\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 191, pp. 1--7.\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-2015/191\hfil Cordes nonlinear operators]
{Cordes nonlinear operators in Carnot groups}

\author[G. Di Fazio, M. S. Fanciullo \hfil EJDE-2015/191\hfilneg]
{Giuseppe Di Fazio, Maria Stella Fanciullo}

\address{Giuseppe Di Fazio \newline
Dipartimento di Matematica e Informatica,
Universit\`a di Catania,
Viale A. Doria 6, 95125, Catania, Italy}
\email{difazio@dmi.unict.it}

\address{Maria Stella Fanciullo \newline
Dipartimento di Matematica e Informatica,
Universit\`a di Catania, Viale A. Doria 6, 95125, Catania, Italy}
\email{fanciullo@dmi.unict.it}

\thanks{Submitted April 8, 2015. Published July 20, 2015.}
\subjclass[2010]{35H20}
\keywords{Cordes condition; Carnot groups; nonlinear equations}

\begin{abstract}
 Our aim is to obtain $L^p$ estimates for the second-order horizontal derivatives
 of the solutions for a nondivergence form nonlinear equation in Carnot groups.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}


The $W^{2,p}$ estimates for elliptic differential equations and systems is a 
very interesting problem and  many Authors have given several contributions 
to this problem from several different points of view (see \cite{cfl,cam,d})
using different approaches.  There are essentially two main approaches to 
the problem: assuming on the coefficients of the equation the Cordes 
condition or the VMO condition.
 The first one consists of a geometric condition on the eigenvalues of linear 
operators. Cordes condition was introduced  in \cite{c} and studied by many 
authors (in the cases of nonlinear nonvariational equations and systems we quote 
\cite{cam1,cam}).
The second technique consists in assuming the coefficients of the operator to
 be in $\operatorname{VMO}$-type classes 
(see \cite{cfl,d,dfz1,dfz2}, and for more general setting see 
\cite{bb2,bf}).

Here we obtain $W^{2,p}$ estimates for the following nonlinear nondivergence 
form equation
$$
a(x,u,Xu,X^2u)=f,
$$
where $X=(X_1,X_2,\dots,X_l)$  is a system of H\"ormander's vector fields on 
a Carnot group, and we assume a condition that in the particular
case of a linear equation gives back the Cordes condition (see \cite{df1} 
for the case of the Heisenberg group).

Namely, we show that there exists a critical exponent $p_0 > 2$  such that 
if the datum $f$ belongs to $L^p$, with $2<p<p_0$, then the second derivatives 
$X^2u$ of the solutions $u$ have the same integrability as $f$.

\section{Preliminaries}

Let $\mathcal{G}$ be a finite-dimensional, stratified, nilpotent Lie algebra. 
We assume  $\mathcal{G}=\oplus_{i=1}^sV_i$, where
$[V_i,V_j]\subset V_{i+j}$ for $i+j\le s$ and $[V_i,V_j]=0$ for $i+j>s$. 
Let $X_1, \dots,X_l$ be a basis for $V_1$ and suppose that $X_1,\dots,X_l$ 
generate $\mathcal{G}$ as a Lie algebra. Then for $2\le j\le s$ we choose 
a basis $\{X_{ij}\}$, $1\le i\le k_j$, for $V_j$ consisting of commutators 
of length $j$. We set $X_{i1}=X_i$, $i=1,\dots,l$ and $k_1=l$, and we call 
$X_{i1}$ a commutator of length $1$. If $\mathbb G$  is the simply connected 
Lie group  associated to $\mathcal{G}$ then $\mathbb G$  is called Carnot group. 
It is well known that the exponential mapping is a global diffeomorphism from 
$\mathcal{G}$ to $\mathbb G$ and then for any $g\in \mathbb G$ there exists 
$x=(x_{ij})\in \mathbb{R}^n$, $1\le i\le k_j$, $1\le j\le s$, $n=\sum_{j=1}^sk_j$,  
such that $g=\exp (\sum x_{ij}X_{ij})$.

We now recall the definition of polynomials on the Carnot group $\mathbb G$ 
given by Folland and Stein in \cite{fs}.

A function $P$ on $\mathbb G$ is said to be a polynomial on  $\mathbb G$ if 
$P\circ$ exp is a polynomial on the Lie algebra $\mathcal{G}$.

Let $X_1,X_2,\dots, X_{n}$ be a basis of $\mathbb G$ and 
$\xi_1,\xi_2,\dots,\xi_{n}$ be the dual basis for $\mathcal{G}^*$   we set 
$\eta_i=\xi_i\circ {\rm exp}^{-1}$. Each $\eta_i$ is a polynomial on 
$\mathbb G$, and $\eta_1,\eta_2,\dots,\eta_{n}$  form a system of global 
coordinates on $\mathbb G$. Then every polynomial on $\mathbb G$ can be 
written uniquely as
$$
P(x)=\sum_Ia_I\eta^I(x),\quad  \eta^I=\eta_1^{i_1}\cdots\eta_{n}^{i_{n}},\;
a_I\in \mathbb{R}
$$
where all but finitely many of the coefficients $a_I$ vanish.
Clearly $\eta^I$ is homogeneous of degree $d(I)=\sum_{j=1}^{n}i_jd_j$, where 
$d_j$ is the length of $X_j$ as a commutator. We define the homogeneous degree 
of the polynomial $P$ as max$\{d(I):a_I \neq 0 \}$.

Here we recall the definition of the Carnot-Carath\'eodory metric.
An absolutely continuous  curve $\gamma:[0,\tau]\to \mathbb{R}^{n}$ is called subunitary 
if there exists a measurable function $c=(c_1,c_2,\dots ,c_{l}):[0,\tau]\to \mathbb{R}^{l}$ 
such that $\gamma'(t)=\sum_{j=1}^{l} c_j(t)X_j(\gamma(t))$,
for a.e. $t\in [0,\tau]$, and $\|c\|_\infty\le 1$. The Carnot-Carath\'eodory 
distance $d(x',x'')$ is defined as the infimum of those $\tau >0$ for which 
there exists a subunitary curve $\gamma :[0,\tau]\to \mathbb{R}^n$ with $\gamma(0)=x'$ 
and $\gamma(\tau)=x''$.

We set $B_r(x_0)=\{x\in \mathbb{R}^{n}: d(x,x_0)<r\}$. When it is clear from the setting 
we will omit $x_0$ or $r$. It is well known that the Carnot-Carath\'eodory 
balls satisfy a doubling condition, that is
$$
|B_{2r}(x_0)|\le 2^Q |B_r(x_0)|
$$
for all $r>0$ and $x_0\in \mathbb{R}^{n}$. The constant $Q$ is the homogeneous dimension 
of $\mathbb G$ .

We define the intrinsic Sobolev spaces for a bounded domain $\Omega$ in $\mathbb{R}^n$. 
Let $k\in \mathbb{N}$ and $p\ge 1$ we set
$$
W^{k,p} (\Omega ) = \big\{ u : \Omega \to {\mathbb R} 
:   u  ,   X_{i_1} \dots   X_{i_j}  u \in L^p (\Omega ),
\; 1 \leq j \leq k  \bigr\} 
$$
endowed with the norm
$$
\|u\|_{W^{k,p}(\Omega)}=\|u\|_{L^p(\Omega)}
+\sum_{h=1}^k\sum_{i_j=1}^{l} \|X_{i_1}X_{i_2}\dots X_{i_h}u\|_ {L^p(\Omega)}  \,.
$$
We define $W^{k,p}_0 (\Omega )$ as the closure of $C_0^{\infty} (\Omega )$ in 
$W^{k,p} (\Omega )$ with respect to the above norm.

For $I=(i_1,i_2,\dots ,i_{n})$ we denote the differential operator 
$X^{i_1}_{1}X^{i_2}_{2}\dots X^{i_{n}}_{n}$  by $X^I$ and  $d(I)$ 
the homogeneous degree of $X^I$.
We denote by $Xu$ the gradient of $u$ $(X_1u,X_2u,\dots,X_{l})$ and by $X^2u$ 
the hessian matrix $\{X_{ij}u\}_{i,j=1,\dots,l}$.

In \cite{lu} the existence of approximation polynomials of Sobolev functions  
in Carnot groups and related Poincar\'{e}-type inequalities have been obtained. 
Here we state some results \cite[Theorems 2.7 and 5.1]{lu}, that we will use 
in the sequel.

\begin{theorem}\label{existencepolynomials}
Let $k$ be a positive integer and $u$ a function in $W^{k,1}(\Omega) $. 
Then there exists a polynomial P of degree less than $k$ such that 
$\int_\Omega X^I(u-P)dx=0$ for any $0\le d(I)<k$.
\end{theorem}

 Choosing first $q_{10}=p=2$  and  $q_{21}=2$, $p=\frac{2Q}{Q+2}$ 
in \cite[Theorem 5.1]{lu}, we obtain the following two inequalities 
that we collect in the same statement.

\begin{theorem} \label{thm2.2}
Let $B_r$ be a ball of $\mathbb{R}^n$ and $u\in W^{2,\frac{2Q}{Q+2}}(\overline B_r)$. 
Then there exists a polynomial of degree $\le 1$ such that
\begin{gather}\label{poincareI}
\int_{B_r} |u-P|^2dx \le  cr^2\int_{B_r} |X(u-P)|^2dx,\\
\label{poincareII}
\int_{B_r} |X(u-P)|^2dx 
\le  c\Big(\int_{B_r} |X^2u|^{\frac{2Q}{Q+2}}dx\Big)^{\frac{Q+2}{Q}},
\end{gather}
where the constant $c$ is independent of $B_r$ and $u$. (The polynomial $P$ 
is the same as  in Theorem \ref{existencepolynomials}).
\end{theorem}

The following Theorem has been proved in \cite{f}, 
(for different cases see \cite{dm,ddfm,df}).

\begin{theorem}\label{cz} 
There exists a constant $C_{G}\ge 1$ such that for every 
$u\in W^{2,2}_0(\Omega)$ the following inequality holds 
\begin{equation}\label{eqcz}
\int_\Omega |X^2u|^2dx \le C_G\int_\Omega |\Delta u|^2dx\,,
\end{equation}
where $\Delta u = \sum_{i=1}^{l} X_iX_iu$.
\end{theorem}

\section{Caccioppoli-type inequality and $W^{2,p}$ estimates}

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Let 
$a(x,u,p,m): \Omega\times \mathbb{R}\times \mathbb{R}^{l}\times \mathbb{R}^{l^2 }\to \mathbb{R}$ 
be a Carath\'eodory function satisfying the condition
\begin{itemize}
\item[(A)]   there exist  three positive constants, $\alpha$, $\gamma$ and 
$\delta$ such that  $ C_G \gamma+\delta<1$,  for all 
$M = \{m_{ij}\}_{i,j=1,\dots, l} \in\mathbb{R}^{l}\times\mathbb{R}^{l}$, 
$u\in \mathbb{R}$, $p\in \mathbb{R}^{l}$,
$$
\Big|\sum_{i=1}^{l}  m_{ii}- \alpha  [a(x,u,p,m)] \Big|^2 
\leq \gamma |M|^2+\delta \Big|\sum_{i=1}^{l} m_{ii}\Big|^2 ,
\quad \text{a.e. }  x \in \Omega \, .
$$
\end{itemize}
We consider the nonlinear nonvariational elliptic equation
\begin{equation}\label{equation}
a(x,u,X,X^2u)=f\,,
\end{equation}
 where $f\in L^2(\Omega)$.

\begin{definition} \label{def3.1}\rm
A function $u\in W^{2,2}(\Omega)$ is called a solution of
 \eqref{equation}  if $u$ satisfies \eqref{equation} for  a.e. $x$ in $\Omega$.
\end{definition}

\begin{remark} \label{rmk3.1} \rm 
In the case of linear equation, i.e.
$$
\sum_{i,j=1}^{l}a_ {ij}(x)X_iX_ju (x) =f
$$   
condition (A) is stronger than the following Cordes condition 
(see \cite{c,t} for a comparison between condition $(A)$ and Cordes 
condition in Euclidean setting).
\end{remark}

\begin{definition} \label{def3.2} \rm
The linear operator $L\equiv a_ {ij}(x)X_iX_j $ satisfies the Cordes 
condition $K_{\epsilon, \sigma , \theta}$ if there exist
$\epsilon\in (0,1]$, $\sigma >0$ and $\theta > 0$ such that for
a.e. $x\in\Omega$,
$\sum_{i=1}^{l} a_{ii} (x) > 0$
and
$$
0< \frac{1}{\sigma} \leq \sum_{i,j=1}^{l}a^2_{ij}(x)\le
\frac{1}{l-1+\epsilon}\Big(\sum_{i=1}^{l}a_{ii} (x) \Big)^2
\leq \frac{\theta^2}{l-1+\epsilon} \, .
$$
 \end{definition}


Now we prove a Caccioppoli type inequality for solutions of \eqref{equation}.

\begin{theorem} \label{thm3.1}
Let condition  $(A)$ hold true and $f\in L^2(\Omega)$. Then for any 
$u\in W^{2,2}(\Omega)$ solution of \eqref{equation}, for any $r>0$ 
such that $B_{2r}\Subset \Omega$, there exists a polynomial $P$ of degree 
less than $2$ such that $\int_{B_{2r}} X^I(u-P)dx=0$ for any $0\le d(I)<2$, and
\begin{equation}\label{cacci}
\int_{B_r}|X^2u|^2dx\le c r^{-2}\int_{B_{2r}}|X(u-P)|^2dx + c\int_{B_{2r}}f^2dx\,.
\end{equation}
\end{theorem}

\begin{proof}
Let $B_{2r}\Subset \Omega$. From Theorem \ref{existencepolynomials}
there exists  a polynomial $P$ of degree less than two such that
$\int_{B_{2r}}X^I (u-P)dx=0$, for $I$ with $d(I)<2$.

Let $\eta $ be a $C^\infty_0(\mathbb{R}^{n})$ with the properties $0\le \eta\le 1$, 
$\eta=1$ in $B_r$, $\eta=0$ in $\mathbb{R}^{n}\setminus B_{2r}$ and 
$|X \eta|\le \frac{c}{ r}$.

If we set $\mathcal U= \eta(u-P) \in W^{2,2}_0(B_{2r})$, since 
$X^2P=0$ (see the proof of \cite[Theorem 2.7]{lu}), we have 
$X^2(u-P)=X^2u$ and $X^2\mathcal U=X^2 u$ in $B_r$.
We have 
$$
\eta \Delta u =\eta (\Delta u - \alpha a(x,u,Xu,X^2u))+\eta\alpha f \, ,
$$
which implies
\begin{align*}
|\eta\Delta u|
&\le \eta |\Delta u -\alpha a(x,u,Xu,X^2u)|+|\eta\alpha f| \\
&\le \eta[\gamma |X^2u|^2+\delta |\Delta u|^2]^{1/2}+\eta \alpha |f|.
\end{align*}
Note that
\begin{gather}\label{Delta}
\Delta \mathcal U =\eta \Delta u + A(u-P),\\
\label{X}
\eta X^2u=X^2\mathcal U-B(u-P),
\end{gather}
where
\begin{equation}\label{AA}
A(u-P)= (u-P)\Delta \eta+2\sum X_i\eta X_i(u-P)
\end{equation}
and  
$$
B(u-P)=\{(u-P)X_i X_j \eta+X_i\eta X_j(u-P)+X_j\eta X_i(u-P)\}_{ij} \, .
$$
Then for $x\in B_{2r}$,
$$
|\Delta\mathcal U |\le |\eta \Delta u|+|A(u-P)|
\le \eta (\gamma |X^2u|^2+\delta|\Delta u|^2)^{1/2}+\eta \alpha |f|+|A(u-P)|,
$$
from which it follows that for all $\epsilon>0$,
\begin{align*}
|\Delta\mathcal U |^2
&\le \eta^2(\gamma |X^2u|^2 +\delta|\Delta u|^2)+ ( \eta \alpha |f|+|A(u-P)|   )^2\\
&\quad +2\eta (\gamma |X^2u|^2+\delta|\Delta u|^2)^{1/2} ( \eta \alpha |f|+|A(u-P)|   )  \\
&\le \eta^2(\gamma |X^2u|^2+\delta|\Delta u|^2)+( \eta \alpha |f|+|A(u-P)|   )^2\\
&\quad +\epsilon \eta^2 (\gamma |X^2u|^2+\delta|\Delta u|^2) 
 +\frac{1}{\epsilon} ( \eta \alpha |f|+|A(u-P)|   )^2\\
&=(1+\epsilon)\eta^2 (\gamma |X^2u|^2+\delta|\Delta u|^2)
 + \big(1+\frac{1}{\epsilon}\big) ( \eta \alpha |f|+|A(u-P)|   )^2\\
&\le (1+\epsilon)\eta^2 (\gamma |X^2u|^2+\delta|\Delta u|^2)
 +2\big(1+\frac{1}{\epsilon}\big) ( \eta^2 \alpha^2 |f|^2+|A(u-P)|^2   ).
\end{align*}
Then from \eqref{Delta} and \eqref{X},
\begin{align*}
|\Delta\mathcal U |^2
&\le (1+\epsilon)\gamma |X^2(\mathcal U)-B(u-P)|^2+(1+\epsilon)\delta |\Delta\mathcal U
 -A(u-P)|^2\\
&\quad +2\big(1+\frac{1}{\epsilon}\big)\eta^2\alpha^2 |f|^2+2\big(1+\frac{1}{\epsilon}\big)|A(u-P)|^2 \\
&\le (1+\epsilon)\gamma (|X^2(\mathcal U)|^2+|B(u-P)|^2+2|X^2(\mathcal U)||B(u-P)|)\\
&\quad +(1+\epsilon)\delta (|\Delta\mathcal U|^2+|A(u-P)|^2+2|\Delta\mathcal U||A(u-P)|)\\
&\quad +2\big(1+\frac{1}{\epsilon}\big)\eta^2\alpha^2 |f|^2+2\big(1+\frac{1}{\epsilon}\big)
 |A(u-P)|^2 \\
&\le (1+\epsilon)\gamma \big[(1+\epsilon)|X^2(\mathcal U)|^2
 +\big(1+\frac{1}{\epsilon}\big)|B(u-P)|^2\big]\\
&\quad +(1+\epsilon)\delta \big[(1+\epsilon)|\Delta\mathcal U|^2
 +\big(1+\frac{1}{\epsilon}\big)|A(u-P)|^2\big]\\
&\quad +2\big(1+\frac{1}{\epsilon}\big)\eta^2\alpha^2 |f|^2
 +2\big(1+\frac{1}{\epsilon}\big)|A(u-P)|^2 \\
&\le (1+\epsilon)^2\gamma |X^2(\mathcal U)|^2+(1+\epsilon)^2\delta |\Delta\mathcal U|^2\\
&\quad +c(\epsilon, \alpha,\gamma,\delta)[ |A(u-P)|^2+ |B(u-P)|^2+|f|^2].
\end{align*}

We integrate on $B_{2r}$ and apply \eqref{eqcz} in  Theorem \ref{cz} to obtain
\begin{align*}
\int_{B_{2r}}|\Delta\mathcal U|^2dx 
&\le (1+\epsilon)^2\gamma \int_{B_{2r}}|X^2(\mathcal U)|^2dx+(1+\epsilon)^2\delta
 \int_{B_{2r}}|\Delta\mathcal U|^2dx \\
&\quad +c \int_{B_{2r}}(|f|^2+|A(u-P)|^2+ |B(u-P)|^2)dx \\
&\le (1+\epsilon)^2 (\gamma C_G+\delta)\int_{B_{2r}}|\Delta\mathcal U|^2dx\\
&\quad +c\int_{B_{2r}}(|f|^2+|A(u-P)|^2+ |B(u-P)|^2)dx.
\end{align*}
It follows that
\begin{align*}
&[1-(1+\epsilon)^2(\gamma C_G+\delta)]\int_{B_{2r}}|\Delta\mathcal U|^2dx  \\
&\le c\int_{B_{2r}}(|f|^2+|A(u-P)|^2+ |B(u-P)|^2)dx,
\end{align*}
and then
$$
\int_{B_{2r}}|\Delta\mathcal U|^2dx \le c\int_{B_{2r}}(|f|^2+|A(u-P)|^2
+ |B(u-P)|^2)dx .
$$
Finally, we get that
\begin{align*}
\int_{B_{r}}|X^2u|^2dx
&\le \int_{B_{2r}}|X^2\mathcal U|^2dx\le C_G\int_{B_{2r}}|\Delta\mathcal U|^2dx \\
&\le c\int_{B_{2r}}(|f|^2+|A(u-P)|^2+ |B(u-P)|^2)dx.
\end{align*}
Now we observe that from \eqref{AA} and  Poincar\'e inequality \eqref{poincareI}
 we obtain
\begin{align*}
&\int_{B_{2r}}|A(u-P)|^2dx \\
&\le c \int_{B_{2r}}|\Delta \eta|^2|u-P|^2dx 
 +c \int_{B_{2r}}\sum |X_i\eta|^2|X_i(u-P)|^2dx \\
&\le c r^{-2}\Big\{r^{-2}\int_{B_{2r}}|u-P|^2dx+\int_{B_{2r}}|X(u-P)|^2dx  \Big\}\\
&\le c r^{-2}\int_{B_{2r}}|X(u-P)|^2 dx.
\end{align*}
In the same way, we find that
$$
\int_{B_{2r}}|B(u-P)|^2dx \le c r^{-2}\int_{B_{2r}}|X(u-P)|^2dx,
$$
from which the Caccioppoli type inequality follows.
\end{proof}

Next  we state \cite[Theorem 3.3]{z} which is a generalization of the
 Gehring lemma \cite{g}.

\begin{lemma}\label{ggm}
Let $U$ and $G$ be non-negative functions in $\Omega$ such that
$$
U\in L^t_{\rm loc}(\Omega), \quad G\in L^s_{\rm loc}(\Omega), \quad 1<t<s.
$$
If there exists $c>1$ such that for every $B_{2r}\Subset \Omega$, $r <1$,
$$
{\int\hspace{-10pt}-}_{B_{r}}U^tdx\le c 
\Big({\int\hspace{-10pt}-}_{B_{2r}}Udx\Big)^t
+c{\int\hspace{-10pt}-}_{B_{2r}}G^tdx,
$$
then there exists $\epsilon\in (0,s-t]$ such that $U\in L^p_{\rm loc}(\Omega)$, 
for all $p \in [t,t+\epsilon)$ and, for every $B_{2r}\Subset \Omega$, with $r <1$,
we have
$$
\Big({\int\hspace{-10pt}-}_{B_{r}}U^pdx\Big)^{1/p}
\le K\Big[\Big({\int\hspace{-10pt}-}_{B_{2r}}U^tdx\Big)^{1/t}
+\Big({\int\hspace{-10pt}-}_{B_{2r}}G^pdx\Big)^{1/p}\Big],
$$
where the constant $K$ depends on $c, t$ and $Q$.
\end{lemma}

Our main Theorem is now an easy consequence of Caccioppoli-type inequality 
\eqref{cacci} and Lemma \ref{ggm}.

\begin{theorem} \label{thm3.2}
Let $u\in W^{2,2}(\Omega)$ be a solution of \eqref{equation} then there exists 
$p_0>2$ such that, if $f\in L^p(\Omega)$, with $2\le p<p_0$, then 
$u\in W^{2,p}_{\rm loc}(\Omega)$ and for all $B_{2r}\subset\subset \Omega$ we have
$$
\Big(\int\hspace{-10pt}-_{B_{r}}|X^2u|^p dx\Big)^{1/p}
\le c \Big(\int\hspace{-10pt}-_{B_{2r}}|X^2u|^2dx\Big)^{1/2}
+\Big(\int\hspace{-10pt}-_{B_{2r}} |f|^pdx\Big)^{1/p}.
$$
 \end{theorem}

\begin{proof}
Let $B_{2r}\subset\subset \Omega$, from the Caccioppoli-type inequality \eqref{cacci} 
and Poincar\'e inequality \eqref{poincareII} it follows
$$
\int_{B_{r}}|X^2u|^2dx
\le c r^{-2}\Big(\int_{B_{2r}}|X^2u|^{\frac{ 2Q}{Q+2}}dx\Big)^{\frac{Q+2}{Q}}
+\int_{B_{2r}} f^2dx\,, 
$$
from which
\begin{equation}\label{2}
\int\hspace{-10pt}-_{B_{r}}|X^2u|^2dx
\le c \Big(\int\hspace{-10pt}-_{B_{2r}}|X^2u|^{\frac{ 2Q}{Q+2}}dx\Big)^{\frac{Q+2}{Q}}
+\int\hspace{-10pt}-_{B_{2r}} f^2dx .
\end{equation}
Now we can apply Lemma \ref{ggm} with $U=|X^2u|^{\frac{2Q}{Q+2}}$, 
$t=\frac{Q+2}{Q}$, $G=|f|^{\frac{2Q}{Q+2}}$ and $s= \frac{p(Q+2)}{2Q}$, 
to obtain the thesis.
\end{proof}

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