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 $m\ge2$ that satisfy $m>n/p$,

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