\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
\begin{equation} \label{e1}
  \begin{split}
   \frac{\partial F}{\partial s} (x,s)& = Z F (x,s)\\
   F(x,0) & =x \; .
  \end{split}
\end{equation}
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|<s_K} \Big( \int_{\Omega} w^{\alpha} (Xf,s,x)
\frac{|\Delta_{Z,s}
f(x)|^{q}}{|s|^{\theta q}} \; dx \Big)^{1/q}  \,,\\
\|f\|_{Z,\alpha,p,q}^{\theta,2}  =  \|f\|_{L^P(\Omega)} +
\sup_{0<|s|<s_K} \Big( \int_{\Omega} w^{\alpha}(Xf,s,x)
\frac{|\Delta_{Z,s}^2 f(x)|^{q}}{|s|^{\theta q}} \; dx \Big)^{1/q}\,.
\end{gather*}

Define the following function spaces which help us to handle the
fractional derivatives in the $Z$ direction:
$$
B_{Z,\alpha,p,q}^{\theta,1} (K , \Omega) = \big\{ f \in XW^{1,p}
(\Omega ) :\mathop{\rm supp} f \subset K  \mbox{ and }
 \|f\|_{Z,\alpha,p,q}^{\theta,1} < \infty  \big\} \, ,
$$
 and
$$
B_{Z,\alpha,p,q}^{\theta,2} (K , \Omega) = \big\{ f \in XW^{1,p}
(\Omega ) :\mathop{\rm supp} f \subset K \mbox{ and }
 \|f\|_{Z,\alpha,p,q}^{\theta,2} < \infty  \big\} \, .
$$
If $\alpha = 0$ then these are linear normed spaces. Also, in the
elliptic case, for $\alpha =0$, $q =p$ we get similar spaces to
the fractional order Besov spaces \cite{pee,tri0}
 $$
B_{p,\infty}^{\theta} (\Omega ) = \big\{ f \in L^p
(\Omega ) :  \|f\|_{L^p (\Omega)}
+ \sup_{0\neq \|z\| \leq \delta,\; z \in {\mathbb R}^{n}}
 \frac{\|\triangle_z^2 f\|_{L^p (\Omega_z )}}{|z|^{\theta}}
< \infty \big\} \,, $$
 where
$\triangle_z^2 f (x)= f(x+z) + f(x-z) - 2f(x) $, and $\Omega_z
= \bigl\{ x \in \Omega : x+z \in \Omega  \bigr\}$.
 In the elliptic case  the vector fields
$\frac{\partial}{\partial x_i}$ generate a commuting family of
strongly continuous semigroup of operators and by their isotropic
nature, we can have a uniform treatment of the difference
quotients in every direction. In the subelliptic case, using the
Carnot-Carath\'{e}odory metric, a generalization of the elliptic
setting is possible \cite{dan}. However, this approach does not
allow us to study fractional derivatives in the direction of one
vector field at a time.

Let us list a few evident properties of our function spaces:
\begin{enumerate}
\item[(i)] By   \cite[Theorem 4.3]{ho},  if $Z$ is a commutator
of length $k$ of the horizontal vector fields $X_i$, then
$$
XW^{1,p} (\Omega ) \subset B_{Z,0,p,p}^{\frac{1}{k},1}
 (K , \Omega) \,.
$$

 \item[(ii)] By \cite[Lemma 2.3]{cap}, if
 $f \in B_{Z,0,p,p}^{1,1} (K , \Omega)$ then
 $Zf \in L^p (K)$.

 \item[(iii)] Using the fact that $\Delta_{Z,s}^2 f(x) = \Delta_{Z,s} f(x) - \Delta_{Z,-s} f(x)$
we easily get that
$$
B_{Z,\alpha,p,q}^{\theta,1} (K , \Omega) \subset
B_{Z,\alpha,p,q}^{\theta,2} (K , \Omega) \,.
$$
The reversed inclusion is not elementary, and for the proof
we use a method of  Zygmund  \cite{zyg} which already proved to be
useful in the Heisenberg group \cite{dom}.
\end{enumerate}

\begin{theorem}\label{thm1} \quad
\begin{enumerate}
\item[(a)] For $0<\theta <1$ we have
 $ B_{Z,\alpha,p,q}^{\theta,2} (K , \Omega)
 \subset B_{Z,\alpha,p,q}^{\theta,1} (K , \Omega)$.
\item[(b)] For every $0< \gamma <1$ we have $
B_{Z,\alpha,p,q}^{1,2} (K , \Omega)
 \subset B_{Z,\alpha,p,q}^{\gamma,1} (K , \Omega)$.
\item[(c)] For $1 < \theta <2$ we have
 $ B_{Z,\alpha,p,q}^{\theta,2} (K , \Omega)
 \subset B_{Z,\alpha,p,q}^{1,1} (K , \Omega)$.
\end{enumerate}
\end{theorem}

\begin{proof}
(a) Let $ f \in B_{Z,\alpha,p,q}^{\theta,2} (K , \Omega)$. Then
\[
\int_{\Omega} \bigl( 1 + |Xf(x)|^2 + |Xf(e^{sZ} x)|^2
\bigr)^{\alpha/2} | f (e^{sZ} x ) +
f (e^{-sZ}x ) - 2 f (x) |^q dx
\leq M^q |s|^{\theta q}
\]
for all $0<|s|<s_K$.
 Therefore,
\begin{equation*}
\int_{\Omega} \bigl( 1 +  |Xf(e^{sZ} x)|^2
\bigr)^{\alpha/2}  | f (e^{sZ} x ) + f (e^{-sZ}x )
- 2 f (x) |^q dx  \leq M^q |s|^{\theta q}
\end{equation*}
and then changing $s$ to $-s/2$ we get
\begin{equation*}
\int_{\Omega} \bigl( 1 +  |Xf(e^{-\frac{s}{2}Z} x)|^2
\bigr)^{\alpha/2} | f (e^{\frac{s}{2}Z} x ) + f
(e^{-\frac{s}{2}Z}x ) - 2  f (x) |^q dx  \leq
\frac{M^q}{2^{\theta q}} |s|^{\theta q} \, .
\end{equation*}
We use now the change of variables $x \mapsto e^{\frac{s}{2}Z} x$
to get
\begin{equation*}
\int_{\Omega} \bigl( 1 +  |Xf(x)|^2 \bigr)^{\alpha/2} \;
\left| f (e^{sZ} x ) + f (x ) - 2 f (e^{\frac{s}{2}Z} x) \right|^q
dx \leq \frac{M^q}{a^q 2^{\theta q}} |s|^{\theta q} \, .
\end{equation*}
In this way we have obtained the inequality
\begin{equation*}
\int_{\Omega} \bigl( 1 +  |Xf(x)|^2 \bigr)^{\alpha/2}
\left| \triangle_{Z,s} (f) (x) - 2 \triangle_{Z,\frac{s}{2}} (
f)(x) \right|^q dx  \leq \frac{M^q}{a^q 2^{\theta q}}
|s|^{\theta q}
\end{equation*}
and repeating $n$-times the process of changing $s$ to
$s/2$ and multiplying the inequality by $2^q$ we get
\begin{align*}
&\int_{\Omega}
\bigl( 1 +  |Xf(x)|^2 \bigr)^{\alpha/2}  \big|
2^{n-1}\triangle_{Z,\frac{s}{2^{n-1}}} f (x) - 2^n
\triangle_{Z,\frac{s}{2^n}} f(x) \big|^q dx\\
&\leq \frac{M^q}{a^q 2^{\theta q}} |s|^{\theta q}\,
2^{(1-\theta)q(n-1)}\, .
\end{align*}
 These inequalities give
\begin{equation} \label{e2}
\Big( \int_{\Omega} \bigl( 1 +  |Xf(x)|^2
\bigr)^{\alpha/2}  \left| \triangle_{Z,s} f (x) - 2^n
\triangle_{Z,\frac{s}{2^n}}
f(x) \right|^q dx \Big)^{1/q}
\leq \frac{M}{a 2^{ \theta}} |s|^{\theta} \sum_{k=0}^{n-1} 2^{(1 -
 \theta)k} \,
\end{equation}
and hence by our assumptions on $q$, $p$ and $\alpha$ it follows
that, for a constant $C>0$  depending on the $XW^{1,p}$ norm of
$f$, we have
\begin{equation} \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}
\end{equation}
For all $h$ 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,
$$
\sup_{0<|h|<s_K} \Big(\int_{\Omega} \bigl( 1 +
|Xf(x)|^2 \bigr)^{\alpha/2}  \frac{\left|
\triangle_{Z,h}
f(x) \right|^q}{|h|^{\theta q}} dx \Big)^{1/q}
\leq C \, .
$$
 The change of variables $x \mapsto e^{-hZ}x$ shows
that, for a possible different $C$ and sufficiently small $h$, we
have
$$
\Big( \int_{\Omega} \bigl( 1 +  |Xf(e^{-hZ}x)|^2
\bigr)^{\alpha/2} \; \frac{\left| \triangle_{Z,-h} f(x)
\right|^q}{|h|^{q \theta}} dx \Big)^{1/q} \leq C \, .
$$
 Changing $h$ to $-h$ gives
\begin{equation}
\Big( \int_{\Omega} \bigl( 1 +  |Xf(e^{hZ}x)|^2
\bigr)^{\alpha/2} \; \frac{\left| \triangle_{Z,h} f(x)
\right|^q}{|h|^{q \theta}} dx \Big)^{1/q} \leq C \, .
\end{equation}
 and therefore,
$$
\sup_{0<|h|<s_K} \Big( \int_{\Omega} \bigl( 1 + |Xf(x)|^2+
 |Xf(e^{hZ}x)|^2
\bigr)^{\alpha/2}  \frac{\left| \triangle_{Z,h} f(x)
\right|^q}{|h|^{q \theta}} dx \Big)^{1/q} \leq C \, .
$$

(b) Let $f \in B_{Z,\alpha,p,q}^{1,2} (K , \Omega)$ and start in a
similar way to the proof of the part (a). Inequality \eqref{e2} for
$\theta =1$  gives
\begin{equation}
\Big( \int_{\Omega} \bigl( 1 +  |Xf(x)|^2
\bigr)^{\alpha/2} \; \left| \triangle_{Z,s} f (x) - 2^n
\triangle_{Z,\frac{s}{2^n}}
f(x) \right|^q dx \Big)^{1/q}
\leq \frac{M}{a 2^{ \theta}} |s| n .
\end{equation}
Again, for $0< |h| < s_K/2$ consider
 $n \in {\mathbb N}$ and $s \in {\mathbb R}$ such that
$|s| \in [ s_K/2 , s_K ]$ and $h =s/2^n$ and get
\begin{equation*}
\frac{1}{|h|^{\gamma}} \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-\gamma}}{s_K}
 + |h|^{1-\gamma} |\ln h| \Big) \, .
\end{equation*}
This leads to $f \in B_{Z,\alpha,p,q}^{\gamma,1} (K , \Omega)$.

(c) Let $ f \in B_{Z,\alpha,p,q}^{\theta,2} (K , \Omega)$. Taking
into consideration that we suppose now $1 < \theta <2$, inequality
\eqref{e2} has the form
\begin{equation}
\left( \int_{\Omega} \bigl( 1 +  |Xf(x)|^2
\bigr)^{\alpha/2} \; \left| \triangle_{Z,s} f (x) - 2^n
\triangle_{Z,\frac{s}{2^n}}
f(x) \right|^q dx \right)^{1/q}
\leq \frac{M}{a 2^{ \theta}} |s| .
\end{equation}
and this leads to
\begin{equation*}
\frac{1}{|h|} \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 \frac{1}{s_K} \left( 1 + s_K^{\theta -1}  \right) \, .
\end{equation*}
It easily follows now that $f \in B_{Z,\alpha,p,q}^{1,1} (K ,\Omega)$.
\end{proof}

\begin{remark} \label{rmk1} \rm
As will be shown in the Examples \ref{exa1} and \ref{exa2} below, slight variations
of these weighted function spaces might also appear. To define
them consider the pseudo-norms:
\begin{gather*}
\|f\|_{XZ,\alpha,p,q}^{\theta,1}  =  \|f\|_{L^P(\Omega)} +
\sup_{0<|s|<s_K} \Big( \int_{\Omega} w^{\alpha} (Xf,s,x)
\frac{|\Delta_{Z,s}
Xf(x)|^{q}}{|s|^{\theta q}} \; dx \Big)^{1/q}  \,,\\
\|f\|_{XZ,\alpha,p,q}^{\theta,2}  =  \|f\|_{L^P(\Omega)} +
\sup_{0<|s|<s_K} \Big( \int_{\Omega} w^{\alpha}(Xf,s,x)
\frac{|\Delta_{Z,s}^2 Xf(x)|^{q}}{|s|^{\theta q}} \; dx
\Big)^{1/q} \,,
\end{gather*}
and the function spaces
$$
XB_{Z,\alpha,p,q}^{\theta,1} (K , \Omega)
= \big\{ f \in XW^{1,p}
(\Omega ) :\mathop{\rm supp} f \subset K \;  \; \mbox{and} \;
 \|f\|_{XZ,\alpha,p,q}^{\theta,1} < \infty  \big\} \, ,
$$
 and
$$
XB_{Z,\alpha,p,q}^{\theta,2} (K , \Omega) = \big\{ f \in XW^{1,p}
(\Omega ) :\mathop{\rm supp} f \subset K  \mbox{ and }
 \|f\|_{XZ,\alpha,p,q}^{\theta,2} < \infty  \big\} \, .
$$
If we follow the proof of Theorem \ref{thm1}, we realize that it remains
valid in the case of  $XB_{Z,\alpha,p,q}^{\theta,1} (K , \Omega)$
and $XB_{Z,\alpha,p,q}^{\theta,2} (K , \Omega)$, too. Another
inclusion  which will be used in Examples \ref{exa1} and \ref{exa2}
 is that if $ f \in XB_{Z,p-2,p,2}^{\theta,1} (K , \Omega)$
 then $f \in XB_{Z,0,p,p}^{\frac{2\theta}{p},1} (K , \Omega)$
 (see also the proof of  \cite[Lemma 3.1]{dom}) .
\end{remark}


In the following two examples we show that our function spaces
naturally appear when we study the regularity of the minimizers to
the problem
\begin{equation} \label{e7}
\min_{u \in XW^{1,p}(\Omega )} \; \int_{\Omega} \left( 1 + |Xu
(x)|^2 \right)^{p/2}
 \, dx
\end{equation}
subject to a boundary condition of type $u - v \in XW^{1,p}_0
(\Omega )$, where $v \in XW^{1,p} (\Omega )$ is fixed. A
minimizing function $u$ is a  weak solutions of
 the following nondegenerate $p$-Laplacian equation
\begin{equation}
 \sum_{i=1}^m \; X_i \Bigl( \bigl(1 + |Xu|^2 \bigr)^{\frac{p-2}{2}}
 \, X_i u \Bigr)  = 0 \,, \quad \mbox{in } \Omega
\end{equation}
which means that
\begin{equation}
\int_{\Omega} \bigl(1 + |Xu|^2 \bigr)^{\frac{p-2}{2}}  \,
 X_1 u \,X_1 \varphi
 + \bigl(1 + |Xu|^2 \bigr)^{\frac{p-2}{2}} \, X_2 u  \,
 X_2 \varphi \; dx = 0 \,,
\end{equation}
for all $\varphi \in XW^{1,p} (\Omega )$ with support
compactly included in $\Omega$.


\begin{example} \label{exa1} \rm
In this example we refer to the proof
of  \cite[Lemma 3.1]{dom}. Consider the the Heisenberg group
${\mathbb H}$ as ${\mathbb R}^3$ endowed with the group
multiplication
\begin{equation*}
(x_1 , x_{2}, t ) \cdot (y_1 , y_{2} , s ) = \Bigl( x_1 + y_1 ,
x_{2} + y_{2} , t + s - \frac{1}{2} ( x_{2} y_1 -  x_1 y_{2} )
\Bigr) \, .
\end{equation*}
 The horizontal vector fields are
\[
X_1 = \frac{\partial}{\partial x_1} - \frac{x_{2}}{2}
\frac{\partial}{\partial t} \, ,\quad
X_{2} = \frac{\partial}{\partial x_{2}} + \frac{x_1}{2}
\frac{\partial}{\partial t} \, .
\]
 Denote
$$
T = \frac{\partial}{\partial t}
$$
and observe that $[X_1, X_{2}] = T$. To study the regularity of
weak solutions first we have to  prove the differentiability in
the direction of $T$. The vector fields $X_1, X_2$ and $T$ span
the tangent space at every point and according to
\cite[Theorem 4.3]{ho} we have
 $$
\eta^2 u \in B^{\frac{1}{2},1}_{T,0,p,p} (\Omega )
$$
for every  $\eta \in C_0^{\infty} (\Omega )$.
 Use now a test function
 $$
\varphi = \frac{\triangle_{T,-s}}{s^{1/2}} \Big(
\frac{\triangle_{T,s} (\eta^2 u)}{s^{1/2}} \Big)
$$
 to get
$$
\eta^2 u \in XB_{T,p-2,p,2}^{\frac{1}{2},1}  (\mathop{\rm supp} \eta , \Omega).
$$
  This implies that
$$
\eta^2 u \in XB_{T,0,p,p}^{\frac{1}{p},1} (\mathop{\rm supp} \eta , \Omega)
$$
 and by the fact that $T$ commutes with the horizontal vector fields
 $X_1$ and $X_2$ we can use again   \cite[Theorem 4.3]{ho} to get
$$
\eta^2 u \in B_{T,0,p,p}^{\frac{1}{2}+\frac{1}{p},2} (\mathop{\rm supp} \eta , \Omega) \,.
$$
For $p=2$ we have $\eta^2 u \in B_{T,p-2,p,2}^{1,2}
 (\mathop{\rm supp} \eta , \Omega)$ which implies
 $$
\eta^2 u \in B_{T,p-2,p,2}^{\gamma,1}  (\mathop{\rm supp} \eta , \Omega)
$$
for any $\frac{1}{2} <\gamma <1 $. Restarting
 our proof on the bases of the previous line we get
$$
\eta^2 u \in B_{T,0,p,p}^{\frac{1}{2}+\gamma,2}  (\mathop{\rm supp} \eta , \Omega) \,,
$$
 and this leads to $Tu \in L^p_{\rm loc} (\Omega )$.\\
For $p>2$, 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 $Tu \in L^p_{\rm loc} (\Omega )$.

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\end{document}