Riesz transforms and elliptic PDEs with VMO coefficients

T. Iwaniec, C. Sbordone

Research output: Contribution to journalArticlepeer-review

67 Scopus citations

Abstract

The present paper is concerned with Lp-theory of the uniformly elliptic differential operator ℒu = Σij∂/∂Cursive Greek chii(ai,j (Cursive Greek chi)∂u/∂Cursive Greek chij) in ℝn with coefficients of vanishing mean oscillation. Recent estimates for the Riesz transform combined with Fredholm index theory enable us to establish invertibility of the map ℒ: W1,p(ℝn) → W-1,p(ℝn), for every 1< p < ∞. As a side benefit, we obtain the existence and uniqueness theorem for the equation ℒu = μ with a signed measure in the right hand side. Within the framework of quasiconformal mappings we give a fairly general method of constructing solutions to the homogeneous equation ℒu = 0.

Original languageEnglish (US)
Pages (from-to)183-212
Number of pages30
JournalJournal d'Analyse Mathematique
Volume74
DOIs
StatePublished - Jan 1 1998

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

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