# A product property of Sobolev spaces with application to elliptic estimates

### Henry C. Simpson

University of Tennessee, Knoxville, USA### Scott J. Spector

Southern Illinois University, Carbondale, USA

## Abstract

In this paper a Sobolev inequality, which generalizes the ordinary Banach algebra property of such spaces, is established; for $p∈[1,∞)$, $n,m∈Z_{+}$, and $m≥2$ that satisfy $m>n/p$,

$∥ϕψ∥_{m,p,Ω}≤K[(Ω_{s}sup ∣ϕ∣)∥ψ∥_{m,p,Ω}+(∥ψ∥_{m−1,q,Ω}+∥ψ∥_{m−1,p,Ω})]∥ϕ∥_{m,p,Ω}$

for all $ϕ,ψ∈W_{m,p}(Ω)$ that satisfy $sptψ⊂Ω_{s}⊂Ω$ and domains $Ω⊂R_{n}$ that are nonempty, open, and satisfy the cone condition. Here $q=p$ if $p>n$, $q∈(n/Υ,pn/(n−p)]$ if $n>p$, $q∈(n/Υ,∞)$ if $p=n$, $K=K(n,p,m,q,C)$, where $C$ is the cone from the cone condition, and $Υ:=[[n/p]]$, the largest integer less than or equal to $n/p$.

## Cite this article

Henry C. Simpson, Scott J. Spector, A product property of Sobolev spaces with application to elliptic estimates. Rend. Sem. Mat. Univ. Padova 131 (2014), pp. 67–76

DOI 10.4171/RSMUP/131-5