Analytic and Signed Riesz Capacities in BMO, Lip(s) and L∞ Spaces

Abstract

This thesis develops a unified distributional framework for the study of analytic capacity and its generalisation to higher dimensions via the $\alpha$-Riesz transform. The work is organised around four main themes.

In the first part, the Cauchy transform is extended to compactly supported distributions, yielding a flexible reformulation of analytic capacity. Within this framework, the classical $L^\infty$ Hausdorff content estimate for analytic capacity, due to Painlevé, the BMO and $\mathrm{Lip}(s)$ variants due to Verdera [Ver86] and Melnikov [Mel69], are presented with unified proofs written in distributional language. These arguments serve as an explicit template for the subsequent chapters.

In the second part, we introduce the $\alpha$-Riesz transform and study criteria for its $L^2(\mu)$-boundedness with respect to non-doubling Radon measures. For $0 < \alpha \leq 1$, we present a proof of a $T1$ theorem by combining the Good Lambda method [Tol14] with a Menger-type curvature $c_\alpha$ arising from symmetrisation of the kernel. A $Tb$ theorem, which relaxes the testing condition to a suitable function $b \in L^\infty(\mu)$, is also stated and will play a key role in the final chapter.

The third part introduces the $\alpha$-Riesz capacity $\gamma_\alpha$ and presents, following Prat [Pra04b], its Hausdorff content characterisation in the $L^\infty$ setting. We then construct a family of self-similar Cantor sets with $0 < \mathcal{H}^{\alpha}(E) < \infty$ but $\gamma_\alpha(E) = 0$, providing a negative answer to the question of whether positive $\mathcal{H}^{\alpha}$ measure implies positive capacity. Building on this distributional framework, we prove complete characterisations of the BMO and $\mathrm{Lip}(s)$ variants, $\gamma_{\alpha,\mathrm{BMO}}$ and $\gamma_{\alpha,\mathrm{Lip}(s)}$, in terms of Hausdorff content via purely real-variable methods.

Finally, following Prat [Pra04b], the case $0 < \alpha < 1$ is studied in depth. Using the $Tb$ theorem together with curvature divergence and kernel regularisation arguments, it is shown that every compact set $E \subset \mathbb{R}^d$ with $\mathcal{H}^{\alpha}(E) < \infty$ satisfies $\gamma_\alpha(E) = 0$.