Journal d’Analyse Mathématique

, Volume 74, Issue 1, pp 183–212 | Cite as

Riesz transforms and elliptic PDEs with VMO coefficients

  • T. Iwaniec
  • C. Sbordone
Article

Abstract

The present paper is concerned withLp-theory of the uniformly elliptic differential operator
$$Lu = \sum\limits_{i,j} {\frac{\partial }{{\partial x_i }}(a_{i,j} } (x)\frac{\partial }{{\partial x_i }}\,)$$
inRn 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:W-1,pRnW1,pRn, for every 1<p<∞. As a side benefit, we obtain the existence and uniqueness theorem for the equationLu=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 equationLu=0.

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Copyright information

© Hebrew University of Jerusalem 1998

Authors and Affiliations

  • T. Iwaniec
    • 1
  • C. Sbordone
    • 2
  1. 1.Department of MathematicsSyracuse UniversitySyracuseUSA
  2. 2.Dip. di Matematica e ApplR. CaccioppoliNapoliItaly

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