Geometric conditions for the L²-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in ℝ^d

2015/05/27 by Girela-Sarrión, Daniel
#42B20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1505.07264

Abstract

Let μ be a finite Radon measure in ℝ^d with polynomial growth of degree n, although not necessarily n-AD-regular. We prove that under some geometric conditions on μ that are closely related to rectifiability and involve the so-called β-numbers of Jones, David and Semmes, all singular integral operators with an odd and sufficiently smooth Calderón-Zygmund kernel are bounded in L²(μ). As a corollary, we obtain a lower bound for the Lipschitz harmonic capacity of a compact set in ℝ^d only in terms of its metric and geometric properties.

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