$q>{N}/\alpha$. Therefore in this case $${T}_{{u}_{0}}(R,0)
  \simeq
  R^{\alpha(p-2)+p}
  \,,$$ and the estimate in essentially amounts to the decay rate $t^{-\alpha/(\alpha(p-2)+p)}$, which is the expected one in view of the results of [@Tedeev:1991].

2\) One can treat also other examples of product graphs; for instance if $H$ is a finite connected graph we let $G=H\times {{\boldsymbol}{Z}}^{{N}}$ and recover results similar to the ones of the previous example.

3\) All examples where the Faber-Krahn function is estimated for $p=2$ yield also examples in our case of $p>2$, as it follows from applying Hölder’s inequality; see e.g., [@Coulhon:Grigoryan:1998], [@Barlow:Coulhon:Grigoryan:2001].

Preliminary material {#s:prelim}
====================

We use for $f:V\to {{\boldsymbol}{R}}$ the notation $${D}_{y}f(x)
  =
  f(y)-f(x)
  =
  -
  {D}_{x}f(y)
  \,,
  \qquad
  x\,,y\in V
  \,.$$

Caccioppoli type inequalities {#s:prelim_cacc}
-----------------------------

\[l:monot\] Let $q>0$, $p>2$, $h\ge 0$, ${u}$, ${v}:V\to{{\boldsymbol}{R}}$. Then for all $x$, $y\in V$ $$\begin{gathered}
    \label{eq:monot_n}
    \big(
    {\lvert{D}_{y}{u}(x)\rvert}^{p-2}
    {D}_{y}{u}(x)
    -
    {\lvert{D}_{y}{v}(x)\rvert}^{p-2}
    {D}_{y}{v}(x)
    \big)
    {D}_{y}{{\ifthenelse{\equal{*}{*}}{({u}(x)-{v}(x)-h)_{+}}{\left({u}(x)-{v}(x)-h\right)_{+}}}}^{q}
    \\
    \ge
    \gamma_{0}
    {\left|
      {D}_{y}{{\ifthenelse{\equal{*}{*}}{({u}(x)-{v}(x)-h)_{+}}{\left({u}(x)-{v}(x)-h\right)_{+}}}}^{\frac{q-1+p}{p}}
    \right|}^{p}
    \,.
  \end{gathered}$$

First we remark that we may assume $h=0$, by renaming $\tilde{v}={v}+h$. The corresponding version of clearly holds true if ${D}_{y}{u}(x)={D}_{y}{v}(x)$.

If ${D}_{y}{u}(x)\not={D}_{y}{v}(x)$ the left hand side of with $h=0$ can be written as, on appealing also to a classical elementary result in monotone operators, see 