JournalsrsmupVol. 131falsepp. 67–76

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

Abstract

In this paper a Sobolev inequality, which generalizes the ordinary Banach algebra property of such spaces, is established; for p[1,)p\in[1,\infty), n,mZ+n,m\in\mathbb Z^+, and m2m\ge2 that satisfy m>n/pm>n/p,

ϕψm,p,ΩK[(supΩsϕ)ψm,p,Ω+(ψm1,q,Ω+ψm1,p,Ω)]ϕm,p,Ω\left\Vert\phi\psi\right\Vert_{m,p,\Omega}\le K \left[ \left( \sup_{\Omega_s} \left\vert \phi \right\vert \right) \left\Vert\psi\right\Vert_{m,p,\Omega} + \Bigl(\left\Vert\psi\right\Vert_{m-1,q,\Omega}+\left\Vert\psi\right\Vert_{m-1,p,\Omega}\Bigr)\right] \left\Vert\phi\right\Vert_{m,p,\Omega}

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