Estimates for the Multiplicative Square Function of Solutions to Nondivergence Elliptic Equations

Rivera-Noriega, Jorge

doi:https://doi.org/10.1155/2007/92354

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2007 | Article ID 092354 | https://doi.org/10.1155/2007/92354

Estimates for the Multiplicative Square Function of Solutions to Nondivergence Elliptic Equations

Jorge Rivera-Noriega¹

Academic Editor: Vesa Mustonen

Received26 Jun 2006

Accepted18 Dec 2006

Published19 Feb 2007

Abstract

We prove distributional inequalities that imply the comparability of the Lp norms of the multiplicative square function of u and the nontangential maximal function of logu, where u is a positive solution of a nondivergence elliptic equation. We also give criteria for singularity and mutual absolute continuity with respect to harmonic measure of any Borel measure defined on a Lipschitz domain based on these distributional inequalities. This extends recent work of M. González and A. Nicolau where the term multiplicative square functions is introduced and where the case when u is a harmonic function is considered.

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Copyright

Copyright © 2007 Jorge Rivera-Noriega. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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