devoted to preliminary material. Proposition \[p:linf\_meas\] and its Corollary \[co:linf\_int\] are proved in Section \[s:linf\], while Section \[s:l1\] contains the proof of Theorem \[t:l1\] and Section \[s:bbl\] deals with Theorem \[p:bbl\] and Corollary \[p:bbl2\]. Finally Theorem \[t:decay\] is proved in Section \[s:decay\].

Examples {#s:examples}
--------

1\) As a first example we consider the case of the standard lattice $G={{\boldsymbol}{Z}}^{{N}}$, where one can take ${\Lambda_p}(v)=\gamma_{0}v^{-p/{N}}$, according to the results of [@Wang:Wang:1977], [@Ostrovskii:2005]. This is the case where comparison with the Cauchy problem for the continuous $p$-Laplacian is more straightforward. In this case $$\label{eq:dcf_euc}
  {\psi}_{r}(s)
  =
  \gamma_{0}
  s^{\frac{{N}(p-2)+pr}{{N}r}}
  \,,
  \qquad
  s>0
  \,,$$ and for example estimate becomes $$\label{eq:l1_nn_grid}
  {{\lVert{u}(t)\rVert}_{\ell^{\infty}(V)}}
  \le
  \gamma
  {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}^{\frac{p}{{N}(p-2)+p}}
  t^{-\frac{{N}}{{N}(p-2)+p}}
  \,,$$ while the critical radius for expansion of mass in amounts to $$\label{eq:bbl_nn_grid}
  R\ge
  \gamma
  {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}^{\frac{p-2}{{N}(p-2)+p}}
  t^{\frac{1}{{N}(p-2)+p}}
  \,.$$ We remark that both results formally coincide with the corresponding ones for the continuous $p$-Laplacian in ${{{\boldsymbol}{R}}^{{N}}}$, see [@DiBenedetto:Herrero:1989].\
Next we apply Theorem \[t:decay\] to the following initial data: for $x=(x_{1},\dots,x_{{N}})\in{{\boldsymbol}{Z}}^{{N}}$ set ${u}_{0}(x)=({\lvertx_{1}\rvert}+\dots+{\lvertx_{{N}}\rvert})^{-\alpha}$ for a given $0<\alpha<{N}$. Let us write here $a(s)\simeq b(s)$ if $\gamma_{0}a(s)\le b(s)\le \gamma a(s)$ for two constants independent of $s$. One can see that $${{\lVert{u}_{0}\rVert}_{\ell^{1}(B_{R}(0))}}
  \simeq
  R^{{N}-\alpha}
  \,;
  \qquad
  {{\lVert{u}_{0}\rVert}_{\ell^{q}(V\setminus B_{R}(0))}}
  \simeq
  R^{{N}-\alpha q}
  \,,$$ for all 