)/\gamma_1;k,p,q}+C t^{-(h+k-s_1)/\gamma_1+(s_2-\varepsilon)/\gamma_1},
\]

Multiplying both sides by $t^{(k+h-s_1)/\gamma_1}$ and taking the supremem 
over $t$, we finally get
\[
\|u\|_{(k+h-s_1)/\gamma_1;k+h, p, q}\leq C\delta \|u\|_{(k-s_1)/\gamma_1;k,p,q}+C \sup_{t\leq T} t^{s_2/\gamma_1},
\]  
which completes the induction argument. We remark that $\delta$ is chosen 
after beginning the induction step (and thus can be absorbed into the constant), 
while the appropriate value of $h$ is fixed by the known parameters
 $n, p, s_1, s_2$, and $\gamma_2$.
\end{proof}


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\end{document}