A product property of Sobolev spaces with application to elliptic estimates

Abstract

In this paper a Sobolev inequality, which generalizes the ordinary Banach algebra property of such spaces, is established; for $p\in [1,\infty )$, $n,m\in {\mathbb{Z}}^{+}$, and $m\ge 2$ that satisfy $m>n/p$,

for all $\varphi ,\psi \in W^{m,p}(\Omega )$ that satisfy $\mathrm{spt}\psi \subset {\mathrm{\Omega }}_{\mathrm{s}}\subset \mathrm{\Omega }$ and domains $\Omega \subset {\mathbb{R}}^n$ that are nonempty, open, and satisfy the cone condition. Here $q=p$ if $p>n$, $q\in (n/{\rm Y},pn/(n-p)]$ if $n>p$, $q\in (n/{\rm Y},\infty )$ if $p=n$, $K=K(n,p,m,q,\mathcal{C})$, where $\mathcal{C}$ is the cone from the cone condition, and ${\rm Y}:=[[n/p]]$, the largest integer less than or equal to $n/p$.