A sublinear Sobolev inequality for $p$-superharmonic functions

Abstract:

We establish a “sublinear” Sobolev inequality of the form \[ \left (\int _{\mathbb {R}^n} u^{\frac {nq}{n-q}} dx\right )^{\frac {n-q}{nq}}\leq C \left (\int _{\mathbb {R}^n}|D u|^{q} dx\right )^{\frac {1}{q}}\] for all global $p$-superharmonic ($1<p<2$) functions $u$ in $\mathbb {R}^n$, $n\geq 2$, with $\inf _{\mathbb {R}^n} u=0$ and $p-1<q<1$. The same result also holds for the class of $\mathcal {A}$-superharmonic functions. More general sublinear trace inequalities, where Lebesgue measure is replaced by a general measure, are also considered.