Radial two weight inequality for maximal Bergman projection induced by a regular weight

TL;DR

This paper characterizes the boundedness of the maximal Bergman projection between weighted spaces using a precise integral condition on regular weights, extending understanding of weighted inequalities in complex analysis.

Contribution

It provides a new necessary and sufficient integral condition for the boundedness of the maximal Bergman projection induced by regular weights, linking it to the weights' properties.

Findings

Boundedness characterized by a specific integral condition.

Equivalence of boundedness for $P_ u$ and $P_ u^+$ under certain conditions.

Provides criteria for regular weights ensuring bounded projections.

Abstract

It is shown in quantitative terms that the maximal Bergman projection \begin{equation*} P^{+}_\omega(f)(z)=\int_\mathbb{D} f(\zeta)|B^\omega_z(\zeta)|\omega(\zeta)\,dA(\zeta), \end{equation*} is bounded from $L^p_\nu$ to $L^p_\eta$ if and only if \begin{equation*} \sup_{0<r<1}\left(\int_0^r\frac{\eta(s)}{\left(\int_{s}^1\omega(t)\,dt\right)^p}\,ds\right)^{\frac{1}{p}} \left(\int_r^1\left(\frac{\omega(s)}{\nu(s)^\frac{1}{p}}\right)^{p'}ds\right)^{\frac{1}{p'}}<\infty, \end{equation*} provided $\omega,\nu,\eta$ are radial regular weights. A radial weight $\sigma$ is regular if it satisfies $\sigma(r)\asymp\int_{r}^1\sigma(t)\,dt/(1-r)$ for all $0\leq r<1$. It is also shown that under an appropriate additional hypothesis involving $\omega$ and $\eta$, the Bergman projection $P_\omega$ and $P^+_\omega$ are simultaneously bounded.