Concentration of analytic functions

Abstract

In this work we study different problems concerning the characterization of those measurable sets that, among all sets having a prescribed measure, can capture the largest possible energy fraction of an analytic function in both the Euclidean and hyperbolic settings. In other terms, considering as spaces of analytic functions the Fock space $\mathcal{F}^2\left(\mathbb{C}^n\right)$, with $n \geq 1$, and the Bergman space $\mathcal{A}_\alpha^2(\mathbb{D})$, with $\alpha>1$, we show that given some measurable sets $\Omega \subset \mathbb{C}$ and $\Omega^{\prime} \subset \mathbb{D}$, with some fixed measure $c>0$, the concentration quantities and

$$ & \max _{F \in \mathcal{F}^2\left(\mathbb{C}^n\right) \backslash\{0\}}\left\{\frac{\int_{\Omega}|F(z)|^2 e^{-\pi|z|^2} d m_{2 n}(z)}{\left.\int_{\mathbb{C}^n}|F(z)|^2 e^{-\pi|z|^2 d m_{2 n}(z)}\right\}}\right. \\ & \max _{f \in \mathcal{A}_\alpha^2(\mathbb{D}) \backslash\{0\}}\left\{\frac{\int_{\Omega^{\prime}}(\alpha-1)|f(z)|^2\left(1-|z|^2\right)^\alpha d m_h(z)}{\int_{\mathbb{D}}(\alpha-1)|f(z)|^2\left(1-|z|^2\right)^\alpha d m_h(z)}\right\} $$

are maximized when considering the sets to be a ball (in each respective geometry) with the same measure $c>0$. Specifically, we give a sharp upper bound for each of the previous problems and characterize not only the subsets but also the functions where the maxima are attained.