Bergman and Szego projections, Extremal Problems, and Square Functions

TL;DR

This paper investigates Hardy space estimates for analytic projections, providing new conditions for Bergman and Szeg"o projections to belong to Hardy spaces, and explores implications for extremal problems and approximation in $L^p$ spaces.

Contribution

It establishes a sufficient condition for the Bergman projection to be in Hardy spaces and proves a converse to Ryabykh's theorem, advancing understanding of extremal functions and projections.

Findings

Bergman projection belongs to Hardy space under new conditions

Converse to Ryabykh's theorem for extremal functions

Best $L^p$ approximation also lies in $L^q$ under certain conditions

Abstract

We study estimates for Hardy space norms of analytic projections. We first find a sufficient condition for the Bergman projection of a function in the unit disc to belong to the Hardy space $H^p$ for $1 < p < \infty$. We apply the result to prove a converse to an extension of Ryabykh's theorem about Hardy space regularity for Bergman space extremal functions. We also prove that the $H^q$ norm of the Szeg\"{o} projection of $F^{p/2} \overline{F}^{(p/2)-1}$ cannot be too small if $F$ is analytic, for certain values of $p$ and $q$. We apply this to show that the best analytic approximation in $L^p$ of a function in both $L^p$ and $L^q$ will also lie in $L^q$, for certain values of $p$ and $q$.