Unique continuation at the boundary for harmonic functions in C1C^1 domains and Lipschitz domains with small constant

Let $\Omega\subset\mathbb R^n$ be a $C^1$ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if $u$ is a function harmonic in $\Omega$ and continuous in $\overline \Omega$ which vanishes in a relatively open subset $\Sigma\subset\partial\Omega$ and moreover the normal derivative $\partial_\nu u$ vanishes in a subset of $\Sigma$ with positive surface measure, then $u$ is identically $0$.Comment: Minor adjustments and more details in the appendix about the Whitney cube

Painleve's problem and the semiadditivity of analytic capacity

Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.Comment: 42 page

Characterization of the atomic space H1H^1 for non doubling measures in terms of a grand maximal operator

Let $\mu$ be a Radon measure on $R^d$, which may be non doubling. The only condition that $\mu$ must satisfy is $\mu(B(x,r))\leq C r^n$, for all $x,r$ and for some fixed $0<n\leq d$. Recently we introduced spaces of type $BMO(\mu)$ and $H^1(\mu)$ which proved to be useful to study the $L^p(\mu)$ boundedness of Calder\'on-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic space $H^1(\mu)$ in terms of a grand maximal operator $M_\Phi$ is given. It is shown that $f$ belongs to $H^1(\mu)$ iff $f\in L^1(\mu)$, $\int f d\mu=0$ and $M_\Phi(f)\in L^1(\mu)$, as in the usual doubling situation. The lack of any regularity condition on $\mu$, apart from the size condition stated above, is one of the main difficulties that appears when one tries to extend the classical arguments to the present situation.Comment: 47 page