\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 66, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/66\hfil Strong maximum principle]
{Remarks on the strong maximum principle for
nonlocal operators}
\author[J. Coville \hfil EJDE-2008/66\hfilneg]
{J\'er\^ome Coville}
\address{J\'er\^ome Coville \newline
Max Planck Institute for mathematical science\\
Inselstrasse 22, D-04103 Leipzig, Germany}
\email{coville@mis.mpg.de}
\thanks{Submitted January 25, 2008. Published May 1, 2008.}
\thanks{Supported by the Ceremade-Universit\'e Paris Dauphine,
and CMM-Universidad de Chile \hfill\break\indent
through an Ecos-Conicyt project.}
\subjclass[2000]{35B50, 47G20, 35J60}
\keywords{Nonlocal diffusion operators; maximum principles;
\hfill\break\indent Geometric condition}
\begin{abstract}
In this note, we study the existence of a
strong maximum principle for the nonlocal operator
$$
\mathcal{M}[u](x) :=\int_{G}J(g)u(x*g^{-1})d\mu(g) - u(x),
$$
where $G$ is a topological group acting continuously on a
Hausdorff space $X$ and $u \in C(X)$.
First we investigate the general situation and derive a pre-maximum
principle. Then we restrict our analysis to the case of homogeneous
spaces (i.e., $ X=G /H$). For such Hausdorff spaces, depending
on the topology, we give a condition on $J$ such that a strong
maximum principle holds for $\mathcal{M}$. We also revisit the classical
case of the convolution operator (i.e.
$G=(\mathbb{R}^n,+), X=\mathbb{R}^n, d\mu =dy$).
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction and Main results}
This note is devoted to the study of the strong maximum
principle satisfied by an operator
\begin{equation}\label{pajc-gen}
\mathcal{M}[u] := \int_{G}J(g)u(x*g^{-1})d\mu(g) -u(x),
\end{equation}
where $G,*,X,J,d\mu$ satisfy the following assumptions:
\begin{itemize}
\item[(H1)] $X$ is a Hausdorff space,
\item[(H2)] $G$ is a topological group acting continuously on $X$
with the operation $*$,
\item[(H3)] $d\mu$ is a Borel measure on $G$ such that for all nonempty
open sets $A\subset G$ we have $d\mu(A)>0$,
\item[(H4)] $J\in C(G,\mathbb{R})$ is a non-negative function of unit mass
with respect to $d\mu$.
\end{itemize}
Such kind of operators have been recently introduced in various
models where long range interactions play an important role, see for
example \cite{BFRW,CF,CD2, DOPT1, MMP, Sch}.
A first example of such models is given by the well known nonlocal
reaction diffusion equation below,
\begin{equation}
\frac{\partial u}{\partial t}= \int_{\mathbb{R}^n}J(x-y) u(y)\,dy - u + u(1-u)
\quad\text{in } \mathbb{R}^+\times\mathbb{R}^n. \label{pajc.nl}
\end{equation}
The above equation models the evolution of a population density
through a homogeneous environment with a constant rate of birth and death.
In this case, we have $(G,*)=(\mathbb{R}^n,+)$, $X=\mathbb{R}^n$, $J\in C(\mathbb{R}^n)$ and
$d\mu=dy$ is the Lebesgue measure.
Such an equation, with a different type of nonlinearity, appears also
in some Ising models and in ecology, see for example
\cite{BFRW,CD2,DGP,Sch} and their many references.
Other examples are given by the following two discrete versions
of \eqref{pajc.nl},
\begin{gather}
\frac{\partial u}{\partial t}=\frac{1}{2}[u(x+1)+u(x-1)-2u(x)]+ f(u)
\quad\text{in }\mathbb{R}^+\times\mathbb{R}, \label{pajc.nld1} \\
\frac{\partial u}{\partial t}=\frac{1}{2}[u(p+1)+u(p-1)-2u(p)]+ f(u)
\quad\text{in }\mathbb{R}^+\times\mathbb{Z}. \label{pajc.nld2}
\end{gather}
In both situations the discrete diffusion operator can be reformulated
in terms of a nonlocal operator $\mathcal{M}$ defined in \eqref{pajc-gen}.
Indeed, in these two cases, by taking $(G,*)=(\mathbb{Z},+)$, $d\mu$
the counting measure and $J\in C(\mathbb{Z},\mathbb{R})$ defined as follows:
$$
J(p):=\begin{cases}
\frac{1}{2}&\text{if } p=-1 \text{ or } p=1,\\
0 &\text{otherwise,}
\end{cases}
$$
it follows that for any $x$ in the Hausdorff space $\mathbb{R}$ or $\mathbb{Z}$ we have
$$
\frac{1}{2}[u(x+1)+u(x-1)-2u(x)]=\int_{G}J(g)u(x*g^{-1})d\mu(g) -u(x).
$$
As for their continuous version \eqref{pajc.nl},
equations \eqref{pajc.nld1} and \eqref{pajc.nld2} appear in
discrete reaction diffusion models describing a wide variety of
phenomenon, ranging from combustion to nerve propagation and phase
transitions. We point the interested reader to \cite{CF,CG,HHZ}
and the many references cited therein.
Another example comes from the following size structured population model,
recently introduced by Perthame \textit{et al.} in \cite{MMP,PR},
\begin{equation}
\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}
=\int_{0}^{+\infty}u(\frac{x}{y})b(y)dy - u(x)
\quad\text{in }\mathbb{R}^+\times\mathbb{R}^+. \label{pajc.fg}
\end{equation}
In such case, we have $(G,*)=(\mathbb{R}^{+}\setminus\{0\},\cdot)$,
$X=\mathbb{R}^{+}$
and $d\mu(y) = dy$ is the Lebesgue measure.
In all these examples, depending on the group and the measure considered,
the properties satisfied by the corresponding operator $\mathcal{M}$ show significant
differences. However, as for the classical Laplace operator ($\Delta$),
they all satisfy the following positive maximum principle.
\begin{definition}[Courr\`ege Positive maximum principle \cite{BCP}]
An operator $A\in \mathcal{L}(C(X))$ is said to satisfy the positive maximum
principle if for all $f \in C(X)$ and $x\in X$ such that
$f(x)=\sup (f)$ we have $A(f)(x)\le 0$.
\end{definition}
For the Laplace operator ($\Delta$), in addition to the above property,
it is well known, see \cite{GT,PW}, that a sub-harmonic functions
satisfies a strong maximum principle:
\begin{theorem}[Elliptic Strong maximum principle] \label{thm1.2}
Let $u\in C^2(\mathbb{R}^n)$ be such that
$\Delta u\ge 0$ in $\mathbb{R}^n$.
Then $u$ cannot achieve a global maximum without being constant.
\end{theorem}
In this note, we investigate the conditions on $(G,*),X,J$ and $d\mu$
in order to achieve such strong maximum principles for $\mathcal{M}$.
More precisely , we are interested in finding simple conditions on
$(G,*),X,J$ and $d\mu$ for the strong maximum principle to hold:
\begin{theorem}[Strong maximum principle] \label{thm1.3}
Let $u\in C(X,\mathbb{R})$ be such that
$\mathcal{M}[u]\ge 0$ in $X$.
Then $u$ cannot achieve a global maximum without being constant.
\end{theorem}
In the analysis of nonlinear elliptic equations, the strong maximum
principle plays a very important role in proving key \textit{ a priori }
estimates. It is expected that such a strong maximum property for $\mathcal{M}$
will play a similar role in the analysis of nonlinear equations
involving nonlocal operators. It is therefore of great interest
to investigate the conditions on $G, X, d\mu$ and $J$ in order that
a strong maximum principle hold for $\mathcal{M}$.
In this direction, we first establish a generic result satisfied by all
operators $\mathcal{M}$. More precisely, we show the following result.
\begin{theorem}[Pre-maximum principle] \label{pajc.theo.1}
Let $(G,*,X,J,d\mu)$ be such that $(H1-H4)$ are satisfied and let
$u\in C(X,\mathbb{R})$ be such that
$$
\mathcal{M}[u] \ge 0\quad \quad ( \text{resp.} \le 0 ).
$$
Assume that $u$ achieves a global maximum (resp. minimum) at some
point $x_0\in X$ and let $F_{x_0}$ denote the smallest closed subset
of $X$ such that
\begin{itemize}
\item $x_0\in F_{x_0}$,
\item $F_{x_0}*\{g^{-1} \in G|J(g)>0\}\subset F_{x_0}$.
\end{itemize}
Then $u\equiv u(x_0)$ in $F_{x_0}$.
\end{theorem}
Our next result is a characterization of the set $F_{x_0}$ defined
in the above Theorem \ref{pajc.theo.1}.
\begin{proposition} \label{pajc.prop.descr}
Let $(G,*,X,J,d\mu)$ be such that {\rm (H1)--(H4)} are satisfied and
let $F_{x_0}$ be the set defined in Theorem \ref{pajc.theo.1}.
Then
$$
F_{x_0}=\overline{\bigcup_{n\in \mathbb{N}}F_n},
$$
where the $F_n$ are defined by induction as follows
\[
F_0=\{x_0\},\quad\text{and}\quad
\forall n\ge 0 \quad F_{n+1}:= F_n*\{g^{-1} \in G|J(g)>0\}.
\]
\end{proposition}
In view of the above generic result, in order to get a strong maximum
principle for $\mathcal{M}$, we need to find conditions on $(G,*),X,d\mu$ and
$J$ which imply that $F_{x_0}=X$. Note that, from the characterization
of the set $F_{x_0}$, the condition $F_{x_0}=X$ implies that
$X=F_{x_0}\subset \overline{orb(x_0)}:=\overline{\{x*g^{-1}|g\in G\}}\subset X$,
which means that $orb(x_0)$ is a dense set in $X$.
Observe that for the discrete diffusion operator considered in
\eqref{pajc.nld1}, the set $orb(x)$ is never dense in $\mathbb{R}$.
Therefore, we cannot expect to have a strong maximum principle in
such situation. On the contrary, for the same diffusion operator
considered in \eqref{pajc.nld2}, the set $orb(x)$ is always dense in $\mathbb{Z}$.
Moreover we can easily see that in this situation the discrete operator
satisfies a strong maximum principle.
Considering the above remarks, in what follows we restrict our attention
to the case of Hausdorff homogeneous spaces $X$
(i.e. $X:=G/H$, where $H$ is a closed subgroup of $G$).
For such Hausdorff spaces, the set $orb(x)$ is always dense in $X$
and sufficient conditions on $(G,*),X,J$ and $d\mu$ for the strong
maximum principle to hold reduce to find some simple conditions on $J$.
In this direction, we first give a sufficient condition on $J$ to ensure
that $\mathcal{M}$ satisfies the strong maximum principle. Namely, we have
the following result.
\begin{theorem} \label{pajc.theo.2}
Let $X$ be a connected homogeneous space and let $(G,*),J,d\mu$ be as
in Theorem \ref{pajc.theo.1}. Let $e$ be the unit element of $G$
and assume that $J(e)>0$. Then $\mathcal{M}$ satisfies the strong maximum principle.
\end{theorem}
When $X$ is a compact connected homogeneous spaces, we can generalize
the previous statement to the following result.
\begin{theorem} \label{pajc.theo.3}
Let $X$ be a connected compact homogeneous space and $(G,*),J,d\mu$ as in
Theorem \ref{pajc.theo.1}. Then $\mathcal{M}$ satisfies the strong maximum principle.
\end{theorem}
Next, we state optimal condition on $J$ in two special cases.
Namely, we first retrieve the Markov necessary and sufficient
condition for the convolution operator (i.e.
$(G,*)=(\mathbb{R}^n,+), X=\mathbb{R}^n, d\mu=dy$), which is well known among experts
in stochastic processes.
\begin{theorem}[Markov condition]\label{pajc.theo.4}
Assume that $(G,*)=(\mathbb{R}^n,+), X=\mathbb{R}^n$ and $d\mu=dy$. Then $\mathcal{M}$ satisfies
the strong maximum principle iff the convex hull of
$\{y\in \mathbb{R}^n|\, J(y)>0\}$ contains $0$.
\end{theorem}
As a consequence of the above Markov condition, we derive the following
optimal condition when $(G,*)=(\mathbb{R}^+\setminus \{0\},\bullet), X=\mathbb{R}^+$ and
$d\mu=dy$:
\begin{corollary}\label{pajc.theo.5}
Assume that $(G,*)=(\mathbb{R}^+\setminus \{0\},\bullet), X=\mathbb{R}^+$ and $d\mu=dy$.
Then $\mathcal{M}$ satisfies the strong maximum principle iff there exists $2$
points $x_1$ and $x_2$ such that $J(x_i)>0$ and $0< x_1\le 1\le x_2)$.
\end{corollary}
\subsection{General comments}
We first note that, provided an extra assumption on the sign of the
maximum (minimum) is made, we can easily extend the above results
to operators $\mathcal{M}[u] +c(x)u$ with non-positive zero order term
(i.e. $c(x)\le 0$).
As for $\mathcal{M}$, the operator $\mathcal{M} +c(x)$ satisfies a Courr\`eges
positive maximum principle \cite{BCP}, which in this case state
the following definition.
\begin{definition}[Positive maximum principle] \rm
An operator $A\in \mathcal{L}(C(X))$ is said to satisfy the positive maximum
principle if for all $f \in C(X)$ and $x\in X$ such that $f(x)\ge 0$
and $f(x)=\sup (f)$ we have $A(f)(x)\le 0$.
\end{definition}
In our investigation of homogeneous spaces, we also observe that to obtain
a strong maximum principle for $\mathcal{M}$, we only need the inequality
$\mathcal{M}[u]\ge 0$ at points where the function $u$ achieves its global
maximum. As a consequence, in the two situation investigated above
(Theorems \ref{pajc.theo.2} and \ref{pajc.theo.3} ), we have
the following characterization:
\begin{proposition} \label{prop1.11}
Let $(G,*), X, d\mu$ and $J$ be as in Theorem \ref{pajc.theo.2}
or \ref{pajc.theo.3}. Then for all $u\in C(X)$ and $x\in X$ such
that $u(x)=\sup (u)$ we have the following alternatives: Either
\begin{itemize}
\item there exists $y\in X$ such that $u(y)=u(x)$ and
$\mathcal{M}[u](y)<0$, or
\item $u$ is a constant.
\end{itemize}
\end{proposition}
We also want to point out that, although the Markov condition is well
known among experts in stochastic analysis, we present here a simple
analytical proof, which we believe is new. Using such a point of
view allows us to relate a simple recovering problem with the
conditions for the strong maximum principle.
The outline of this note is the following.
In the two first Sections (Sections \ref{pajc.s.pre} and
\ref{pajc.s.premp}), we recall some basic topological results and prove
the pre-maximum principle and the characterization of $F_x$
(Theorems \ref{pajc.theo.1} and Proposition \ref{pajc.prop.descr}).
Then in Section \ref{pajc.s.smp}, we establish the strong maximum
principle (Theorems \ref{pajc.theo.2} and \ref{pajc.theo.3}).
Finally, in the last section, we prove the optimal conditions
(Theorems \ref{pajc.theo.4} and \ref{pajc.theo.5}).
\medskip
\section{Preliminaries\label{pajc.s.pre}}
In this section, we first present some definitions and notation that we
will use in this paper. Then we establish a useful proposition.
Let us first define some notations:
\begin{itemize}
\item $\Sigma:=\{g^{-1}\in G|J(g)>0\}$.
\item For a function $u$, we define $\Gamma_{y}:=\{x\in X|u(y)=u(x)\}$.
\end{itemize}
Let us now introduce the following two definitions:
\begin{definition} \label{def2.1} \rm
Let $A\subset X$ and $B\subset G$ be two sets, then we define $A*B\subset X$ as follows
$$
A*B:=\{a*b\;|\; a\in A \text{ and } b\in B\}.
$$
\end{definition}
\begin{definition} \label{def2.2} \rm
Let $A\subset X$ and $B\subset G$ be two sets, then
we say that $A$ is $B*$ stable if $$A*B\subset A.$$
\end{definition}
Next, let us recall the following basic property of $*$ stable sets.
\begin{proposition} \label{pajc.prop.stabi}
Let $A\subset X$ and $B\subset G$ be two sets. If $A$ is $B*$ stable,
then $\bar A$ is $B*$ stable, where $\bar A$ denotes the closure of $A$.
\end{proposition}
\begin{proof}
Let $y\in \bar A*B$ and $V(y)$ be an open neighbourhood of $y$.
By definition, we have $y:=x_1*b_1$ for some $x_1 \in \bar A$ and $b_1\in B$.
Since the operation $*$ is continuous,
the following map $T$ is continuous:
\begin{align*}
T:X &\to X\\
z&\mapsto z*b_1.
\end{align*}
Therefore, $T^{-1}(V(y))$ is a open neighbourhood of $x_1$.
Since $\bar A$ is a closed set and $x_1\in \bar A$, we have
$T^{-1}(V(y))\cap A\neq\emptyset$.
By definition of $T^{-1}(V(y))$, using the stability of $A$,
it follows that for all $z\in T^{-1}(V(y))\cap A$, $z*b_1\in A$.
Therefore,
$$
z*b_1\in V(y)\cap A\quad \text{for all }
z \in T^{-1}(V(y))\cap A,
$$
and yields
$ V(y)\cap A\neq \emptyset$.
The above argumentation, being independent of the choice of $V(y)$, shows that $y\in \bar A$.
Now, since $y$ is arbitrary, we end up with
$ \bar A*B \subset \bar A$.
\end{proof}
\section{Pre-maximum principle and Characterizations of $F_x$
\label{pajc.s.premp}}
In this Section we prove Theorem \ref{pajc.theo.1} and Proposition
\ref{pajc.prop.descr}.
Let us first start with the proof of the pre-maximum principle.
\begin{proof}[Proof of Theorem \ref{pajc.theo.1}]
The proof is rather simple.
Let us first recall the definition of $\Gamma_{x_0}$:
\begin{equation}
\Gamma_{x_0}:=\{x\in X|u(x)=u(x_0)\}.\label{defG}
\end{equation}
Since $u$ is continuous, $\Gamma_{x_0}$ is a closed subset of $X$.
Now observe that $\Gamma_{x_0}$ is $\Sigma*$ stable
(i.e. $\Gamma_{x_0}*\Sigma\subset \Gamma_{x_0}$).
Indeed, choose any $\bar x\in \Gamma_{x_0}$. At $\bar x, u$
satisfies
\begin{equation*}
0\le \mathcal{M}[u](\bar x)
=\int_{G}J(g)u(\bar x*g^{-1})\,d\mu-u(\bar x)
=\int_{G}J(g)[u(\bar x*g^{-1})-u(\bar x)]\,d\mu \le 0.
\end{equation*}
Therefore,
\begin{equation}\label{pajc.eq.zero}
\int_{G}J(g)[u(\bar x*g^{-1})-u(\bar x)]\,d\mu=0.
\end{equation}
Using that $J\ge 0$ and that for all
$g\in G, [u(\bar x*g)-u(\bar x)]\le 0$, \eqref{pajc.eq.zero} yields
$$
u(\bar x*g^{-1})=u(\bar x)\quad\text{ for all }\quad g \in \Sigma.
$$
Thus, we have
\begin{align*}
&u(y)=u(x_0) \quad \text{ for all }\quad y\in \{\bar x\}*\Sigma.
\end{align*}
Hence,
$\{\bar x\}*\Sigma\subset\Gamma_{x_0}$.
Since this computation holds for any element $\bar x$ of $\Gamma_{x_0}$,
we have $\Gamma_{x_0}*\Sigma\subset\Gamma_{x_0}$.
Recall now that $F_{x_0}$ is the smallest closed subset of $X$ such that
\begin{itemize}
\item $x_0\in F_{x_0}$,
\item $F_{0}*\Sigma\subset F_{x_0}$.
\end{itemize}
Since $\Gamma_{x_0}$ satisfies the above conditions, we then have
$F_{x_0}\subset\Gamma_{x_0}$.
\end{proof}
Note that $\Gamma_{x_0}$ is independent of the choice of the point where $u$
takes its global maximum. Indeed, we easily see that $\Gamma_{x_0}=\Gamma_{y}$
for any $y\in \Gamma_{x_0}$. On the contrary, the set $F_{x_0}$ strongly
depends on $x_0$ and there is no reason to always have $F_{x_0}=F_{y}$.
Indeed, for $X=G=\mathbb{R}$, if $\Sigma=\mathbb{R}^+$ then for $x_00\}$ contains $0$.
\end{theorem}
\begin{proof}
Let us start with the necessary condition.
Assume that the Markov condition fails. We will show
that $\mathcal{M}$ does not satisfy the strong maximum principle.
To this end, we construct a non constant function $u$ that achieves
a global maximum and satisfies $\mathcal{M}[u] \ge 0$.
Let us denote $\mathop{\rm conv}(\{y\in\mathbb{R}^n| J(y)>0\} )$ the convex hull of
$\{y\in\mathbb{R}^n| J(y)>0\}$. By assumption,
$0\not\in \mathop{\rm conv}(\{y\in\mathbb{R}^n| J(y)>0\} )$.
Using the Hahn-Banach Theorem, there exists a hyperplane $H$ such that
$\mathop{\rm conv}(\{y\in\mathbb{R}^n| J(y)>0\})\subset H^+$, where
$H^+:=\{x\in\mathbb{R}^n|x_n\ge0\}$ in an orthonormal basis $(e_1; e_2;\ldots; e_n)$.
Let $v$ be a non-increasing function that is constant in $\mathbb{R}^-$,
and let us compute
$\mathcal{M}[u]$ with $u(x):=v(x_n)$.
Since the Lebesgue measure is invariant under rotation and
$\mathop{\rm supp}(J)\subset H^+$ we have
\begin{align*}
\mathcal{M}[u]&=\int_{\mathbb{R}^{n-1}}\int_{\mathbb{R}}J(t,x_n-y_n) [v(y_n)-v(x_n)]\,dx_n \,dt\\
&=\int_{\mathbb{R}^{n-1}}\int_{-\infty}^{x_n}J(t,x_n-y_n) [v(y_n)-v(x_n)]\,dx_n
\,dt.
\end{align*}
Therefore, since $v$ is non increasing we end up with
$\mathcal{M}[u]\ge 0$.
Since $u$ achieves a global maximum without being constant,
$u$ is our desired function.
Let us now turn our attention to the sufficient condition. Assume that
$0\in \mathop{\rm conv}(\{y\in\mathbb{R}^n| J(y)>0\})$, then there exists a simplex
$S(p_i)$ formed by $n+1$ points of $\mathbb{R}^n$ such that $0\in S$ and $J(p_i)>0$.
By continuity, we can always assume that $(p_1,\dots, p_n)$ is a
basis of $\mathbb{R}^n$. Let us now rewrite $x_0$ in the basis $(p_1,\dots,p_n)$:
$$
x_0=-a_1p_1\dots-a_np_n \quad\text{with } a_i\ge 0.
$$
Observe now that for $\mathbb{R}^n$ equipped with the sup norm associated to
the base $(p_1,\dots,p_n)$, there exists $r>0$ so that
$B(x_0,r)\subset\{J>0\}$.
Now for all integer $m>0$, set $ y_m=mp_0+[ma_1]p_1+\dots+[ma_n]p_n$, where
$[\cdot]$ denotes the integer part.
Now let $u$ be a continuous function satisfying
$\mathcal{M}[u]\ge 0$ and that achieves a global maximum at some point $z\in \mathbb{R}$.
Without loss of generality, we may always assume that $z=0$.
Indeed, if $z\neq 0$, we consider the function $u_{z}(x):=u(x-z)$,
instead of $u$.
We easily see that $u_z$ achieves a global maximum at $0$ and satisfies
$\mathcal{M}[u_z]\ge 0$.
Using now Theorem \ref{pajc.theo.1}, we see that for all $m\in \mathbb{N}$,
$$
\|y_m\|<1 \quad \text{and}\quad B(y_m;mr)\subset \Gamma_0.
$$
Therefore,
$$
\bigcup_{m\in\mathbb{N}} B(y_m;mr) \subset \Gamma_0.
$$
Hence, $\mathbb{R}^n\subset \Gamma_0$.
\end{proof}
The above necessary and sufficient condition for the convolution operator
can be weakened depending on the underlying topological structure of
the space. In particular, we have in mind the following setting.
Since $\mathcal{M}$ is translation invariant, $\mathcal{M}$ is also an operator on the
set of periodic functions. On this set of functions, the strong
maximum principle always holds.
This condition is not so surprising since the additional periodic
structure will in some sense compactify the homogeneous space $\mathbb{R}^n$.
\subsection*{Another special case:
$X=\mathbb{R}^+,(G,*)=(\mathbb{R}^{+}\setminus\{0\},\cdot)$
and $d\mu=dy$}
In this situation,
$$
\mathcal{M}[u]:=\int_{\mathbb{R}^+}J(y)u\big(\frac{x}{y}\big)\,dy -u,
$$
and the above operator has essentially the same property as the usual
convolution operator. Indeed,
let us make the following change of variables $x:=e^t$. Then we have
$$
\mathcal{M}[u](e^t)=\int_{\mathbb{R}}\tilde J(t-s)u(e^s)\,ds-u(e^t),
$$
where $\tilde J(t):=J(e^t)e^t$.
Therefore, letting $v(t)=u(e^t)$, we end up with
$$
\mathcal{R}[v](t)=\tilde J \star v(t) -v(t)\quad \text{in } \mathbb{R},
$$
with $\int_{\mathbb{R}}\tilde J(t)dt=1$.
Hence, the optimal condition to achieve a strong maximum principle for
such a kind of operator will be of the same type as the one used for
the convolution operator.
Namely, there exists two points $a<1**0$ and $J(b)>0$.
This condition corresponds to the one given for the convolution operator
which is the existence of two points $a'<0< b'$ such that $\tilde J(a')>0$
and $\tilde J(b')>0$.
The above observation proves Corollary \ref{pajc.theo.5}.
\subsection*{Acknowledgments}
The author would like to warmly thanks Dr. Pascal Autissier for
suggesting me this problem and is constant support.
\begin{thebibliography}{10}
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\end{document}
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