Geometric conditions for the $L^2$-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in $\mathbb{R}^d$

Abstract

Let $\mu$ be a finite Radon measure in $\mathbb{R}^d$ with polynomial growth of degree $n$, although not necessarily $n$-AD-regular. We prove that under some geometric conditions on $\mu$ that are closely related to rectifiability and involve the so-called $\beta$-numbers of Jones, David and Semmes, all singular integral operators with an odd and sufficiently smooth Calder\'on-Zygmund kernel are bounded in $L^2(\mu)$. As a corollary, we obtain a lower bound for the Lipschitz harmonic capacity of a compact set in $\mathbb{R}^d$ only in terms of its metric and geometric properties.