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

Abstract

It is shown in quantitative terms that the maximal Bergman projection \begin{equation*} P^{+}_ω(f)(z)=\int_\mathbb{D} f(ζ)|B^ω_z(ζ)|ω(ζ)\,dA(ζ), \end{equation*} is bounded from $L_{\nu }^p$ to $L_{\eta }^p$ if and only if \begin{equation*} \sup_{0<r<1}\left(\int_0^r\frac{η(s)}{\left(\int_{s}^1ω(t)\,dt\right)^p}\,ds\right)^{\frac{1}{p}} \left(\int_r^1\left(\frac{ω(s)}{ν(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 $0\le r<1$. 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.