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

Abstract

Let μ be a finite Radon measure in ℝd 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 Calderon-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 ℝd only in terms of its metric and geometric properties.