Sharp stability of convex functionals on weighted Bergman spaces with applications

Abstract

Recently, Kulikov (\cite{Ku}) has shown that certain convex functionals on weighted Bergman spaces are maximized by reproducing kernels. We show a sharp quantitative stability of these estimates with the optimal norm and the exponent and an explicit constant asymptotically sharp in both directions ($\alpha\rightarrow -1$ and $\alpha\rightarrow +\infty$). Several applications of this result include recovering the appropriate result for Fock spaces, interpretation to Cauchy wavelets, and the Hardy space counterpart for functionals induced by increasing function. In addition, we prove a higher-dimensional analog of the main result assuming that all convex functionals on the weighted Bergman space $\mathcal{A}^2_{\alpha}(\mathbb{B}_n)$ attain their extrema in reproducing kernels.