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Riesz transforms and elliptic PDEs with VMO coefficients

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Abstract

The present paper is concerned withL p-theory of the uniformly elliptic differential operator

$$Lu = \sum\limits_{i,j} {\frac{\partial }{{\partial x_i }}(a_{i,j} } (x)\frac{\partial }{{\partial x_i }}\,)$$

inR n 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,pRn→W 1,pRn, for every 1<p<∞. As a side benefit, we obtain the existence and uniqueness theorem for the equationL u=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 equationL u=0.

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Authors and Affiliations

  1. Department of Mathematics, Syracuse University, 13244, Syracuse, NY, USA

    T. Iwaniec

  2. Dip. di Matematica e Appl, R. Caccioppoli, Via Cintia, 80126, Napoli, Italy

    C. Sbordone

Authors
  1. T. Iwaniec
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  2. C. Sbordone
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Additional information

The research of the first author was supported by NSF Grant DMS-9401104. The research of the second author was carried out during his visit to Syracuse University and was supported by NSF Grant DMS-9401104 and by GNAFA-CNR Florence.

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Iwaniec, T., Sbordone, C. Riesz transforms and elliptic PDEs with VMO coefficients. J. Anal. Math. 74, 183–212 (1998). https://doi.org/10.1007/BF02819450

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  • DOI: https://doi.org/10.1007/BF02819450

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