text:
\begin{document}
\title{A compactness result for the Sobolev embedding \\ via potential theory
\footnote{AMS Subject Classification: 46E35, 35K70, 35B45, 35D30.}}
\author{{\sc{Filippo Camellini, Michela Eleuteri and Sergio Polidoro}}\\
Dipartimento di Scienze Fisiche Informatiche e Matematiche
\thanks{Modena via Campi 2013/b.
E-mail: michela.eleuteri@unimore.it, sergio.polidoro@unimore.it}
\thanks{Investigation supported by I.N.d.A.M.}
}

\maketitle

\footnotesize
\begin{abstract}
In this note we give a proof of the Sobolev and Morrey embedding theorems based on the representation of functions
in terms of the fundamental solution of suitable partial differential operators. We also prove the compactness of the
Sobolev embedding. We first describe this method in the classical setting, where the fundamental solution of the
Laplace equation is used, to recover the classical Sobolev and Morrey theorems. We next consider degenerate Kolmogorov
equations. In this case, the fundamental solution is invariant with respect to a non-Euclidean translation group and
the usual convolution is replaced by an operation that is defined in accordance with this geometry. We recover some
known embedding results and we prove the compactness of the Sobolev embedding. We finally apply our regularity results
to a kinetic equation.

\medskip\noindent
{\it Keywords: Sobolev spaces, Sobolev embedding, Morrey embedding, Compactness, Fundamental solution, Kolmogorov
equation.}
\end{abstract}

\normalsize

\section{Introduction}
\setcounter{section}{1} \setcounter{equation}{0}
\setcounter{theorem}{0}

Sobolev and Morrey embedding theorems are fundamental tools in the regularity theory for Elliptic and Parabolic second
order Partial Differential Equations (\emph{PDEs} in the sequel). In particular, they play a crucial role in the
natural setting for the study of uniformly elliptic PDEs in divergence form, that is the Sobolev space $W^{1,p}$.

There are several proofs of the Sobolev and Morrey embedding theorems, all of them rely on some integral representation
of a general function $u \in W^{1,p}$ in terms of its gradient. Here we focus in particular on representation formulas
based on the fundamental solution of the Laplace equation.

Consider a function $u \in C_0^\infty(\mathbb{R}^n)$. By the very definition of fundamental solution $\Gamma$, the
following identity holds
\begin{equation} \label{eq-fund-sol1}
  u(x) =  -  \int_{\mathbb{R}^n}\Gamma(x-y) \Delta u(y) \, d y, \qquad  \text{for every} \ x \in \mathbb{R}^n,
\end{equation}
and an integration by parts immediately gives
\begin{equation} \label{eq-fund-sol2}
  u(x) =   \int_{\R^n} \left\langle \nabla_y \Gamma(x-y), \nabla u(y) \right\rangle d y, \qquad
  \text{for every} \ x \in \R^n,
\end{equation}
where $\langle \cdot, \cdot \rangle$ and $\nabla$ denote the usual inner product in $\R^n$ and the gradient,
respectively. We recall that the gradient of the fundamental solution of the Laplace equation writes as follows
\begin{equation} \label{eq-grad-fund-sol-lapl}
  \nabla \Gamma(x-y)= - \frac{1}{n \omega_n \abs{x-y}^n} (x-y), \qquad x \ne y,
\end{equation}
where $\omega_n$ is the measure of the $n$-dimensional unit ball. In particular, $\nabla \Gamma$ is an homogeneous
function of degree $-n+1$, and there exists a positive constant $c_n$ such that
\begin{equation} \label{9Gamma}
\abs{\nabla \Gamma(x-y)}\leq c_n \abs{x-y}^{1-n},
\end{equation}
thus \eqref{eq-fund-sol2} yields the following inequality:
\begin{equation} \label{eq-conv-grad}
\abs{u(x)} \le c_n \int_{\R^n}{\abs{x-y}^{1-n}\abs{\nabla u(y)}} d {y}.
\end{equation}
The Young inequality for convolution with homogeneous kernels (see, for instance, Theorem 1, p. 119 in \cite{undici})
then gives
\begin{equation} \label{eq-Young}
  \| \nabla \Gamma * \nabla u \|_{L^{p^*}(\R^{n})} \le C_{p} \, \| \nabla u \|_{L^{p}(\R^{n})},  \qquad 1 < p < n,
\end{equation}
where $p^* = \frac{pn}{n-p}$ is the Sobolev conjugate of $p$, and $C_p$ is a positive constant which only depends on
$p$ and on the dimension $n$. Here and in the sequel the dependence on $n$ will be often omitted. As a
consequence we find
\begin{equation} \label{eq-pre-Sobolev}
  \| u \|_{L^{p^*}(\mathbb{R}^n)} \le C_{p} \, \| \nabla u \|_{L^{p}(\mathbb{R}^n)},
  \qquad \text{for every} \ u \in C_0^{\infty}(\mathbb{R}^n), \quad 1 < p < n.
\end{equation}
From the above inequality  we plainly obtain the following Sobolev inequality for any open set
$\Omega \subseteq \R^{n}$
\begin{equation} \label{eq-Sobolev}
  \| u \|_{L^{q}(\Omega)} \le C_{p, q} \, \| u \|_{W^{1,p}(\Omega)},
  \qquad \text{for every} \ u \in W_0^{1,p}(\Omega),
\end{equation}
with $1 < p < n$ and  $p \le q \le p^*$. Here $C_{p,q}$ is a positive constant which only depends on $p, q$ and $n$.
By a standard argument \eqref{eq-Young} also gives the Sobolev embedding theorem for $W^{1,p}(\Omega)$ provided
that the boundary of $\Omega$ is sufficiently smooth.

The Morrey inequality (see Theorem \ref{e} below) can be obtained by the representation formula \eqref{eq-fund-sol2},
by using the following fact: there exists a positive constant $M_n$, only depending on $n$, such that

with

\begin{equation}\label{eq-grad-hom}
  | \partial_{x_j} \Gamma (x) - \partial_{x_j} \Gamma (y)  | \le M_n \frac{|x-y|}{|x|^{n}},
  \quad \text{for} \ j= 1, \dots, n,
\end{equation}
for every $x,y \in \R^{n} \setminus \big\{ 0 \big\}$ such that $|x-y| \le |x|/2$. Indeed, a rather simple argument
based on \eqref{eq-grad-hom} provides us with the following bound: if $u \in W_0^{1,p}(\Omega)$,
with $p > n$, then
\begin{equation} \label{eq-Morrey}
  | u(x) - u(y) | \le \widetilde C_{p} \, \| \nabla u \|_{L^{p}( \Omega)} |x-y|^{1 - \frac{n}{p}},
  \qquad \text{for every} \ x, y \in\Omega,
\end{equation}
for some positive constant $\widetilde C_{p}$ only depending on $p$ and $n$.

It is worth noting that the inequality \eqref{eq-grad-hom} can be also used to prove the compactness of the Sobolev
embedding \eqref{eq-Sobolev} for $p < q < p^*$, if $\Omega$ is a bounded open set. As we will see in the sequel, the
following estimates holds for $p < q < p^*$: there exists a positive constant $\widetilde C_{p, q}$ such that
\begin{equation} \label{eq-Sobolev-unif}
  \| u ( h + \cdot) - u \|_{L^{q}(\Omega)} \le \widetilde C_{p, q} \, \| \nabla u \|_{L^{p}(\Omega)} \
  |h|^{n \left( \frac{1}{q}- \frac{1}{p^*} \right)},
\end{equation}
for every $u \in C_0^{\infty}(\Omega)$ and for every $h \in \R^n$ sufficiently small. Note that the exponent in the
right hand side of \eqref{eq-Sobolev-unif} belongs to the interval $]0,1[$ if, and only if, $p < q < p^*$,
then in this case we have
\begin{equation*}
  \| u ( h + \cdot) - u \|_{L^{q}(\Omega)} \to 0 \qquad \text{as} \qquad |h| \to 0.
\end{equation*}
This inequality provides us with the \emph{integral uniform continuity}, which  is needed for the compactness in the
$L^q$ spaces. We also observe that $n \Big( \dfrac{1}{q}- \dfrac{1}{p^*} \Big) \to 1$ as $q \rightarrow p$. We
then retrieve a known result contained for instance in \cite{undici}, Chapter V, Section 3.5.

\medskip

The advantage of the method described above, with respect to other ones, is in that it only requires the
existence of a \emph{fundamental solution} and its \emph{homogeneity} properties. In particular, it applies to the
function spaces introduced by Folland \cite{nove} for the study second order linear differential operators that satisfy
the H\"ormander's condition (see \cite{dieci}). It should be noticed that this approach has also a drawback, in
that it does not provide us with the Sobolev inequality for $p=1$. On the other hand it is unifying, as it gives
the Sobolev and Morrey embedding theorems and a compactness result by using a single representation formula.

To clarify the use of this method to the study of the so-called H\"ormander's operators we next focus on the degenerate
Kolmogorov $\L_0$ on $\R^{2n+1}$, which is one of the simplest examples belonging to this class. Let $\Omega$ be an
open
subset of $\R^{2n+1}$ and let $u$ be a smooth real valued function defined on $\Omega$. We denote the variable of
$\R^{2n+1}$ as follows $z = (x,y,t) \in \R^n \times  \R^n \times  \R$, and we set
\begin{equation}\label{e-Kolmo-0}
  \L_0 u := \Delta_{x} u + \langle x, \nabla_y u \rangle - \partial_t u, \qquad
  \Delta_{x} u := \sum_{j=1}^n \partial_{x_j}^2 u.
\end{equation}
As we will see in the sequel (see equation \eqref{eq-fund-sol-K} below) the function $\Gamma$ defined as
\begin{equation*}
\begin{cases}
\displaystyle
\Gamma(x,y,t) = \frac{\widetilde c_n}{t^{2n}} \exp\left( - \tfrac{|x|^2}{t} - 3 \tfrac{\langle x, y
\rangle}{t^2} - 3 \tfrac{|y|^2}{t^3}  \right), &  \text{for} \ (x,y,t) \in \R^{2n} \times ]0, + \infty[, \\
\Gamma(x,y,t) = 0, & \text{for} \ (x,y,t) \in \R^{2n} \times ]- \infty, 0],
\end{cases}
\end{equation*}
is the fundamental solution of $\L_0$. Here $\widetilde c_n = \frac{{3}^{n/2}}{(2 \pi)^n}$. In particular, in analogy
with the heat equation, we have that the function $u$ defined as
\begin{equation} \label{eq-fund-sol-K0}
\begin{split}
 u(x,y,t) = & \int_{\R^{2n}} \Gamma(x - \xi,y + t \xi -\eta,t- t_0) \varphi(\xi, \eta) d \xi \, d \eta \, - \\
 & \int_{\R^{2n} \times ]t_0,t[}  \Gamma(x - \xi,y + (t- \tau) \xi -\eta,t- \tau)
 f(\xi, \eta, \tau) d \xi \, d \eta \, d \tau
\end{split}
\end{equation}
is a solution to the following Cauchy problem
\begin{equation*}
\begin{cases}
 \L_0 u = f  & \text{in} \quad \R^{2n} \times ]t_0, + \infty[, \\
 u_{\mid t=t_0} = \varphi & \text{in} \quad \R^{2n}.
\end{cases}
\end{equation*}
whenever $f$ and $\varphi$ are bounded continuous functions.

A remarkable fact is that a kind of convolution is hidden in the expression \eqref{eq-fund-sol-K0}. More specifically,
we define the operation ``$\circ$'' by setting
\begin{equation} \label{eq-group-K0}
  (x,y,t) \circ (\xi, \eta, \tau) := (x + \xi, y + \eta + \tau x, t + \tau), \qquad (x,y,t), (\xi, \eta, \tau) \in
\R^{2n+1},
\end{equation}
and we note that $\left( \R^{2n+1}, \circ \right)$ is a non commutative group. The identity of the group is $(0,0,0)$
and the inverse of $(x,y,t)$ is $(-x, -y + xt, -t)$. With this notation, it is easy to check that the expression
appearing in \eqref{eq-fund-sol-K0} can be written as follows
\begin{equation*}
  (x - \xi,y + (t- \tau) \xi -\eta,t- \tau) =  (\xi, \eta, \tau)^{-1} \circ (x,y,t).
\end{equation*}
Moreover, the group $\left( \R^{2n+1}, \circ \right)$ is homogeneous with respect to the dilation defined as $d_r
(x,y,t) := \left( r x, r^3 y, r^2 t\right)$, in the sense that
\begin{equation} \label{eq-dil-K0}
  d_r \big( (x,y,t) \circ (\xi, \eta, \tau) \big) = d_r (x,y,t) \circ d_r (\xi, \eta, \tau),
  \qquad (x,y,t), (\xi, \eta, \tau) \in \R^{2n+1}, r>0.
\end{equation}
This algebraic structure was introduced and studied by Lanconelli and Polidoro in \cite{dodici}. In \cite{dodici}
it was also noticed that $\Gamma$ is homogeneous of degree $- 4n$ with respect to $\left( d_t \right)_{r>0}$, that is
\begin{equation} \label{eq-dilGamma-K0}
  \Gamma \big(d_r (x,y,t) \big) = \frac{1}{r^{4n}} \Gamma (x,y,t), \qquad (x,y,t) \in \R^{2n+1}, r>0.
\end{equation}
Moreover, if we let $z = (x,y,t), \zeta = (\xi, \eta, \tau)$, then \eqref{eq-fund-sol-K0} can be written as follows
\begin{equation} \label{eq-fund-sol-K0-bis}
 u(z) = \int_{\R^{2n}} \Gamma((\xi,\eta,t_0)^{-1} \circ z) \varphi(\xi, \eta) d \xi \, d \eta \, - \int_{\R^{2n}
\times ]t_0,t[}  \Gamma(\zeta^{-1} \circ z)  f(\zeta) d \zeta.
\end{equation}
In particular, if $u \in C_0^\infty(\R^{2n+1})$ and supp$(u) \subset \big\{ t > t_0 \big\}$, then we have that
\begin{equation} \label{eq-fund-sol1-bis}
  u(z) =  -  \int_{\mathbb{R}^{2n+1}}\Gamma(\zeta^{-1} \circ z) \L_0 u(\zeta) \,
  d \zeta, \quad \text{for every} \ z \in \R^{2n+1},
\end{equation}
which is analogous to \eqref{eq-fund-sol1}. Summarizing: the operation in \eqref{eq-fund-sol1-bis} is considered here
as
a \emph{convolution} with respect to the non-Euclidean operation ``$\circ$'' defined in \eqref{eq-group-K0}, with a
kernel $\Gamma$ that is homogeneous whit respect to the \emph{anisotropic} dilation $d_r$. Based on this representation
formula, we prove Sobolev and Morrey theorems for solutions to Kolmogorov equations in divergence form $\L u =
\text{\rm
div}_{x} F + f$, where
\begin{equation}\label{e-Kolmo-1}
  \L u := \text{\rm div}_{x} \left(A(z) \nabla_x u \right) + \langle x, \nabla_y u \rangle - \partial_t u.
\end{equation}
Here $A$ is a $n \times n$ symmetric matrix with bounded and measurable coefficients and, for every vector field $F
\in C^1(\R^{2n+1}, \R^n)$ we denote $\text{\rm div}_{x} F(x,y,t) := \sum_{j=1}^n \partial_{x_j} F_j(x,y,t)$.
In order to simplify our treatment, we suppose that $F = 0$ and $f=0$, so that $u$ is a solution of $\L u = 0$.
In this case we have that $\L_0 u = \text{\rm div}_{x}(I_n - A) \nabla_x u$, where $I_n$ denotes the $n \times n$
identity matrix. Then, an integration by parts in \eqref{eq-fund-sol1-bis} gives
\begin{equation} \label{eq-fund-sol2-bis}
  u(z) =   \int_{\mathbb{R}^{2n+1}} \langle (I_n - A(\zeta)) \nabla_\xi \Gamma(\zeta^{-1} \circ z),
  \nabla_{\xi} u(\zeta)\rangle \, d \zeta,
\end{equation}
for every solution $u$ to $\L u = 0$. It is known that the derivatives $\partial_{\xi_1} \Gamma, \dots,
\partial_{\xi_n} \Gamma$ are homogeneous functions of degree $- (2n+1)$ with respect to the dilation $\left( d_r
\right)_{r>0}$. Moreover, the coefficients of the matrix $I_n - A$ are bounded, then the above identity provides us
with
the analogous of \eqref{eq-fund-sol2} for the solutions $u$ to the equation $\L u = 0$.

We point out that only the derivatives with respect to the first $n$ variables of the gradient of $u$ appear in
the representation formula \eqref{eq-fund-sol2-bis}, then a Sobolev inequality holding for all functions cannot
be obtained from \eqref{eq-fund-sol2-bis}, because of the lack of information on the remaining $n$ direction.
Nevertheless, this formula is used by Cinti, Pascucci and Polidoro in  \cite{sei,sette} to prove a Sobolev embedding
theorem for \emph{solutions} to the Kolmogorov equation $\L u = 0$.  Indeed, in \cite{sei,sette} the Sobolev theorem
for
\emph{solutions} is combined  with a Caccioppoli inequality, still for \emph{solutions}, in order to apply the Moser's
iterative method and prove an $L^\infty_{{\rm loc}}$ estimate for the solutions to $\L u = 0$.
We also recall that a Morrey result for the solutions to $\L u = \text{\rm div}_{x} F$ was proven by Manfredini and
Polidoro in \cite{otto}, and later by Polidoro and Ragusa in \cite{otto.1} by the same method used here.

In this note we are concerned with the compactness of the Sobolev embedding for the solutions to $\L u = 0$ for a
family of degenerate Kolmogorov equations, defined on $\mathbb{R}^{N+1}$, that will be still denoted by $\L$. As
we will see in Section 3, the operator \eqref{e-Kolmo-1} is the prototype of this family of degenerate operators, and
in this case, $N = 2n$. In Section 3 we introduce the notation that will be used in the following part of this
introduction, and we will state the conditions (H.1) and (H.2) that ensure that the \emph{principal part} $\L_0$ of
$\L$ has a smooth fundamental solution $\Gamma$, which is invariant with respect to a translation analogous to
\eqref{eq-group-K0}, and homogeneous of degree $-Q$, with respect to a dilation analogous to \eqref{eq-dil-K0}. We will
refer to the positive integer $Q+2$ as \emph{homogeneous dimension} of the space $\rnn$ and plays the role of $n$ in
the
Euclidean setting $\R^n$ where the elliptic operators are studied. In the sequel $p^*$ and $p^{**}$ denote the positive
numbers such that
\begin{equation} \label{eq-p-pstar}
  \frac{1}{p^*} = \frac{1}{p} - \frac{1}{Q+2}, \qquad \frac{1}{p^{**}} = \frac{1}{p} - \frac{2}{Q+2}.
\end{equation}
Clearly, $p^*$ and $p^{**}$ are finite and positive whenever $1 \le p < Q+2$ and $1 \le p < \dfrac{Q+2}{2}$,
respectively.

Our main result is the following Theorem. It provides us with some estimates of the convolution of a function
belonging to some $L^p$ space with the fundamental solution $\Gamma$ and with its derivatives $\partial_{x_j} \Gamma$,
$j = 1, \dots, m_0$, with $m_0 \le N$. These estimates, applied to the representation formula for solutions to
$\L u = 0$ given in Theorem \ref{th-rappr}, yield Sobolev theorems, Morrey theorems and the compactness of the Sobolev
embedding.

\begin{theorem} \label{main-th-1}
Let $\L$ be an operator in the form \eqref{ultraparabolica}, satisfying the hypotheses {\rm (H.1)} and {\rm (H.2)} in
Section 3, and let $\Gamma$ be the fundamental solution of its \emph{principal part}. Let also $Q+2$ be the homogeneous
dimension of the space $\mathbb{R}^{N+1}$, and let $p$ be such that $1 \le p < + \infty$. For every $f, g_j \in
L^p(\rnn)$ we let $u, v_j$ be defined as follows
\begin{equation*}
  u(z) = \int_{\rnn} \Gamma(\zeta^{-1} \circ z) f(\zeta) \, {\rm d} \zeta,
  \quad v_j(z) = \int_{\rnn} \partial_{x_j} \Gamma(\zeta^{-1} \circ z) g_j (\zeta) \, {\rm d} \zeta,
  \quad j = 1, \dots, m_0.
\end{equation*}
Then, for every $j = 1, \dots, m_0$ we have:
\begin{itemize}
  \item (Sobolev) if $1 < p < Q+2$, then there exists a positive constant $C_p$ such that
\begin{equation*}
  \| v_j \|_{L^{p^*}(\rnn)} \le C_{p} \, \|g_j \|_{L^{p}(\rnn)},
\end{equation*}
 \item (Compactness) if moreover $p < q < p^*$, then there exists a positive constant
$\widetilde C_{p, q}$ such that
\begin{equation*}
  \| v_j ( \cdot \circ h) - v_j \|_{L^{q}(\rnn)} \le \widetilde C_{p, q} \, \| g_j \|_{L^{p}(\rnn)} \
  \|h\|^{(Q+2) \left( \frac{1}{q}- \frac{1}{p^*} \right)},
\end{equation*}
for every $h \in \rnn$,
  \item (Morrey) if  $p > Q+2$, then there exists a positive constant $\widetilde C_p$ such that
\begin{equation*}
   | v_j(z) - v_j(\zeta) | \le \widetilde C_{p} \, \|  g_j \|_{L^{p}( \rnn)} \|\zeta^{-1} \circ z\|^{1 -
\frac{Q+2}{p}}, \quad \text{for every} \ z, \zeta \in \rnn.
\end{equation*}
 \end{itemize}
 We also have
\begin{itemize}
  \item (Sobolev) if $1 < p < \dfrac{Q+2}{2}$, then there exists a positive constant $C_p$ such
that
\begin{equation*}
  \| u \|_{L^{p^{**}}(\rnn)} \le C_{p} \, \| f \|_{L^{p}(\rnn)},
\end{equation*}
  \item (Compactness) if $p^* < q < p^{**}$, then there exists a positive constant $\widetilde
C_{p, q}$ such that
\begin{equation*}
  \| u ( \cdot \circ h) - u \|_{L^{q}(\rnn)} \le \widetilde C_{p, q} \, \| f \|_{L^{p}(\rnn)} \
  \|h\|^{(Q+2) \left( \frac{1}{q}- \frac{1}{p^{**}} \right)},
\end{equation*}
for every $h \in \rnn$,
  \item (Morrey) if $\dfrac{Q+2}{2} < p < Q+2$, then there exists a positive constant $\widetilde C_p$
such that
\begin{equation*}
     | u(z) - u(\zeta) | \le \widetilde C_{p} \, \|  f \|_{L^{p}( \rnn)} \|\zeta^{-1} \circ z\|^{2 -
\frac{Q+2}{p}},   \qquad \text{for every} \ z, \zeta \in \rnn.
\end{equation*}
\end{itemize}
\end{theorem}

From the above result and a representation formula for the solution to $\L u = 0$ we obtain the following result.

\begin{theorem} \label{main-th-2} Let $\Omega$ be an open set of $\rnn$, and let $u$ be a weak solution to
$\L u = 0$ in $\Omega$. Suppose that $u, \partial_{x_1} u, \dots, \partial_{x_{m_0}} u \in L^p
(\Omega)$. Then for every compact set $\K \subset \Omega$, there exist a positive constant $\widetilde \varrho$
such that we have:
\begin{itemize}
  \item (Sobolev embedding) if $1 < p < Q+2$, then there exists a positive constant $C_p$ such that
\begin{equation*}
   \| u \|_{L^{p^*}(\K)} \le \,  C_{p} \bigg( \| u \|_{L^{p}(\Omega)} + \sum_{j=1}^{m_0}\| \partial_{x_j} u
\|_{L^{p}(\Omega)} \bigg),
\end{equation*}
 \item (Compactness) if moreover $p < q < p^*$, then there exists a positive constant $\widetilde C_{p, q}$ such that
\begin{equation*}
  \| u ( \cdot \circ h) - u \|_{L^{q}(\K)} \le \widetilde C_{p, q}
  \bigg( \| u \|_{L^{p}(\Omega)} + \sum_{j=1}^{m_0}\| \partial_{x_j} u \|_{L^{p}(\Omega)} \bigg) \
  \|h\|^{(Q+2) \left( \frac{1}{q}- \frac{1}{p^*} \right)},
\end{equation*}
for every $ h \in \rnn$ such that $\| h \| \le \widetilde \varrho$,
\item (Morrey embedding) if  $p > Q+2$, then there exists a positive constant $\widetilde C_p$ such that
\begin{equation*}
   | u (z) - u (\zeta) | \le  \widetilde C_{p}
   \bigg( \| u \|_{L^{p}(\Omega)} + \sum_{j=1}^{m_0}\| \partial_{x_j} u \|_{L^{p}(\Omega)} \bigg)
   \|\zeta^{-1} \circ z\|^{1 - \frac{Q+2}{p}},
\end{equation*}
for every $\ z, \zeta \in \K$ such that $\|\zeta^{-1} \circ z\| \le \widetilde \varrho$.
 \end{itemize}
\end{theorem}

\medskip

The following Theorem is related to the main result of the article \cite{quattordici} by Bouchut, where the regularity
of the solution of the kinetic equation
\begin{equation} \label{eq-kinetic}
  \partial_t f + \langle v, \nabla_x f \rangle = g, \qquad (t,x,v) \in \Omega \subseteq \R \times \R^{n} \times \R^{n},
\end{equation}
is considered. Note that the differential operator appearing in the left hand side of \eqref{eq-kinetic} agrees
with the first order part of $\L$ defined in \eqref{e-Kolmo-1}. Actually, the notation of the following result refers
to this operator, and, in particular, the homogeneous dimension of the space $\R^{2n + 1}$ is in this case $Q+2 = 4
n + 2$.

\medskip

\begin{theorem} \label{main-th-3} Let $\Omega$ be an open set of $\R^{2n+1}$, and let $f \in L^2_\text{\rm
loc}(\Omega)$ be a weak solution to  \eqref{eq-kinetic}. Suppose that $g, f, \partial_{v_1} f, \dots, \partial_{v_{n}}
f \in L^p
(\Omega)$. Then for every compact set $\K \subset \Omega$, there exist a positive constant
$\widetilde \varrho$
such that we have:
\begin{itemize}
  \item (Sobolev embedding) if $1 < p < 4n+2$, then there exists a positive constant $C_p$ such that
\begin{equation*}
   \| f \|_{L^{p^*}(\K)} \le \,  C_{p} \bigg( \| g \|_{L^{p}(\Omega)} + \| f \|_{L^{p}(\Omega)} + \sum_{j=1}^{n}\|
\partial_{v_j} f \|_{L^{p}(\Omega)} \bigg),
\end{equation*}
 \item (Compactness) if moreover $p < q < p^*$, then there exists a positive constant $\widetilde C_{p, q}$ such that
\begin{equation*}
  \| f ( \cdot \circ h) - f \|_{L^{q}(\K)} \le \widetilde C_{p, q}
  \bigg( \| g \|_{L^{p}(\Omega)} + \| f \|_{L^{p}(\Omega)} + \sum_{j=1}^{n}\| \partial_{v_j} f \|_{L^{p}(\Omega)}
\bigg)
  \ \|h\|^{(4n + 2) \left( \frac{1}{q}- \frac{1}{p^*} \right)},
\end{equation*}
for every $ h \in \R^{2n+1}$  such that $\| h \| \le \widetilde \varrho$,
\item (Morrey embedding) if  $p > 4n + 2$, then there exists a positive constant $\widetilde C_p$ such that
\begin{equation*}
   | f (z) - f (\zeta) | \le  \widetilde C_{p}
   \bigg( \| g \|_{L^{p}(\Omega)} + \| f \|_{L^{p}(\Omega)} + \sum_{j=1}^{n}\| \partial_{v_j} f \|_{L^{p}(\Omega)}
   \bigg)  \|\zeta^{-1} \circ z\|^{1 - \frac{4n + 2}{p}},
\end{equation*}
for every $\ z, \zeta \in \K$ such that $\|\zeta^{-1} \circ z\| \le \widetilde \varrho$.
 \end{itemize}
\end{theorem}

The proof of Theorems \ref{main-th-2} and \ref{main-th-3} is given in Section 4.

\medskip

We next give some comments to our main results. We still refer here to the notation relevant to the operator $\L$
defined in \eqref{e-Kolmo-1}, and to the representation formula \eqref{eq-fund-sol2-bis}. As we said above, it
holds for solutions to $\L u = 0$ then, for this reason, it seems to be weaker than the usual Sobolev inequality. On
the other hand, due to the strong degeneracy of the operator $\L$, its natural Sobolev space $W^{1,p}_{\L}$ is the space
of the functions $u \in L^p$ with weak derivatives $\partial_{x_1} u, \dots, \partial_{x_n} u \in L^p$. In particular,
it is impossible to prove a Sobolev inequality unless some information is given on $u$ with respect to the remaining
variables $y_1, \dots, y_n$ and $t$. We obtain this missing information from the fact that $u$ is a solution to $\L u =
0$ (or, in a more general case, to $\L u = \text{\rm div}_{x} F + f$). We also note that the regularity property of the
operator $\L$ is quite unstable. Indeed, let us fix any $x_0 \in \R^n$ and consider the operator $\widetilde \L_0$,
acting on $(x,y,t) \in \R^{2n+1}$ as follows
\begin{equation*}
  \widetilde \L_0 u := \Delta_{x} u + \langle x_0, \nabla_y u \rangle - \partial_t u.
\end{equation*}
Its natural Sobolev spaces agrees with that of $\L$, however it is known that a fundamental solution for $\widetilde
\L_0$ does not exists and our method for the proof of the Sobolev inequality fails in this case. Actually, it is not
difficult to check that the Sobolev inequality does not hold for the solutions to $\widetilde \L_0 u = 0$.

We conclude this discussion with a simple remark. Also when we consider the more familiar uniformly parabolic
equations, we find that the natural Sobolev space only contains the \emph{spatial} derivatives, and it is not possible
to find a simple natural space for the time derivative. As a matter of facts, several regularity results for parabolic
equations depend on some \emph{fractional Sobolev spaces}. The situation becomes more complicated when we consider
second order PDEs with non-negative characteristic form analogous to $\L$. An alternative approach to our method, that
only relies on a representation formula in terms of the fundamental solution, is the use of \emph{fractional Sobolev
spaces} (we refer to the articles by Bochut \cite{quattordici}, see also Golse, Imbert, Mouhot and Vasseur
\cite{tredici}) to recover the missing information with respect to the variables $y_1, \dots, y_n$ and $t$.

\medskip

This article is organized as follows. In Section 2 we give a comprehensive proof of the Sobolev embedding, of its
compactness, and the Morrey embedding, following the method above outlined. In Section 3 we recall the tools of the
Real Analysis on Lie groups we need to prove Theorem \ref{main-th-1}, and we give its proof. In Section 4 we discuss
some  applications of Theorem \ref{main-th-1} to the solutions of $\L u = 0$. Section 5 contains some comments about
the possible extension of Theorem \ref{main-th-1} to a family of more general operators considered by Cinti and
Polidoro in \cite{quindici}.

\section{Continuous and compact embeddings: the Euclidean case}
\setcounter{section}{2} \setcounter{equation}{0}
\setcounter{theorem}{0}

In this Section we give a comprehensive proof of the Sobolev embedding \eqref{eq-Sobolev}, the Morrey embedding
\eqref{eq-Morrey}, and of the inequality \eqref{eq-Sobolev-unif} from which the compactness of the Sobolev
embedding follows. As said in the Introduction, all these results rely on the representation formula
\eqref{eq-fund-sol2}, which gives the bound \eqref{eq-conv-grad} that we recall below
\begin{equation*}
  \abs{u(x)} \le c_n \int_{\R^n}{\abs{x-y}^{1-n}\abs{\nabla u(y)}} \, d {y},
  \qquad \text{for every} \ u \in C_0^\infty (\R^n).
\end{equation*}
With this aim, we first recall the weak Young inequality that gives the Sobolev embedding, then we prove
\eqref{eq-grad-hom} and we deduce from this and \eqref{eq-conv-grad} the Morrey embedding
\eqref{eq-Morrey}, and that stated in the inequality \eqref{eq-Sobolev-unif}.

\subsection{Some preliminary results}

For a given positive $\alpha$
we denote by $K_\alpha$ any continuous homogeneous function of degree $- \alpha$, that is a function satisfying
\begin{equation*}
  K_\alpha(r x)= r^{- \alpha } K_\alpha (x), \quad
  \text{for every} \  x \in \mathbb{R}^n \setminus \{0\}, \ \text{and} \ r >0.
\end{equation*}
We easily see that
\begin{equation} \label{eq-calpha}
  \left| K_\alpha(x) \right| \le \frac{c_\alpha}{|x|^{\alpha}},
  \quad \text{for every} \  x \in \mathbb{R}^n \setminus \{0\},
\end{equation}
where $c_\alpha := \max_{|x| = 1} \left| K_\alpha(x) \right|$.

In particular, $K_\alpha$ belongs to the space $L^q_{\rm weak}(\R^n)$, for $q = \dfrac{n}{\alpha}$, that is
\begin{equation} \label{eq-Lpw}
  \text{\rm meas} \big\{ x \in \R^n \mid | K_\alpha (x) | \ge \lambda  \big\} \le \left( \frac{C}{\lambda}\right)^q,
\quad \text{for every} \quad \lambda >0,
\end{equation}
for some non-negative constant $C$. Here {\rm meas} $E$ denotes the Lebesgue measure of the set $E$. Moreover we
define the \emph{seminorm} of $K_\alpha$ as follows
\begin{equation*}
  \left\| K_\alpha \right\|_{L^q_{\rm weak}(\R^n)} := \inf \big\{ C \ge 0 \mid \text{\eqref {eq-Lpw} holds} \big\}.
\end{equation*}
From \eqref{eq-calpha} it plainly follows that $C \le c_\alpha \omega_n^{\alpha/n}$. We next recall two elementary
inequalities that will be useful in the sequel. For every $R>0$ we have that:
  \begin{itemize}
    \item $K_\alpha \in L^q\big(\big\{ x \in \R^n \mid |x| \le R \big\}\big)$ if, and only if, $q < \dfrac{n}{\alpha}$.
Moreover, there exists a positive constant $c_{\alpha,q}$, only depending on $K_\alpha, n$ and $q$, such that
\begin{equation}\label{eq-pa>n}
  \left\| K_\alpha \right\|_{L^q(\{ x \in \R^n \mid |x| \le R\})} \le
  c_{\alpha,q} R^{\frac{n}{q} - \alpha},
\end{equation}
    \item $K_\alpha \in L^q\big(\big\{ x \in \R^n \mid |x| \ge R \big\}\big)$ if, and only if, $q > \dfrac{n}{\alpha}$.
Moreover, there exists a positive constant $c_{\alpha,q}$, only depending on $K_\alpha, n$ and $q$, such that
\begin{equation}\label{eq-pa<n}
  \left\| K_\alpha \right\|_{L^q(\{ x \in \R^n \mid |x| \ge R \})} \le
  c_{\alpha,q} R^{\frac{n}{q} - \alpha}.
\end{equation}
  \end{itemize}

The following weak Young inequality holds (see Theorem 1, p. 119 in \cite{undici}, where this result is referred to
as Hardy-Littlewood-Sobolev theorem for fractional integration).
\begin{theorem} \label{Th-weak-Young}
Let $K_\alpha$ be a continuous  homogeneous function of degree $- \alpha$, with $0 < \alpha < n$. Let $p, q$ be such
that $1 \le p < q < + \infty$ and that $1 + \dfrac{1}{q} = \dfrac{1}{p} + \dfrac{\alpha}{n}$. Then, for every $f \in
L^p(\R^n)$ the integral $K_\alpha * f (x)$ is convergent for almost every $x \in \R^n$. Moreover,

\noindent - if $p>1$, then there exists $C_{\alpha, p} >0$ such that $\norm{K_\alpha * f}_{L^q (\mathbb{R}^n)} \leq
C_{\alpha, p} \norm{f}_{L^p (\mathbb{R}^n)}$,

\noindent - if $p=1$, then there exists $C_{\alpha,1} >0$ such that $\left\| K_\alpha * f \right\|_{L^q_{\rm
weak}(\R^n)} \le C_{\alpha, 1} \norm{f}_{L^1 (\mathbb{R}^n)}$.

\end{theorem}

In order to prove the Morrey embedding \eqref{eq-Morrey} and the compactness estimate \eqref{eq-Sobolev-unif}, we
state and prove the following lemma.

\begin{lemma} \label{14}
Let $K_\alpha \in C^1(\mathbb{R}^n\backslash \lbrace0\rbrace)$ be any homogeneous function of degree $- \alpha$, with
$0<\alpha<n$. Then there exists a positive constant $M_\alpha$ such that
\[
\abs{K_\alpha(x)-K_\alpha(y)}\leq M_\alpha \frac{\abs{x-y}}{\abs{x}^{\alpha+1}}, \quad \text{for every} \quad
x,y\in\mathbb{R}^n\backslash \lbrace 0 \rbrace \ \text{such that} \ \abs{x-y}\leq \dfrac{\abs{x}}{2}.
\]
\end{lemma}
\noindent \textit{Proof.} We first prove the result for $x$ such that $\abs{x}=1$. In this case
$\abs{x-y}\leq\dfrac{1}{2}$
and by the Mean Value Theorem there exists $\theta\in(0,1)$ such that
\begin{equation*}
|K_\alpha(x)-K_\alpha(y)|=|\scalare{(x-y)}{\nabla K_\alpha(\theta x+(1-\theta y))}|\leq M_\alpha\abs{x-y},
\end{equation*}
where
\begin{equation} \label{Emme}
M_\alpha=\max_{\frac{1}{2}\leq |z| \leq \frac{3}{2}}{\abs{\nabla K_\alpha(z)}}.
\end{equation}

Consider now a general choice of $x,y\in\mathbb{R}^n\backslash \lbrace0\rbrace$ with $\abs{x-y}\leq\dfrac{\abs{x}}{2}$.
Being $K_\alpha$ homogeneous of degree $- \alpha$, we obtain
\[
\abs{K_\alpha(x)-K_\alpha(y)}=\frac{1}{\abs{x}^{\alpha}} \left
|K_\alpha\left(\frac{x}{|x|}\right)-K_\alpha\left(\frac{y}{|x|}\right) \right |

{\leq} \frac{M_\alpha}{\abs{x}^{ \alpha}}\left |\frac{x}{|x|} - \frac{y}{|x|} \right |=
M_\alpha\frac{\abs{x-y}}{\abs{x}^{\alpha+1}},
\]
being $M_\alpha$ as in \eqref{Emme}, because $\left| \dfrac{x}{|x|}\right|= 1$. $\Box$

\medskip

In order to prove the Morrey embedding \eqref{eq-Morrey} and the compactness of the Sobolev embedding
for $p < q < p^*$ we rely on the following argument. We choose any $u \in C_0^\infty(\R^n), h \in \R^n$ and
we set
\begin{equation} \label{eq-vx}
  v(x) := u(x + h) - u(x), \qquad \text{for every} \quad x \in \R^n,
\end{equation}
then
\begin{eqnarray}

v(x) &=&\integrale{\lbrace y\in\mathbb{R}^n:\abs{x+h-y} \geq
2\abs{h}\rbrace}{} {\langle \nabla\Gamma(x+h-y)-\nabla\Gamma(x-y), \nabla u(y) \rangle}{y} \nonumber \\
& + & \integrale{\lbrace y\in\mathbb{R}^n:\abs{x+h-y} < 2\abs{h}\rbrace}{}{\langle{\nabla\Gamma(x+h-y)},{\nabla
u(y)} \rangle} {y} \nonumber\\
& + & \integrale{\lbrace y\in\mathbb{R}^n:\abs{x+h-y} < 2\abs{h}\rbrace}{}{- \langle {\nabla\Gamma(x-y)}, {\nabla
u(y)}\rangle}{y} =: I_A (x) + I_B (x)  + I_C (x). \label{usplit}
\end{eqnarray}
We next rely on Lemma \ref{14} and on \eqref{eq-grad-fund-sol-lapl} to estimate the terms $I_A, I_B$ and $I_C$ as
follows
\begin{equation} \label{eq-bound-ABC}
\begin{split}
 | I_A (x) | & \le M |h|   \integrale{\lbrace y\in\mathbb{R}^n:\abs{x+h-y} \geq
2\abs{h}\rbrace}{}{\frac{1}{\abs{x+h-y}^n}\abs{\nabla u(y)}}{y},  \\
 | I_B (x) | & \le c_n  \integrale{\lbrace y\in\mathbb{R}^n:\abs{x+h-y} <
2\abs{h}\rbrace}{}{\frac{1}{\abs{x+h-y}^{n-1}}\abs{\nabla u(y)}}{y}, \\
 | I_C (x) | & \le c_n  \integrale{\lbrace y\in\mathbb{R}^n:\abs{x+h-y} <
2\abs{h}\rbrace}{}{\frac{1}{\abs{x-y}^{n-1}}\abs{\nabla u(y)}}{y},
\end{split}
\end{equation}
where $M := \max\limits_{\frac{1}{2}\leq |z| \leq \frac{3}{2}, j,k = 1 , \dots, n}{\abs{ \partial^2_{x_j x_k}
\Gamma(z)}}$.

\subsection{The Sobolev and Morrey embedding theorems}

As we said in the Introduction, Theorem \ref{Th-weak-Young} combined with \eqref{eq-conv-grad} immediately yields the
following result
\begin{theorem} \label{Th-Sobolev}
Let $1<p<n$. Then there exists $C_p>0$ such that:
\begin{equation*}
\norm{u}_{p^*}\leq C_p \norm{\nabla u}_p  \quad \text{for every} \quad  u \in L^1_{\text {\rm
loc}}(\mathbb{R}^n) \quad  \text{such that} \quad \nabla u \in L^{p} (\mathbb{R}^n),
\end{equation*}
where $p^*=\dfrac{np}{n-p}$.
\end{theorem}

We next turn our attention on the Morrey's Theorem.

\begin{theorem}[Morrey's Theorem] \label{e}
Let $u:\mathbb{R}^n\rightarrow\mathbb{R}$ and $p>n$. If $\nabla u \in \sob{p}$, then $u$ is continuous and
\begin{equation} \label{Morrey-ineq}
\abs{u(x+h)-u(x)}\leq  C_{n,p} \norm{\nabla u}_p\abs{h}^{1-\frac{n}{p}}, \qquad \text{for every} \quad x, h \in \R^n,
\end{equation}
for some positive constant $C_{n,p}$ depending only on $p$ and $n$.

In particular $\sobolev{p}$ is continuously embedded in the space of H\"older continuous
functions $C^\beta(\mathbb{R}^n)$, with $\beta=1-\frac{n}{p}$.
\end{theorem}

\noindent \textit{Proof.} Consider a function $u \in C_0^\infty(\mathbb{R}^n)$, let $x, h \in \R^n$, and $v(x)$ be
the function defined in \eqref{usplit}. We next estimate  $I_A, I_B$ and $I_C$ by using the H\"older inequality.

From the first inequality in \eqref{eq-bound-ABC} and \eqref{eq-pa<n}, with $\alpha = n$ and $q = \dfrac{p}{p-1}$, we
obtain
\begin{equation*}
\begin{split}
    | I_A (x) | \le & \, M |h| \norm{\nabla u}_{L^p(\mathbb{R}^n)}
    \left\|{\frac{1}{|z|^n}}\right\|_{L^q(\{ |z| \ge 2 |h|\})} \\
    = & \, M |h| \norm{\nabla u}_{L^p(\mathbb{R}^n)} c_{n,q} (2|h|)^{\frac{n(p-1)}{p} - n} =
    M_A\norm{\nabla u}_{L^p(\mathbb{R}^n)}\abs{h}^{1-\frac{n}{p}},
\end{split}
\end{equation*}
for some positive constant $M_A$ only depending on $M, p$ and $n$. Moreover, from the second inequality in
\eqref{eq-bound-ABC} and \eqref{eq-pa>n}, with $\alpha = n-1$ and $q = \dfrac{p}{p-1}$,
we find
\begin{equation*}
\begin{split}
    | I_B (x) | \le & \, c_n \norm{\nabla u}_{L^p(\mathbb{R}^n)}
    \left\|{\frac{1}{|z|^{n-1}}}\right\|_{L^q( \{|z| \le 2 |h|\})} \\
    = & \, c_n \norm{\nabla u}_{L^p(\mathbb{R}^n)} c_{n-1,q} (2|h|)^{\frac{n(p-1)}{p} - n+1} =
    M_B \norm{\nabla u}_{L^p(\mathbb{R}^n)}\abs{h}^{1-\frac{n}{p}},
\end{split}
\end{equation*}
where $M_B$ is a positive constant only depending on $p$ and $n$. Finally the last term in \eqref{usplit} can be
estimated similarly to the second one, observing that $\abs{x-y}\leq \abs{x+h-y}+\abs{h}< 3\abs{h}$, thus getting
\begin{equation*}
\begin{split}
    | I_C (x) | \le & c_n \norm{\nabla u}_{L^p(\mathbb{R}^n)}
    \left\|{\frac{1}{|z|^{n-1}}}\right\|_{L^q( \{|z| \le 3 |h|\})} = M_C
\norm{\nabla u}_{L^p(\mathbb{R}^n)}\abs{h}^{1-\frac{n}{p}},
\end{split}
\end{equation*}
for some positive constant $M_C$. We note that  the $L^q$ norms of the functions $z \mapsto |z|^{-n}$ and $z
\mapsto |z|^{-n+1}$ appearing in the above estimates are finite thanks to the assumption $p > n$.
Then \eqref{Morrey-ineq} is obtained with $C_{n,p} := M_A + M_B + M_C$.  This proves our claim for $u \in
C_0^\infty(\R^n)$. The general case follows by a density argument. \hfill $\Box$

\medskip

We next prove the compactness of the Sobolev embedding \eqref{eq-Sobolev-unif} for $p < q < p^*$ starting again from
\eqref{usplit}. Here we use the Young inequality instead of the H\"older inequality.

\begin{theorem} \label{Uniform continuity}
Let $p, q \ge 1$ be such that $p < q < p^* := \frac{np}{n-p}$. Then there exists a positive constant $C_{p,q}$
depending on $n, p, q$ such that
\[
\norm{u(h + \cdot)-u}_{L^q(\mathbb{R}^n)}\leq  C_{p,q} \, \|\nabla u\|_{L^p(\mathbb{R}^n)}
\, \abs{h}^{n \left(\frac{1}{q} - \frac{1}{p^*}\right)}.
\]
for every $u \in C^{\infty}_0(\mathbb{R}^n)$ and for any $h \in \mathbb{R}^n$. In particular, as long as $q < p^*$, we
have
\[
\norm{u(h + \cdot)-u}_{L^q(\mathbb{R}^n)} \rightarrow 0 \qquad \textnormal{as} \qquad |h| \rightarrow 0.
\]
\end{theorem}

\noindent \textit{Proof.} Consider a function $u \in C_0^\infty(\mathbb{R}^n)$, let $x, h \in \R^n$, and $v(x)$ be
the function defined in \eqref{usplit}. We next estimate the $L^q$ norm of $I_A, I_B$ and $I_C$ by using the Young
inequality. To this aim we introduce the exponent $r$ defined by the identity
\begin{equation} \label{esponente-r}
1 + \frac{1}{q} = \frac{1}{r} + \frac{1}{p},

\end{equation}
and we note that
\begin{equation} \label{condizionesur}
1 < r < \frac{n}{n-1} \,\, \Leftrightarrow \,\, p < q < p^*.
\end{equation}
From the first inequality in \eqref{eq-bound-ABC} and \eqref{eq-pa<n}, with $\alpha = n$, we obtain
\begin{equation*}
\begin{split}
    \left\| I_A \right\|_{L^q(\R^n)} \le & \, M |h| \norm{\nabla u}_{L^p(\mathbb{R}^n)}
    \left\|{\frac{1}{|z|^n}}\right\|_{L^r(\{ |z| \ge 2 |h| \})} \\
    = & \, M |h| \norm{\nabla u}_{L^p(\mathbb{R}^n)} c_{n,r} (2|h|)^{\frac{n}{r} - n}
    = C_A(n,r) |h|^{n\left(\frac{1}{q}-\frac{1}{p^*}\right)},
\end{split}
\end{equation*}
From the second inequality in \eqref{eq-bound-ABC} and \eqref{eq-pa>n}, with $\alpha = n-1$, we obtain
\begin{equation*}
\begin{split}
    \left\| I_B \right\|_{L^q(\R^n)} \le & \, c_n \norm{\nabla u}_{L^p(\mathbb{R}^n)}
    \left\|{\frac{1}{|z|^{n-1}}}\right\|_{L^r(\{ |z| \le 2 |h| \})} \\
    = & \, c_n \norm{\nabla u}_{L^p(\mathbb{R}^n)} c_{n-1,r} (2|h|)^{\frac{n}{r} - n+1} =
 C_B(n,r) \norm{\nabla u}_{L^p(\mathbb{R}^n)}\abs{h}^{n\left(\frac{1}{q}-\frac{1}{p^*}\right)},
\end{split}
\end{equation*}
where $C_B(n,r)$ is a constant depending on $n$ and $r$ (and thus on $n, p, q$). The same argument applies to $I_C$, so
that, provided that we consider the norm $\left\|{\frac{1}{|z|^{n-1}}}\right\|_{L^r( \{|z| \le 3 |h|\})}$ instead of
$\left\|{\frac{1}{|z|^{n-1}}}\right\|_{L^r(\{ |z| \le 2 |h|\})}$, as in the proof of the Morrey's theorem. We then find
\begin{equation*}
\begin{split}
    \left\| I_C \right\|_{L^q(\R^n)} \le C_C(n,r) \norm{\nabla
     u}_{L^p(\mathbb{R}^n)}\abs{h}^{n\left(\frac{1}{q}-\frac{1}{p^*}\right)},
\end{split}
\end{equation*}
We note that  the $L^r$ norms of the functions $z \mapsto |z|^{-n}$ and $z \mapsto |z|^{-n+1}$ appearing in the
above estimates are finite if, and only if, the condition \eqref{condizionesur} is satisfied. The thesis is
obtained with $C_{p,q} := C_A(n,r) + C_B(n,r) + C_C(n,r)$ for $u \in C_0^\infty(\R^n)$. The
general case follows by a density argument. \hfill $\Box$

\subsection{A more general compactness result}

We note that the Theorem \ref{Uniform continuity} only applies to a kernel that is homogeneous of degree $-n+1$.
Actually, the method used in its proof also applies to any general homogeneous kernel $K_\alpha$, with $0 < \alpha <
n$, as those considered in Theorem \ref{Th-weak-Young}. In the following statement we denote by $u$ the convolution
$K_{\alpha} * f$, that is
\[
u(x)=\integ{K_{\alpha} (x-y) f(y)}{y}.
\]

\begin{theorem} \label{Uniform continuity-bis}
Let $K_\alpha$ be a $C^1(\R^n \setminus \{ 0 \})$ homogeneous function of degree $- \alpha$, with $1 < \alpha < n$, and
let $p, q \ge 1$ be such that $q > p$ and

\begin{equation} \label{eq-pqalpha}
  1 - \dfrac{\alpha+1}{n} < \dfrac{1}{p} - \dfrac{1}{q} <  1 - \dfrac{\alpha}{n}.
\end{equation}
Then there exists a positive constant $\widetilde{C}_{p,q}$, depending on $n, p, q$, such that
\[
\norm{u(h + \cdot)-u}_{L^q(\mathbb{R}^n)}\leq  \widetilde{C}_{p,q} \, \| f \|_{L^p(\mathbb{R}^n)}
\, \abs{h}^{\frac{n}{r} - \alpha},
\]
for every $f \in \sobb{p}{{\mathbb{R}^{n}}}$ and $h \in \R^n$. Here $r$ is the constant introduced in
\eqref{esponente-r}, that is $1 + \dfrac{1}{q} = \dfrac{1}{r} + \dfrac{1}{p}$. Moreover the exponent
$\dfrac{n}{r} - \alpha$ is strictly positive.
\end{theorem}

\noindent \textit{Proof.} We choose $x, h \in \R^n$ and let $v$ be defined as in \eqref{eq-vx}: $v(x) = u(x+h) - u(x)$,
and we consider three integrals $v(x) = {I}_A(x) + {I}_B(x) + {I}_C(x)$ as in \eqref{usplit}.

We proceed as we did in the proof of Theorem \ref{e}. We find
\begin{equation*}
\begin{split}
 | I_A (x) | & \le M_\alpha |h|   \integrale{\lbrace y\in\mathbb{R}^n:\abs{x+h-y} \geq
2\abs{h}\rbrace}{}{\frac{1}{\abs{x+h-y}^{\alpha + 1}}\abs{f(y)}}{y}, \\
 | I_B (x) | & \le c_\alpha  \integrale{\lbrace y\in\mathbb{R}^n:\abs{x+h-y} <
2\abs{h}\rbrace}{}{\frac{1}{\abs{x+h-y}^{\alpha}}\abs{f(y)}}{y}, \\
 | I_C (x) | & \le c_\alpha  \integrale{\lbrace y\in\mathbb{R}^n:\abs{x+h-y} <
2\abs{h}\rbrace}{}{\frac{1}{\abs{x-y}^{\alpha}}\abs{f(y)}}{y},
\end{split}
\end{equation*}
being $M_\alpha := \max\limits_{\frac{1}{2}\leq |z| \leq \frac{3}{2}}{\abs{ \nabla K_\alpha(z)}}$. In order to use the
Young inequality, we recall that
\begin{equation*}

\left\| \frac{1}{\abs{z}^{\alpha + 1}} \right\|_{L^r(\lbrace y\in\mathbb{R}^n:\abs{z} \geq 2\abs{h}\rbrace)}

= c_{\alpha+1, r} (2|h|)^{\frac{n}{r} - (\alpha + 1)},

\end{equation*}
where $r$ is the exponent introduced in \eqref{esponente-r} and

we note that the above integral converges if, and only if, $({\alpha + 1})r > {n}$. Moreover,
\begin{equation*}

\left\| \frac{1}{\abs{z}^{\alpha}} \right\|_{L^r(\lbrace z \in\mathbb{R}^n:\abs{z} \leq 2\abs{h}\rbrace)}

= c_{\alpha, r} (2|h|)^{\frac{n}{r} - \alpha},

\end{equation*}
and the above integral converges if, and only if, ${\alpha} r <{n}$. Summarizing, the two above integrals are finite
if, and only if,
\begin{equation}
\label{cond-su-r}
\frac{n}{\alpha + 1} < r < \frac{n}{\alpha},
\end{equation}
which is equivalent to \eqref{eq-pqalpha}. We also note that the exponent $\dfrac{n}{r} - \alpha$ is strictly positive.

We next proceed as in the proof of Theorem \ref{e}. By the Young inequality we deduce
\begin{eqnarray*}
& \left\| I_A \right\|_{L^q(\R^n)} \le \widetilde{C}_A(n,\alpha, r) |h|^{\left(\frac{n}{r}- \alpha\right)}
\left\| f \right\|_{L^p(\R^n)}, \\
& \left\| I_B \right\|_{L^q(\R^n)} \le \widetilde{C}_B(n,\alpha, r) |h|^{\left(\frac{n}{r}- \alpha\right)}
\left\| f \right\|_{L^p(\R^n)}, \\
& \left\| I_C \right\|_{L^q(\R^n)} \le \widetilde{C}_C(n,\alpha, r) |h|^{\left(\frac{n}{r}- \alpha\right)}
\left\| f \right\|_{L^p(\R^n)},
\end{eqnarray*}
where $\widetilde{C}_A(n,\alpha, r),  \widetilde{C}_B(n,\alpha, r)$, and $\widetilde{C}_C(n,\alpha, r)$ are positive
constants only depending on $n, p, q$, on $c_\alpha$ and on $M_\alpha$. The thesis is obtained with
$\widetilde{C}_{p,q}
:= \widetilde{C}_A(n,\alpha, r) +  \widetilde{C}_B(n,\alpha, r) + \widetilde{C}_C(n,\alpha, r)$. \hfill $\Box$

\begin{remark} \label{listona}
The statement of Theorem \ref{Uniform continuity-bis} is more involved with respect to that of Theorem
\ref{Uniform continuity} as we don't have a natural counterpart of the Sobolev exponent $p^*$ for any $\alpha \in
]0,n[$. We list here the explicit conditions on $q$ for the validity of \eqref{eq-pqalpha}. We discuss
specifically the case $1 < p < n$ as we are interested
in the compactness of the Sobolev embedding.
  \begin{itemize}
\item[(i)] If \fbox{$0 < \alpha < n-2$} then
\begin{itemize}
\item[(i.1)] If $\displaystyle 1 < p < \frac{n}{n - \alpha}$ then
\[
\frac{np}{n - p (n - 1 - \alpha)} < q < \frac{np}{n - p (n - \alpha)}
\]
\item[(i.2)] If $\displaystyle \frac{n}{n - \alpha} < p < \frac{n}{n - 1 - \alpha}$ then
\[
q > \max \left \{1, \frac{np}{n - p (n - 1 - \alpha)} \right \} = \frac{np}{n - p (n - \alpha)}
\]
\item[(i.3)] If $\displaystyle  \frac{n}{n - 1 - \alpha} < p < n$ then no values of $q$ are available
\end{itemize}
\item[(ii)] If \fbox{$\alpha = n - 2$} then
\begin{itemize}
\item[(ii.1)] If $\displaystyle 1 < p < \frac{n}{2}$ then
\[
p^* = \frac{np}{n - p} < q < \frac{np}{n - 2p}
\]
\item[(ii.2)] If $\displaystyle \frac{n}{2} < p < n$ then
\[
q > \max \left \{1, \frac{np}{n - p} \right \} = \frac{np}{n - p} = p^*
\]
\end{itemize}
\item[(iii)] If \fbox{$n-2 < \alpha < n-1$} then
\begin{itemize}
\item[(iii.1)] If $\displaystyle 1 < p < \frac{n}{n - \alpha}$ then
\[
\frac{np}{n - p (n - 1 - \alpha)} < q < \frac{np}{n - p (n - \alpha)}
\]
\item[(iii.2)] If $\displaystyle \frac{n}{n - \alpha} < p < n$ then
\[
q > \max \left \{1, \frac{np}{n - p (n - \alpha)} \right \} = \frac{np}{n - p (n - 1 - \alpha)}
\]
\end{itemize}
\item[(iv)] If \fbox{$\alpha = n - 1$} then for any $1 < p < n$
\[
p < q < p^* = \frac{np}{n - p}
\]
\item[(v)] If \fbox{$n - 1 < \alpha < n$} then for any $1 < p < n$
\[
\frac{np}{n - p (n - 1 - \alpha)} < q < \frac{np}{n - p (n - \alpha)}.
\]
\end{itemize}
\end{remark}

\section{Continuous and compact embeddings for degenerate Kolmogorov equations}
\setcounter{section}{3} \setcounter{equation}{0}
\setcounter{theorem}{0}

The aim of this section is to prove compactness estimates for weak solutions to a family of degenerate Kolmogorov
operators that includes the one in \eqref{e-Kolmo-1} as the simplest prototype. Specifically, we consider
operators in this form
\begin{equation} \label{ultraparabolica}
\L u(x,t) :=\sum_{i,j=1}^{m_0}{\partial_{x_i}(a_{i,j}(x,t)\partial_{x_j}u(x,t))}+
\sum_{i,j=1}^{N}{b_{i,j}x_i\partial_{x_j} u(x,t)}-\partial_t u(x,t),
\end{equation}
with $(x,t) =(x_1,...,x_N,t) \in \rnn$. Here $m_0\in\mathbb{N}$ is such that $1\leq m_0\leq N$. In the
sequel we will also use the following notation $z:=(x,t)$, and we always assume the following hypotheses:
\begin{itemize}
\item[(H.1)] The matrix $A=(a_{i,j}(z))_{i,j=1,...,m_0}$ is symmetric, with measurable coefficients and there exists a
positive constant  $\mu$ such that
\[
\mu^{-1}\abs{\xi}^2 \leq \sum_{i,j=1}^{m_0}{a_{i,j}(z)\xi_i\xi_j} \leq \mu\abs{\xi}^2
\]
for all $z\in\rnn$ and $\xi\in\mathbb{R}^{m_0}$.
\item[(H.2)] The matrix $B$ has constant coefficients. Moreover there exists a basis of $\rn$ such that the
matrix $B$ can be written in a canonical form:
\begin{equation*}
B=\begin{pmatrix}
0 & B_1 & 0 & ... & 0 \\
0 & 0 & B_2 &... & 0 \\
\vdots & \vdots &  \vdots & \ddots & \vdots \\
0 & 0 & 0 &... & B_r \\
0 & 0 & 0 &... & 0
\end{pmatrix}
\end{equation*}
where $B_k$ is a matrix $m_{k-1}\times m_k$ with rank $m_k$, $k=1,2,...,r$ with
\[
m_0\geq m_1\geq ... \geq m_r\geq 1, \ \ \ \mathrm{e} \ \ \ \sum_{k=0}^r m_k=N.
\]
\end{itemize}
We prove in detail Theorem \ref{main-th-1} along the same techniques outlined in the Introduction for the operator $\L$
in \eqref{e-Kolmo-1}. In particular, we will rely on a representation formula analogous to \eqref{eq-fund-sol2} in
terms
of the fundamental solution to the operator
\begin{equation} \label{ultraparabolica-0}
  \L_0 u :=\sum_{i=1}^{m_0}{\partial_{x_i}^2 u}+\sum_{i,j=1}^{N}{b_{i,j}x_i\partial_{x_j}u}-\partial_t u.
\end{equation}
\textit{Remark 1}.
It is know that Assumption (H.2) is equivalent to the assumption of hypoellipticity of $\L_0$ (see \cite{dodici} and
its bibliography). This means that any function  $u$ which is a distributional solution to $\L_0 u=f$ in some open
subset $\Omega$ of $\rnn$ is a $C^\infty$ function whenever $f$ is $C^\infty$.

In order to simplify our statements, in the sequel we adopt the following compact notation. If $I_{m_0}$ is the
identity matrix
$m_0\times m_0$, we set
\[
\Delta_{m_0}=\sum_{i=1}^{m_0}\partial_{x_i}^2, \quad \quad
Y=\sum_{i,j=1}^{N}{b_{i,j}x_i\partial_{x_j}}-\partial_t,
\quad \quad
A_0=\begin{pmatrix}
I_{m_0} & 0 \\
0 & 0 \\
\end{pmatrix}.
\]
In particular we will write
\[
\L_0 =\Delta_{m_0}+Y =\mathrm{div}(A_0\nabla)+Y.
\]
We next introduce the non-Euclidean geometric setting suitable for the study of $\L$, the fundamental solution of
$\L_0$ and the definition of the convolution with homogeneous kernels.

\subsection{Dilation and translation groups associated to $\L$}

We recall here some invariance properties of the operator $\L_0$. We refer to \cite{dodici} where the definition of
translation group and dilation group for Kolmogorov operators have been given for the first time. Let
\begin{equation} \label{eq-translation}
(x,t)\circ (\xi,\tau)=(\xi+E(\tau)x,t+\tau), \ \ \ E(t)=\exp(-tB^T), \ (x,t),(\xi,\tau)\in\rnn
\end{equation}
\begin{equation} \label{eq-dilation}
  D(\lambda)=\mathrm{diag}(\lambda I_{m_0},\lambda^3 I_{m_1},...,\lambda^{2r+1} I_{m_r},\lambda^2), \ \ \ \lambda>0
\end{equation}
where $I_{m_j}$ denotes the identity matrix $m_j\times m_j$. It is known that $(\rnn,\circ)$ is a
\emph{non commutative} group, and $\L_0$ is invariant with respect to the \emph{left translations} of
$(\rnn,\circ)$, in the following sense: if we choose any $\zeta \in \rnn$ and we set $v(z) := u (\zeta \circ z)$ and
$g(z) := f (\zeta \circ z)$, then the have
\[
\L_0 u (z) = f (z) \quad \Leftrightarrow \quad \L_0 v (z) =g (z).
\]
Moreover, $\L_0$ is invariant with respect to $(D(\lambda))_{\lambda>0}$, with the following meaning:
\[
\L_0 u (z) = f (z) \quad \Leftrightarrow \quad \L_0 w (z) = \lambda^2 h (z).
\]
Now $w(z) := u(D(\lambda) z)$, $h(z) := f(D(\lambda) z)$, and $\lambda$ is any positive constant. The zero of the group
is $(0,0)$ and the inverse of $(\xi, \tau)$ is $(\xi, \tau)^{-1} = (- E(\tau)\xi, - \tau)$. Moreover the following
\emph{distributive property} holds:
\[
   D(\lambda) \left( z \circ \zeta \right) = \left( D(\lambda) z \right) \circ \left( D(\lambda) \zeta \right)
\]
We summarize the above properties by saying that $\L_0$ is invariant with respect to the \emph{homogeneous group}
$\left(\rnn,\circ, (D(\lambda))_{\lambda>0} \right)$.

In the sequel we will use the following notation
\[
   D(\lambda) = \text{diag} (\lambda^{\alpha_1}, \lambda^{\alpha_2}, \dots , \lambda^{\alpha_N}, \lambda^{2}),
\]
where, in accordance with \eqref{eq-dilation} $\alpha_1 = \alpha_2 = \dots = \alpha_{m_0} = 1$,  $\alpha_{m_0 + 1} =
\dots = \alpha_{m_0 + m_1} = 3, \dots , \alpha_{m_0 + m_1 + \dots + m_{r-1} + 1} = \dots = \alpha_{N} = 2 r +1$.

We define now a norm of $\rnn$ homogeneous of degree one with respect to the dilation introduced before.
\begin{definition} \label{def-norm}
\label{normaomogenea}
For all $z\in\rnn\setminus \{0\}$, we define the norm $\norm{z}=\rho$, as the unique positive solution to:
\[
\frac{x_1^2}{\rho^{2\alpha_1}}+\frac{x_2^2}{\rho^{2\alpha_2}}+...+\frac{x_N^2}{\rho^{2\alpha_N}}+\frac{t^2}{\rho^4}=1
\]
and $\norm{0}=0$.
\end{definition}

This norm is homogeneous with respect to the dilation group $(D(\lambda))_{\lambda>0}$ as long as the following
property holds:
\[
\norm{D(\lambda)z}=\lambda\norm{z} \ \ \ \forall z\in\rnn \setminus\{0\}  \ \text{and} \ \lambda>0.
\]
Moreover the following quasi-triangular inequality holds. There exists a constant $c_T \geq 1$ such that
\begin{equation} \label{eq-quasi-norm}
  \norm{z\circ\zeta}\leq c_T(\norm{z}+\norm{\zeta}), \qquad \norm{z^{-1}}\leq c_T \norm{z},
\end{equation}
for every $z, \zeta \in \rnn$. We denote by $d(\zeta, z) := \norm{z^{-1} \circ \zeta}$ the \emph{quasi-distance} of
$z$ and $\zeta$. We denote by $B_\varrho (z)$  the open ball of radius $\varrho$ and center $z$ with respect to the
above quasi-distance
\begin{equation} \label{eq-brho}
  B_\varrho (z) := \big\{ \zeta \in \rnn \mid \norm{z^{-1} \circ \zeta} < \varrho \big\}.
\end{equation}
Note that the topology induced on $\rnn$ from the norm introduced in Definition \ref{def-norm} is equivalent to the
Euclidean one.

\begin{remark} \label{rem-Leb-invariance}
\textnormal{The Lebesgue measure is invariant with respect to the group $(\rnn,\circ)$. Moreover, as long
as $\det D(\lambda)=\lambda^{Q+2}$, where
\[
Q=m_0+3m_1+...+(2r+1)m_r
\]
we have that the following identity holds:
\[
    \textnormal{meas} (B_r(0))=r^{Q+2}\textnormal{meas}(B_1(0)),
\]
where meas($B$) indicates the Lebesgue measure of the set $B$. For this reason, we will refer to $(Q+2)$ as the
\textit{homogeneous dimension} of $\rnn$ with respect to the dilation \eqref{eq-dilation}. }

\textnormal{We also note that, in view of the structure of the matrix $B$ and of the definition of $E(\tau)$, we have
that det$E(\tau) = 1$ for every $\tau$. In particular, the Jacobian determinant of the left translation $(x,t) \mapsto
(\xi, \tau) \circ (x,t)$ agrees with $1$ for every $(\xi, \tau) \in \rnn$. The same is true for the Jacobian
determinant of $(\xi, \tau) \mapsto (\xi, \tau)^{-1}$. As a consequence we have that}
\begin{equation} \label{eq-Leb-invariance}
\begin{split}
    & \int_{A} f(\zeta \circ z) d z =  \int_{\zeta \circ A} f(w) d w,
    \qquad \quad \zeta \circ A := \big\{ w= \zeta \circ z \mid z \in A \big\},\\
    & \int_{A} f\big(z^{-1}\big) d z =  \int_{A^{-1}} f(w) d w,
    \qquad \quad A^{-1} := \big\{ w= z^{-1} \mid z \in A \big\},\\
    & \int_{A} f(z^{-1} \circ \zeta) d z =  \int_{A^{-1}\circ \zeta} f(w) d w,
    \quad A^{-1} \circ \zeta := \big\{ w= z^{-1}\circ \zeta \mid z \in A \big\}.
\end{split}
\end{equation}
\end{remark}

The following results is the analogous of \eqref{eq-pa>n} and \eqref{eq-pa<n} in the setting of the homogeneous Lie
group $\left(\rnn,\circ, (D(\lambda))_{\lambda>0} \right)$.
\begin{lemma} \label{lem-ABC}
Let $K_{\alpha}$ denote any continuous  function which is homogeneous of degree $- \alpha$  with respect to
$(D(\lambda))_{\lambda>0}$, for some $\alpha$ such that  $0 < \alpha < Q + 2$. For every $R>0$ we have that:
  \begin{itemize}
    \item $K_\alpha \in L^q\big(\big\{ z \in \rnn \mid \|z\| \le R \big\}\big)$ if, and only if, $q >
\dfrac{Q+2}{\alpha}$.
Moreover, there exists a positive constant $\widetilde c_{\alpha,q}$, only depending on $K_\alpha,
(D(\lambda))_{\lambda>0}$ and $q$, such that
\begin{equation}\label{eq-pa>Q+2}
  \left\| K_\alpha \right\|_{L^q(\{ z \in \rnn \mid \|z\| \le R\})} \le
  \widetilde c_{\alpha,q} R^{\frac{Q+2}{q} - \alpha},
\end{equation}
    \item $K_\alpha \in L^q\big(\big\{ z \in \rnn \mid \|z\| \ge R \big\}\big)$ if, and only if, $q <
\dfrac{Q+2}{\alpha}$.
Moreover, there exists a positive constant $\widetilde c_{\alpha,q}$, only depending on $K_\alpha,
(D(\lambda))_{\lambda>0}$ and $q$, such that
\begin{equation}\label{eq-pa<Q+2}
  \left\| K_\alpha \right\|_{L^q(\{ z \in \rnn \mid \|z\| \ge R \})} \le
  \widetilde c_{\alpha,q} R^{\frac{Q+2}{q} - \alpha}.
\end{equation}
  \end{itemize}
\end{lemma}

\medskip

\noindent \textit{Proof.} We compute the integrals by using the ``polar coordinates''
\begin{equation*}
\begin{cases}
x_1=\rho^{\alpha_1}\cos{\psi_1}...\cos{\psi_{N-1}}\cos{\psi_N} \\
x_2=\rho^{\alpha_2}\cos{\psi_1}...\cos{\psi_{N-1}}\sin{\psi_N} \\
 \vdots \\
x_N=\rho^{\alpha_N}\cos{\psi_1}\sin{\psi_2} \\
t=\rho^2\sin{\psi_1}.
\end{cases}
\end{equation*}
Note that, in accordance with the Definition \ref{def-norm}, the Jacobian determinant of the above change of coordinate
is homogeneous of degree $Q+1$ with respect to the variable $\rho$, that is  $J(\rho, \psi) = \rho^{Q+1} J(1, \psi)$.
The claim then follows by proceedings as in the Euclidean case. \hfill $\square$

\subsection{Preliminary results on convolutions in homogeneous Lie groups}

We recall some facts concerning the convolution of functions in homogeneous Lie groups. We refer to the work of Folland
\cite{dieci}, and to its bibliography, for a comprehensive treatment of this subject. The first result is a Young
inequality in the non-Euclidean setting
\begin{theorem} \label{Qyoung} Let $p, q, r \in [1,+\infty]$ be such that:
\begin{equation} \label{eq-pqr}
  1 + \frac{1}{q}=\frac{1}{p}+\frac{1}{r}.
\end{equation}
If $f\in\sobb{p}{\rnn}$ and $g\in\sobb{r}{\rnn}$, then the function $f* g$ defined as:
\[
f* g (z) := \integrale{\rnn}{}{f(\zeta^{-1}\circ z)g(\zeta)}{\zeta}
\]
belongs to $\sobb{q}{\rnn}$ and it holds:
\[
\norm{f* g}_{L^q(\rnn)} \leq \norm{f}_{L^p(\rnn)}\norm{g}_{L^r(\rnn)}.
\]
\end{theorem}

The following two theorems are the counterpart of Theorem \ref{Th-weak-Young} and Lemma \ref{14} in Section 2,
respectively.

\begin{theorem} \label{continuitanon} {\rm (Proposition (1.11) in \cite{dieci})}
Let $K_{\alpha}$ be a continuous function, homogeneous of degree $-\alpha$ with $0 < \alpha < Q+2$,  with respect to
the dilation \eqref{eq-dilation}. Then, for every $p \in ]1, + \infty[$, the convolution $u$ of $K_{\alpha}$ with a
function $f \in L^p(\rnn)$
\begin{equation} \label{eq-convolution-sec3}
  u(z) = \int_{\rnn} K_{\alpha}(\zeta^{-1} \circ z) f(\zeta) \, {\rm d} \zeta,
\end{equation}
is defined for almost every $z \in \rnn$ and is a measurable function. Moreover

there exists a constant $\widehat{C}_p = \widehat{C}_p(p, Q)$ such that
\[
\|u\|_{L^q(\rnn)} \leq \widehat{C}_p
\max_{\norm{z}=1}{\abs{K_{\alpha}(z)}}\norm{f}_{L^p(\rnn)},
\]
for every $f\in\sobb{p}{\rnn}$, where  $q$ is defined by
\[
1 + \frac{1}{q}=\frac{1}{p} + \frac{\alpha}{Q+2}.
\]
\end{theorem}

For the proof of the next result we refer to Proposition (1.11) in \cite{dieci} or Lemma 5.1 in \cite{otto}.

\begin{theorem} \label{Qlagrange}
Let $K_\alpha \in C^1(\rnn\setminus\{0\})$ be a homogeneous function of degree $-\alpha$ with respect to
the group $(D(\lambda))_{\lambda>0}$. Then there exist two constants $\kappa > 1$ and $M_\alpha >0$  such that:
\[
\vert K_\alpha(\zeta)-K_\alpha(z)\vert\leq M_\alpha \, \frac{\norm{z^{-1}\circ\zeta}}{\norm{z}^{\alpha+1}}
\]
for all $z,\zeta$ such that $\norm{z}\geq \kappa \norm{z^{-1}\circ \zeta}$.
\end{theorem}

\subsection{Compactness estimates for convolutions with homogeneous kernels}

\begin{theorem} \label{uniformecontinuitanon}
Let $K_{\alpha}$ be a $C^1(\rnn\setminus{\{0\}})$ homogeneous function of degree $-\alpha$ with $0 < \alpha
< Q+2$, with respect to the dilation \eqref{eq-dilation}.

Then for every $p, q \ge 1$ such that $q > p$ and
\begin{equation} \label{eq-pqalpha-bis}
1 - \frac{\alpha + 1}{Q + 2} < \frac{1}{p} - \frac{1}{q} < 1 - \frac{\alpha}{Q + 2},
\end{equation}
there exists a positive constant $\widetilde{C}_{p,q}$ depending on $\alpha, p, q$ and on the dilation group
$\left( D(\lambda) \right)_{\lambda >0}$ such that
\[
\norm{u(\cdot \circ h)-u}_{L^q(\rnn)} \leq \widetilde{C}_{p,q} \norm{h}^{\frac{Q + 2}{r} - \alpha}
\norm{f}_{L^p(\rnn)},
\]
for every $f\in \sobb{p}{{\rnn}}$, and $h \in \rnn$. Here $r$ is the constant defined by \eqref{eq-pqr}, that is
$$
  \displaystyle  1 + \frac{1}{q} = \frac{1}{r} + \frac{1}{p},
$$
and the exponent $\frac{Q + 2}{r} - \alpha$ is strictly positive, because of \eqref{eq-pqalpha-bis}.

\end{theorem}

\noindent \textit{Proof.} We proceed as in the proof of Theorem \ref{Uniform continuity-bis}. We choose $z, h \in \rnn$
and we let
\begin{equation} \label{eq-vz}
  v(z) := u(z \circ h) - u(z).
\end{equation}
By the formula \eqref{eq-convolution-sec3} we have $v(z) = \widetilde{I}_A(z) + \widetilde{I}_B(z) +
\widetilde{I}_C(z)$, where
\begin{eqnarray}
 \widetilde{I}_A(z) &= & \integrale{\lbrace \zeta\in\rnn:\|\zeta^{-1} \circ z \circ h \| \geq
\kappa \|h\|\rbrace}{}{\left( K_{\alpha}(\zeta^{-1} \circ z \circ h)-K_{\alpha}(\zeta^{-1} \circ
z)  \right){f(\zeta)}}{\zeta}, \nonumber \\
 \widetilde{I}_B(z) &= & \integrale{\lbrace \zeta\in\rnn:\|\zeta^{-1} \circ z \circ h \| <
\kappa \|h\|\rbrace}{}{{K_{\alpha}(\zeta^{-1} \circ z \circ h)}{f(\zeta)}}{\zeta},
\label{usplittildeQ}\\
 \widetilde{I}_C(z) &= & \integrale{\lbrace \zeta\in\rnn:\|\zeta^{-1} \circ z \circ h \| <
\kappa \|h\|\rbrace}{}{{- K_{\alpha}(\zeta^{-1} \circ z)}{f(\zeta)}}{\zeta}. \nonumber

\end{eqnarray}
Then, as in the proof of Theorem \ref{Uniform continuity-bis}, we find
\begin{equation*}
\begin{split}
 | \widetilde{I}_A (z) | & \le c_T M_\alpha \|h\|
 \integrale{\lbrace \zeta\in\rnn:\|\zeta^{-1} \circ z \circ h \| \geq
\kappa \|h\|\rbrace}{}{\frac{1}{\|\zeta^{-1} \circ z \circ h\|^{\alpha + 1}}\abs{f(\zeta)}}{\zeta}, \\
 | \widetilde{I}_B (z) | & \le c_\alpha \integrale{\lbrace \zeta\in\rnn:\|\zeta^{-1} \circ
z \circ h \| < \kappa \|h\|\rbrace}{}{\frac{1}{\|\zeta^{-1} \circ z \circ h\|^{\alpha}}\abs{f(\zeta)}}{\zeta}, \\
 | \widetilde{I}_C (z) | & \le c_\alpha \integrale{\lbrace \zeta\in\rnn:\|\zeta^{-1} \circ z
\circ h \| < \kappa \|h\|\rbrace}{}{\frac{1}{\|\zeta^{-1} \circ z\|^{\alpha}}\abs{f(\zeta)}}{\zeta}.
\end{split}
\end{equation*}
The first estimate follows from Theorem \ref{Qlagrange} and the constant $c_T$ is the one appearing in
\eqref{eq-quasi-norm}, while $c_\alpha := \max_{\norm{w}=1} K_\alpha(w)$.

We next compute the $L^r$ norm of the homogeneous functions appearing above. In view of Remark
\ref{rem-Leb-invariance} and Lemma \ref{lem-ABC}, we have that
\begin{equation*}

\left\| \frac{1}{\norm{\zeta^{-1} \circ z \circ h}^{\alpha + 1}} \right\|_{L^r(\lbrace
\zeta\in\rnn:\|\zeta^{-1} \circ z \circ h \| \geq
\kappa \|h\|\rbrace)} = \widetilde{C}_A(r, Q) \|h\|^{\frac{Q+2}{r} - (\alpha + 1)},

\end{equation*}
where $r$ is the exponent introduced in \eqref{esponente-r} and $\widetilde{C}_A(r,Q)$ is a constant depending on $Q$
and $r$  (hence on $Q, p, q$), and on the dilation group $\left( D(\lambda) \right)_{\lambda >0}$.
Using again Remark \ref{rem-Leb-invariance} and Lemma \ref{lem-ABC}, and  we also find
\begin{equation*}

\left\| \frac{1}{\norm{\zeta^{-1} \circ z \circ h}^{\alpha}} \right\|_{L^r(\lbrace
\zeta\in\rnn:\|\zeta^{-1} \circ z \circ h \| \le
\kappa \|h\|\rbrace)} = \widetilde{C}_B(r, Q) \|h\|^{\frac{Q+2}{r} - \alpha},

\end{equation*}
The same argument applies to $\widetilde{I}_C$, by using the quasi-triangular inequality \eqref{eq-quasi-norm}, so that
\begin{equation*}

\left\| \frac{1}{\norm{\zeta^{-1} \circ z \circ h}^{\alpha}} \right\|_{L^r(\lbrace
\zeta\in\rnn:\|\zeta^{-1} \circ z \circ h \| \le
\kappa \|h\|\rbrace)} = \widetilde{C}_C(r, Q) \|h\|^{\frac{Q+2}{r} - \alpha},

\end{equation*}
Note that the three above integrals converge if, and only if,
\begin{equation} \label{cond-su-r-Q}
\frac{Q+2}{\alpha + 1} < r < \frac{Q+2}{\alpha}.
\end{equation}
We note that \eqref{eq-pqalpha-bis} is equivalent to \eqref{cond-su-r-Q} and that the second inequality in
\eqref{cond-su-r-Q} says that the exponent $\frac{Q + 2}{r} - \alpha$ appearing in the statement of this Theorem is
strictly positive. By the Young inequality (Theorem \ref{Qyoung}) we conclude that
\begin{equation*}
\begin{split}
& \left\| \widetilde{I}_A \right\|_{L^q(\rnn)} \le c_T M_\alpha \widetilde{C}_A(r,Q) \|h\|^{\frac{Q+2}{r}- \alpha}
\left\| f \right\|_{L^p(\rnn)}, \\
& \left\| \widetilde{I}_B \right\|_{L^q(\rnn)} \le c_\alpha \widetilde{C}_B(r,Q) \|h\|^{\frac{Q+2}{r} -
\alpha} \left\|f\right\|_{L^p(\rnn)}, \\
& \left\| \widetilde{I}_B \right\|_{L^q(\rnn)} \le c_\alpha \widetilde{C}_C(r,Q) \|h\|^{\frac{Q+2}{r} -
\alpha} \left\|f\right\|_{L^p(\rnn)}.
\end{split}
\end{equation*}
The thesis is obtained with $\widetilde{C}_{p,q} := c_T M_\alpha \widetilde{C}_A(r,Q) +  c_\alpha \widetilde{C}_B(r,Q)
+ c_\alpha \widetilde{C}_C(r,Q)$. \hfill $\Box$

\begin{remark} \label{liston-b}
Similarly as we did in Remark \ref{listona}, we can state the conditions on $q$ for the validity of
\eqref{eq-pqalpha-bis}. They can can be obtained by substituting the \emph{dimension} $n$ with the
\emph{homogeneous dimension} $Q+2$. We explicitly write here the condition for $\alpha = Q$ and $\alpha = Q+1$, which
occur in the representation formulas for the solutions to $\L u = f$. Moreover, when $\alpha = Q+1$, we only consider
the case $1 < p < Q+2$, as we apply Theorem \ref{uniformecontinuitanon} to prove the compactness of the embedding of
Theorem \ref{continuitanon}, which holds only for $p < Q+2$. For the same reason, when $\alpha = Q$, we only consider
the case $\displaystyle 1 < p < \frac{Q + 2}{2}$.
\begin{itemize}
\item[(i)]  If {$\alpha = Q$} and $\displaystyle 1 < p < \frac{Q + 2}{2}$, we have that $p^* < q < p^{**}$,

\item[(ii)] If {$\alpha = Q + 1$} and $1 < p < Q+2$ we have that $p < q < p^*$.

\end{itemize}
\end{remark}

\subsection{Proof of our main result}

\noindent {\it Proof of Theorem \ref{main-th-1}}. The proof of the Sobolev inequality is a direct consequence of
Theorem \ref{continuitanon}, with $\alpha = Q+1$ when considering $v_1, \dots, v_{m_0}$, and $\alpha = Q$ as we
consider $u$.

The compactness of the embedding is a direct consequence of Theorem \ref{uniformecontinuitanon}. As noticed in Remark
\ref{liston-b}, it applies to the derivatives $\partial_{\xi_j}\Gamma$, that are homogeneous functions of degree
$-(Q+1)$, only when $p < q < p^*$. Moreover, a direct computation based on \eqref{eq-pqr} shows that
\begin{equation}
  \frac{Q + 2}{r} - (Q+1) =  (Q+2) \left(\frac{1}{q}- \frac{1}{p^*} \right).
\end{equation}
Analogously, as $\Gamma$ is homogeneous of degree $-Q$, we need to consider $p^* < q < p^{**}$. In this case, by using
again \eqref{eq-pqr} we find
\begin{equation}
  \frac{Q + 2}{r} - Q =  (Q+2) \left(\frac{1}{q}- \frac{1}{p^{**}} \right).
\end{equation}

The proof of the Morrey embedding is obtained by the same argument used in the proof of Theorem \ref{e}.
We consider the  function $v(z)= u(z \circ h) - u(z)$ introduced in  \eqref{eq-vz} and, as in the proof of  Theorem
\ref{uniformecontinuitanon}, we write $v(z) = \widetilde{I}_A(z) + \widetilde{I}_B(z) + \widetilde{I}_C(z)$, where the
functions $\widetilde{I}_A, \widetilde{I}_B, \widetilde{I}_C$ are defined in \eqref{usplittildeQ}. The conclusion of
the
proof is obtained by using the Young inequality stated in Theorem \ref{Qyoung}. \hfill $\Box$

\section{Representation formulas}
\setcounter{section}{4} \setcounter{equation}{0}
\setcounter{theorem}{0}

\subsection{Fundamental solution to $\L_{0}$ and representation formula}

In this Section we focus on the representation formulas for the equation $\L u = 0$, for $\L$ satisfying  the
assumptions (H.1) and (H.2), then we prove Theorem \ref{main-th-2}.

We first recall the definition of \emph{weak solution} to $\L u = 0$, then we recall that,
under these assumptions, the fundamental solution to $\L_0$ has been derived by H\"ormander in \cite{dieci}.
We say that $u$ is a weak solution to $\L u = 0$ in an open set $\Omega\subset\rnn$ if
$u, \partial_{x_1}u, \dots, \partial_{x_{m_0}}u, Yu \in L_{\mathrm{loc}}^2(\Omega)$ and
\[
\integrale{\Omega}{}{-\langle A(z)\nabla u(z),\nabla\psi(z)\rangle +\psi(z) Y u(z)}{z}=0,
\]
for all $\psi\in C_0^\infty(\Omega)$.

With the notations introduced in Section 3, we let

\[
C(t)=\integrale{0}{t}{E(s)A_0 E^T(s)}{s}
\]
The assumptions (H.1) and (H.2) guarantee that the matrix $C(t)$ is strictly positive for all $t>0$. In this case its
inverse $C^{-1}(t)$ is well defined and the fundamental solution to $\L_{0}$ with singularity at the origin of $\rnn$,
is given by
\begin{equation} \label{eq-fund-sol-K}
  \Gamma((x,t),(0,0))=
\begin{cases}
\dfrac{(4\pi)^{-\frac{N}{2}}}{\sqrt{\mathrm{det}C(t)}}\mathrm{exp}
\left(-\frac{1}{4}\left\langle C^{-1}(t)x,x\right\rangle \right) , & \mbox{if } t>0, \\
0, & \mbox{if } t\leq 0.
\end{cases}
\end{equation}
To simplify the notation, in the sequel we will write $\Gamma(x,t,\xi, \tau)$ instead of
$\Gamma((x,t),(\xi,\tau))$, and $\Gamma(x,t)$ instead of $\Gamma(x,t,0,0)$.
The fundamental solution $\Gamma(x,t,\xi,\tau)$ of $\L_{0}$ with pole at $(\xi,\tau)$ is the ``left
translation'' of $\Gamma(\cdot,0,0)$ with respect to the group $(\mathbb{R}^{N+1},\circ)$:
\[
\Gamma(x,t,\xi,\tau)=\Gamma((\xi,\tau)^{-1}\circ(x,t),0,0).
\]

Let us explicitly remark that $\Gamma(\cdot,0,0)$ is homogeneous of degree $-Q$ with respect to the group
$(D(\lambda))_{\lambda>0}$ and $\partial_{x_j}\Gamma(\cdot,0,0)$ is homogeneous of degree $-Q-1$, for $j=1,...,m_0$.
Moreover, also $\partial_{\xi_j}\Gamma(0,0, \cdot)$ is homogeneous of degree $-Q-1$, for $j=1,...,m_0$.

We next represent weak solutions to $\L u = 0$ as convolutions with the fundamental solution $\Gamma$ to $\L_0$ and
to its derivatives $\partial_{\xi_1}\Gamma, \dots, \partial_{\xi_{m_0}}\Gamma$.

Consider any open set $\Omega \subseteq \rnn$ let $u$ be a function such that $u, \partial_{x_1} u, \dots,
\partial_{x_{m_0}} u, Yu  \in L^p_{\text{\rm loc}}(\Omega)$, and $\eta \in C_0^\infty(\Omega)$ is any cut-off
function, then, by an elementary density argument we find
\begin{equation*}
\begin{split}
(\eta u)(z) = & -\integrale{\mathbb{R}^{N+1}}{}{[\Gamma(z,\cdot)\L_0 (\eta u)](\zeta)}{\zeta} = \\
  & \integrale{\mathbb{R}^{N+1}}{}{[\langle A_0 \nabla_\xi \Gamma(z,\cdot),  \nabla (\eta u)
  \rangle](\zeta)}{\zeta}-\integrale{\mathbb{R}^{N+1}}{}{[\Gamma(z,\cdot)Y(\eta u)](\zeta)}{\zeta}.
\end{split}
\end{equation*}
If moreover $u$ is a weak solutions to $\L u = \text{div} (A_0 F) + f$, then $\L_0 u =$ div$\left((A_0 - A\right)
\nabla u + A_0 F) + f$, for some $f \in L^p_\text{\rm loc}(\Omega)$ and some vector valued function $F = (F_1, \dots,
F_{m_0}, 0, \dots, 0)$ with $F_1, \dots, F_{m_0} \in L^p_\text{\rm loc}(\Omega)$, then we obtain the following
representation formula introduced in Theorem 3.1 of \cite{otto}, and used in the proof of Theorem 3.3 of \cite{sei}.
\begin{theorem} \label{th-rappr}
If $u$ is a weak solution to $\L u = \text{ \rm div} (A_0 F) + f $,

in some open set $\Omega \subset \rnn$, with $f, F_1, \dots, F_{m_0} \in L^p_\text{\rm loc}(\Omega)$, and $\eta$ is the
cut-off function defined above,  then:
\begin{equation} \label{rappresentazione}
\begin{split}
(\eta u)(z) = & \integrale{\mathbb{R}^{N+1}}{}{ \! \!  [ \eta \langle \nabla_\xi \Gamma(z,\cdot),
(A_0 - A) \nabla u + A_0 F \rangle](\zeta)}{\zeta} -
\integrale{\mathbb{R}^{N+1}}{}{\! \! [\Gamma(z,\cdot) (\langle A \nabla \eta , \nabla u \rangle + \eta f )]}{\zeta} \\
 + & \integrale{\mathbb{R}^{N+1}}{}{[\langle A_0 \nabla_\xi \Gamma(z,\cdot),  \nabla \eta \rangle u ](\zeta)}{\zeta}-
\integrale{\mathbb{R}^{N+1}}{}{\Gamma(z,\cdot)(Y \eta) u}{\zeta}.
\end{split}
\end{equation}
\end{theorem}

In the following statement $B_{\varrho}(z_0)$ denotes the ball defined in \eqref{eq-brho}, and $c_T$ is the constant in
\eqref{eq-quasi-norm}.

\begin{proposition} \label{prop-estimates} Let $\Omega$ be an open set of $\rnn$, and let $u$ be a weak solution to
$\L
u = \text{div} (A_0 F) + f$ in $\Omega$. Suppose that $u, f, \partial_{x_1} u, \dots, \partial_{x_{m_0}} u, F_1,
\dots, F_{m_0} \in L^p_\text{\rm loc}(\Omega)$. Then for every $z_0 \in \Omega$, and $\varrho, \sigma >0$ such that the
ball $B_{\varrho}(z_0)$ is contained in $\Omega$, and $\sigma < \dfrac{\varrho}{2 c_T}$, we have:
\begin{itemize}
  \item (Sobolev embedding) if $1 < p < Q+2$, then there exists a positive constant $C_p$ such that
\begin{equation*}
\begin{split}
  \| u \|_{L^{p^*}(B_{\sigma}(z_0))} \le \, &  C_{p} \big( \| u \|_{L^{p}(B_{\varrho}(z_0))} +
  \| f \|_{L^{p}(B_{\varrho}(z_0))} \\
  + & \, \| A_0 \nabla u \|_{L^{p}(B_{\varrho}(z_0))} + \|  A_0 F \|_{L^{p}(B_{\varrho}(z_0))} \big),
\end{split}
\end{equation*}
 \item (Compactness) if moreover $p < q < p^*$, then there exists a positive constant
$\widetilde C_{p, q}$ such that
\begin{equation*}
\begin{split}
  \| u ( \cdot \circ h) - u \|_{L^{q}(B_{\sigma}(z_0))} \le \, & \widetilde C_{p, q}
  \big( \| u \|_{L^{p}(B_{\varrho}(z_0))} +  \| f \|_{L^{p}(B_{\varrho}(z_0))} \\
 + & \, \| A_0 \nabla u \|_{L^{p}(B_{\varrho}(z_0))} + \|  A_0 F \|_{L^{p}(B_{\varrho}(z_0))} \big) \
  \|h\|^{(Q+2) \left( \frac{1}{q}- \frac{1}{p^*} \right)},
\end{split}
\end{equation*}
for every $ h \in B_{\sigma}(z_0)$,
\item (Morrey embedding) if  $p > Q+2$, then there exists a positive constant $\widetilde C_p$ such that
\begin{equation*}
\begin{split}
   | u (z) - u (\zeta) | \le \, & \widetilde C_{p}  \big( \| u \|_{L^{p}(B_{\varrho}(z_0))} +
   \| f \|_{L^{p}(B_{\varrho}(z_0))} \\
   + & \,  \| A_0 \nabla u \|_{L^{p}(B_{\varrho}(z_0))} + \|  A_0 F \|_{L^{p}(B_{\varrho}(z_0))} \big)
   \|\zeta^{-1} \circ z\|^{1 - \frac{Q+2}{p}},
\end{split}
\end{equation*}
for every $\ z, \zeta \in B_{\sigma}(z_0)$.
 \end{itemize}
\end{proposition}

\medskip

\noindent {\it Proof}. We apply Theorem \ref{th-rappr} with a function $\eta$ supported in the ball $B_{\varrho}(z_0)$
and such that $\psi(z) = 1$ for every $z \in B_{2 c_T \sigma}(z_0)$. It is not difficult to check that a cut-off
function with the above properties exists (see formula (3.3) in \cite{otto} for instance). Note that the integrals
appearing in the equation \eqref{rappresentazione} involving $\partial_{x_1}u, \dots, \partial_{x_{m_0}}u, F_1, \dots,
F_{m_0}$ are convolutions of $\partial_{\xi_1}\Gamma, \dots, \partial_{\xi_{m_0}}\Gamma$, that are
homogeneous kernels of degree $-(Q+1)$, with functions belonging to $L^p(B_{\varrho}(z_0))$ multiplied by bounded
functions compactly supported in $B_{\varrho}(z_0)$.  For these terms of the representation formula
\eqref{rappresentazione} the thesis then follows from a direct application of Theorem \ref{main-th-1}. Indeed, by our
choice of $\eta$, we have $u(z) = (\eta u)(z)$ for every $z \in B_{\sigma}(z_0)$. Moreover, if $z, h \in
B_{\sigma}(z_0)$ we also have that $z \circ h \in B_{2 c_T \sigma}(z_0)$, by \eqref{eq-quasi-norm}, then also $u(z
\circ h) = (\eta u)(z \circ h)$.

We next consider the terms involving $u$ and $f$. They are convolutions of $\Gamma$, that is a
homogeneous kernel of degree $-Q$, with $u$ and $f$, multiplied by bounded functions compactly supported in
$B_{\varrho}(z_0)$. Moreover, $u$ and $f$ belong to $L^r(B_{\varrho}(z_0))$, for every $r$ such that $1 \le r \le p$.
We then choose $r$ such that
\begin{equation*}
  \frac{1}{r} = \frac{1}{p} + \frac{1}{Q+2}
\end{equation*}
and we apply again Theorem \ref{main-th-1} with $p$ replaced by $r$. This concludes the proof. \hfill $\Box$

\medskip
\noindent {\it Proof of Theorem \ref{main-th-2}}. It follows from Proposition \ref{prop-estimates} by a simple covering
argument. The constant $\widetilde \varrho$ can be chosen as follows. We let
$$
\overline \varrho : = \min \big\{\varrho >0 \mid B_\varrho(z) \subset\Omega \ \text{for every} \ z \in \K \big\},
$$
then $\widetilde \varrho := \dfrac{\overline \varrho}{3 c_T}$, so that we can choose $\sigma = \widetilde \varrho$ in
every ball of the covering of $\K$. \hfill $\Box$

\medskip
\noindent {\it Proof of Theorem \ref{main-th-3}}. If $f$ is a weak solution to \eqref{eq-kinetic}, then it is a weak
solution to
\begin{equation} \label{eq-kinetic-2}
  \partial_t f + \langle v, \nabla_x f \rangle = \Delta_v f + g - \text{\rm div}_v G,
\end{equation}
where $G_j = \partial_{v_j} f, j=1, \dots, n$. Note that the homogeneous dimension of the operator in
\eqref{eq-kinetic-2} is $Q+2 = n + 3n +2$. By our assumptions $G_j \in L^p (\Omega)$ for every $j=1, \dots, n$.
Then the proof can be concluded by the same argument used in the proof of Theorem \ref{main-th-2}. \hfill $\Box$

\section{Conclusion}
\setcounter{section}{5} \setcounter{equation}{0}
\setcounter{theorem}{0}

The method used in this article for Kolmogorov equations can be adapted to the study of a wider family of differential
operators, provided that they have a fundamental solution and that are invariant with respect to a suitable Lie group
on their domain. Sobolev inequalities for operators of this kind have been proven in \cite{quindici}. We recall here
the assumptions on the operators.

Consider a differential operator in the form
\begin{equation}\label{e1-parabolic}
     \L u : = \sum_{i,j=1}^m X_j \left( a_{ij}(x,t) X_i u \right) + X_0 u - \p_t u,
\end{equation}
where $(x,t)=(x_{1},\dots,x_{N},t)$ denotes the point in $\R^{N+1}$, and $1 \le m \le N$. The $X_{j}$'s
in \eqref{e1-parabolic} are smooth vector fields acting on $\R^{N}$, i.e.
\begin{equation*}
     X_{j}(x,t)=\sum_{k=1}^{N} b_{k}^{j}(x,t)\p_{x_{k}}, \qquad j=0,\dots,m,
\end{equation*}
and every $b_{k}^{j}$ is a $C^{\infty}$ function. In the sequel we always denote by $z=(x,t)$ the point
in $\rnn$, and by $A$ the $m \times m$ matrix $A = \left( a_{i,j} \right)_{i,j = 1, \dots, m}$. We also consider the
\emph{elliptic} analogous of $\L$
\begin{equation}\label{e1-elliptic}
     \L u : = \sum_{i,j=1}^m X_j \left( a_{ij}(x) X_i u \right)
\end{equation}
In both cases we assume that the coefficients of the matrix $A$ are bounded measurable functions, and that $A$ is
symmetric and uniformly positive, that is, there is a positive constant $\mu$ such that
\begin{equation*}
      \sum_{i,j=1}^m a_{ij}(x,t) \xi_i, \xi_j \ge \mu |\xi|^2, \qquad \text{for every} \ \xi \in \R^m,
\end{equation*}
and for every $(x,t) \in \rnn$ (or for every $x \in \rn$ as we consider the operator $\L$ in \eqref{e1-elliptic}).

Clearly, the Laplace operator $\Delta$ and the heat operator $\Delta - \partial_t$ write in the form
\eqref{e1-elliptic} and \eqref{e1-parabolic}, respectively, if we choose $X_j := \partial_{x_j}$ for $j=1, \dots, N$,
$X_0 := 0$, and the matrix $A$ agrees with the $N \times N$ identity $I_N$.
In the sequel we will use the following notations:
\begin{equation*}
  X = \left( X_{1}, \dots, X_{m} \right), \qquad Y=X_{0}-\p_{t},
\qquad  \div F = \sum_{j=1}^m X_j F_j,
\end{equation*}
for every vector field $F = (F_{1}, \dots, F_{m} )$, so that the expression $\L u$ reads
\begin{equation*}
  \L u = \div (A Xu) + Y u.
\end{equation*}
Finally, when $A$ is the $m \times m$ identity matrix, we will use the notation
\begin{equation*}
  \LO:= \sum_{k=1}^{m}X_{k}^{2}+Y.
\end{equation*}

config: arxiv


TITLE: Expectation and random sample
QUESTION [2 upvotes]: Let $X_1,X_2,...,X_n$  be a random sample of size $n$ from a population $N(\mu, \sigma^2)$ with $\sigma^2=4$. Further assume that $\mu \sim N(0,1)$.

Find $E[\mu|X]$ and $E[(\mu - E[\mu|X])^2 |X]$.
Find the precision of the normal distribution.
Find a 95% credible interval for $\mu$.

So far I understand what to do for 3, and I know that for 2 it is the inverse of the variance. But I cannot figure out 1. I have looked in my book, which is hard to read and I cannot find anything on how to do 1. If I figure out 1, then I can do 2. Please help with 1. Thank you so much!

REPLY [4 votes]: To find the expectation value and variance of $\mu$ given $X$, you need the conditional probability for $\mu$ given $X$, which is:
$$p(\mu|X)=\frac{p(\mu,X)}{p(X)}\;,$$
with
$$p(X)=\int_{-\infty}^\infty p(\mu,X)\mathrm d\mu\;.$$
For the given distributions, we have
$$p(\mu,X)=\frac1{\sqrt{2\pi}}\mathrm e^{-\mu^2/2}\prod_{i=1}^n\frac1{\sqrt{2\pi}\sigma}\mathrm e^{-(X_i-\mu)^2/(2\sigma^2)}\;.$$
We don't need the constant factors since they drop out when we calculate expectation values. With $S_0:=n$ and $S_k:=\sum\limits_{i=1}^n(X_i)^k$, this becomes
$$
p(\mu,X)\propto
\mathrm e^{-\mu^2/2}
\mathrm e^{-(\mu^2S_0-2\mu S_1+S_2)/(2\sigma^2)}\;,$$
and with $T:=S_0+\sigma^2=n+\sigma^2$ we can rewrite this as
$$
\begin{eqnarray}
p(\mu,X)
&\propto&
\mathrm e^{-(\mu^2T-2\mu S_1+S_2)/(2\sigma^2)}
\\
&=&
\mathrm e^{-T(\mu-S_1/T)^2/(2\sigma^2)}\mathrm e^{-(S_2-S_1^2/T)/(2\sigma^2)}
\\
&\propto&
\mathrm e^{-T(\mu-S_1/T)^2/(2\sigma^2)}\;,
\end{eqnarray}
$$
where again we don't need the second factor because it doesn't depend on $\mu$.
So we have a new normal distribution for $\mu$ given $X$, and we can read off the mean $S_1/T$ and the variance $\sigma^2/T$ without having to carry out any integrations.
Analyzing these results is a good exercise in Bayesianology. If we write
$$\frac{S_1}T=\frac{n\bar X-0}{n+\sigma^2}=\frac{n\sigma^{-2}\cdot\bar X-1\cdot0}{n\sigma^{-2}+1}\;,$$
$$\frac{T}{\sigma^2}=\frac{n+\sigma^2}{\sigma^2}=1+n\sigma^{-2}\;,$$
we can see that the precision of the distribution of the mean has increased by the number of measurements times the precision of the measurements, and the new expectation value for the mean is the weighted average of the prior expectation value and the mean of the measurements, weighted by the respective precisions. Thus, things are nicely linear; each measurement adds its precision to the overall precision and its value to the sum of the values, weighted with its precision, and the prior distribution can be treated as if it were an initial measurement.


TITLE: Is there a difference between the wedge product and the exterior product?
QUESTION [2 upvotes]: I haven't been able to find much on the internet regarding this distinction (if there is one). I suspect they might be different, but I'm not sure.

REPLY [4 votes]: There is no difference: "wedge product" and "exterior product" are synonyms (at least in the context of exterior algebras and differential forms).


TITLE: Finding the image of a complex-valued function and the interpretation
QUESTION [0 upvotes]: Given that $f(z) := \frac{1}{z^4 - 1}$ for all $z\in \mathbb{C} \ \{\pm 1, \pm i
\}$, show that $|f(z)| \leq \frac{1}{15}$ if $|z| \geq 2$.
I can prove this, but the interpretation of this; is it: the range of $f$ will be complex numbers, lying inside the circle with centre at the origin, radius $\frac{1}{15}$? If so, it seems that the ""majority"" of complex-valued functions would output points inside some circle (since we can always manipulate the triangle inequality) given some condition on $z$. Am I correct?
Basically my thought after doing the quesion is: "A lot of complex-valued functions can be bounded in absolute value by some real number, meaning that the function's image lies in some circle".

REPLY [0 votes]: If $|z| \ge 2$, then $|z|^4 \ge 16$. Hence $15 \le |z|^4-1 \le |z^4-1|$.


text:
{\bf Problem.} Let $N$ be the number of consecutive $0$'s at the right end of the decimal representation of the product $1!2!3!4!\cdots99!100!.$ Find the remainder when $N$ is divided by $1000$.

{\bf Level.} Level 5
{\bf Type.} Number Theory
{\bf Solution.} A number in decimal notation ends in a zero for each power of ten which divides it. Thus, we need to count both the number of 5s and the number of 2s dividing into our given expression. Since there are clearly more 2s than 5s, it is sufficient to count the number of 5s.
One way to do this is as follows: $96$ of the numbers $1!,\ 2!,\ 3!,\ 100!$ have a factor of $5$. $91$ have a factor of $10$. $86$ have a factor of $15$. And so on. This gives us an initial count of $96 + 91 + 86 + \ldots + 1$. Summing this arithmetic series of $20$ terms, we get $970$. However, we have neglected some powers of $5$ - every $n!$ term for $n\geq25$ has an additional power of $5$ dividing it, for $76$ extra; every n! for $n\geq 50$ has one more in addition to that, for a total of $51$ extra; and similarly there are $26$ extra from those larger than $75$ and $1$ extra from $100$. Thus, our final total is $970 + 76 + 51 + 26 + 1 = 1124$, and the answer is $\boxed{124}$.

set_name: MATH


TITLE: Multiple eigenvalue solutions problem
QUESTION [1 upvotes]: In a problem regarding multiple eigenvalue solutions (defective eigenvalues, complete eigenvalues,  the like) I have a 4x4 matrix with one complete eigenvalue, and another incomplete eigenvalue with a defect of 1 and a multiplicity of 3.
The associated eigenvector to the latter is defined by the system:
$2a + c = 0$
$2b + d = 0$
Thus, depending on our choices for $a, b$ and $c, d$ respectively, we obtain two eigenvectors:
$u_1 = [1,  0,  -2,  0]^T$
$u_2 = [0,  1,  0,  -2]^T$
Because this eigenvalue has defect $1$, we are tasked with finding a generalized eigenvector of rank $2$.
After solving for $(A + 2I)^2 v_2 = 0$ {the eigenvalue of concern is -2}, we have $v_2 = [0,  0,  1,  -1]^T$ is such a vector and satisfies $(A + 2I)v_2 = v_1$ {therefore $v_2$ is an associated eigenvector}.
Thus {$v_1$, $v_2$} is the length 2 chain we need. Now here is my question (I applaud those who made it this far): The book reads, "The eigenvector $v_1$ is neither of the two eigenvectors found previously, but we observe that $v_1 = u_1 - u_2$."
What is the significance of this observation?
The book continues: "For a length $1$ chain {$w_1$} to complete the picture, we can choose any combination of $u_1$ and $u_2$ that is independent of $v_1$; for instance,
$w_1 = u_1 + u_2$"
How am I able to do this? What does this mean? What leads us to this conclusion? I know that the associated eigenvectors of different eigenvalues have to be linearly independent, but do the associated eigenvectors of a single eigenvalue have to be linearly independent?
Please do excuse my tedious rambling, as I am unable to post a picture of my problem (10 reputation apparently). I also realize this is more than one question, however, I feel the longevity and detail of the problem warrants more than a single question!

REPLY [0 votes]: You need a basis of the generalized eigenspace. Therefore you need to make sure that the vectors are linearly independent.
You have $v_1,v_2$ and now you need to complete them to a basis of the generalized eigenspace by adding a vector from the eigenspace that is independent of the two. (It actually suffices that it is independent of $v_1$ but I find it easier to think about it that way.)

What is the significance of this observation?

This is just said to make clear that $v_1$ is not independent of $u_1,u_2$. So it is in the eigenspace but (necessarily) a linear combination of the two, as the dimension of the eigenspace is two.


TITLE: Proving sequence is Cauchy sequence
QUESTION [1 upvotes]: I need to examine if this sequence is Cauchy sequence using definition.
$$a_n= 1/3 + 2^2/3^2+...+n^2/3^n$$
I start with $m,n \in N, n>m$
$$ \vert(a_n - a_m)\vert = \vert(m+1)^2/3^{(m+1)}+...+n^2/3^n\vert$$
And I don't know what is bigger than this expression and what to compare it to.

REPLY [0 votes]: An idea to make things simpler: take $\;n,\,\,m=n+p\;,\;\;p\in\Bbb N\;$ , so
$$\left|a_{n+p}-a_n\right|=\frac{(n+1)^2}{3^{n+1}}+\ldots+\frac{(n+p)^2}{3^{n+p}}\le p\frac{n^2}{3^n}$$
Since for any $\;p\in\Bbb N\;$ we have that $\;\lim\limits_{n\to\infty}\;p\,\cfrac {n^2}{3^n}=0\;$ , we get that for any $\;\epsilon>0\;$ and for any $\;p\in\Bbb N\;$ there exists $\;N_{p,\epsilon}\in\Bbb N\;$ s.t.
$$n>N_{p,\epsilon}\implies p\frac{n^2}{3^n}<\epsilon$$
Fill in details and complete the proof.


TITLE: First Order Recursion
QUESTION [1 upvotes]: Well I got this problem:
$$T(0) = 0$$
$$T(n) = 3T(n/2) + n, n \geq 1$$
I simplified the equation to the following.
$$3^iT(n/2^i) + 2n((3/2)^i - 1)$$
From this point I am confused how to solve.
Thanks for any help.

REPLY [0 votes]: I claim that $\boxed{T(n) = \Theta(n^{\log_2 3})}$. This magical claim came from Case 1 of the Master Method. To see this, I will prove it by induction (I'll skip the base case).
Induction Hypothesis: Assume that there exist some $a,b,c,d>0$ such that for all $k<n$, we have:
$$
ak^{\log_2 3} + bk \leq T(k) \leq ck^{\log_2 3} - dk
$$
It remains to prove that the inequality holds true for $k=n$. Observe that:
\begin{array}{rcccl}
3T(n/2) + n &=& T(n) &=& 3T(n/2) + n \\
3\left[a\left(\dfrac{n}{2}\right)^{\log_2 3} + b\left(\dfrac{n}{2}\right)\right] + n &\leq& T(n) &\leq& 3\left[c\left(\dfrac{n}{2}\right)^{\log_2 3} - d\left(\dfrac{n}{2}\right) \right] + n\\
3\left[a\left(\dfrac{n^{\log_2 3}}{3}\right) + \left(\dfrac{b}{2}\right)n\right] + n &\leq& T(n) &\leq& 3\left[c\left(\dfrac{n^{\log_2 3}}{3}\right) - \left(\dfrac{d}{2}\right)n \right] + n\\
\left[an^{\log_2 3} + \left(\dfrac{3b}{2}\right)n\right] + n &\leq& T(n) &\leq& \left[cn^{\log_2 3} - \left(\dfrac{3d}{2}\right)n \right] + n\\
\left[an^{\log_2 3} + bn\right] + \left(\dfrac{b}{2} + 1 \right)n &\leq& T(n) &\leq& \left[cn^{\log_2 3} - dn \right] + \left(1-\dfrac{d}{2}\right)n\\
an^{\log_2 3} + bn &\leq& T(n) &\leq& cn^{\log_2 3} - dn\\
\end{array}
provided that $d \geq 2$. This completes the induction.


TITLE: The equation of the wavelength of the bremsstrahlung
QUESTION [1 upvotes]: In my text book, I came across this question:

What is the equation used to determine the shortest wavelenght of the bremsstrahlung spectrum of x-rays?

The answer was:

λ = hc/eV.  (Where: h is planck constant, c is the speed of light, e is the electron charge and V is the voltage across the cathod and the anode in Coolidge tube.

I thought the above equation is used to determine the shortest wavelength of the charecteristic spectrum of x-rays, so why it is used here? Could some ome clarify things for me, please?

REPLY [1 votes]: That represents all the kinetic energy gained by an electron $eV$ becoming a photon of energy $\frac {hc}{\lambda}$.
If electrons do not have enough energy to produce a characteristic X-ray photon then none such photons will be produced.

If the cut off wavelength for the continuous spectrum was $1\,\rm Å$ (corresponding to an electron energy of approximately $12 \,\rm kV$) then the K-lines would be produced from the Copper target but not from the Molybdenum target.
So the shortest wavelength of the continuous spectrum must be less than the wavelength of a characteristic X-ray line.
In the case of Molybdenum the minimum energy of the incoming electron has be be approximately $18 \,\rm kV$ (photon wavelength $0.7\,\rm Å$) to produce a K-line.


TITLE: Parametric equation to find distance from origin to plane
QUESTION [1 upvotes]: The equation $2x_1 - 3x_2 -6x_3 = -4$ defines a plane in $\mathbb{R}^3$
I know the normal vector $\bf a$ for this is $(2,-3,-6)$
I am trying to use the parametric equation of the line that pass through $\bf 0$ with direction vector $\bf a$.
I know how to do this with the projection. I am just using the $\frac{|c|}{||a||}$, which gave me an answer of $\frac{4}{7}$. But I do not know how to get the same answer with parametric equations.

REPLY [2 votes]: Well, that line is given by $\langle x,y,z\rangle=\langle2t,-3t,-6t\rangle.$ We know the origin is a point on this line. What we need, then, is to find $t_0$ such that $2(2t_0)-3(-3t_0)-6(-6t_0)=-4$ (why?), then find the distance from the point $\langle2t_0,-3t_0,-6t_0\rangle$ to the origin.