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

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