Sharp off-diagonal weighted norm estimates for the Bergman projection

TL;DR

This paper establishes sharp weighted off-diagonal estimates for the Bergman projection on the upper-half plane, extending understanding of weighted norm inequalities in complex analysis.

Contribution

It provides the first sharp off-diagonal weighted norm estimates for the Bergman projection in the upper-half plane, including explicit dependence on weight characteristics.

Findings

Proved weighted norm inequalities for the Bergman projection with sharp constants.

Derived conditions on parameters for the inequalities to hold.

Extended the theory of weighted estimates to off-diagonal cases.

Abstract

We prove that for $1<p\le q<\infty$, $qp\geq {p'}^2$ or $p'q'\geq q^2$, $\frac{1}{p}+\frac{1}{p'}=\frac{1}{q}+\frac{1}{q'}=1$, $$\|\omega P_\alpha(f)\|_{L^p(\mathcal{H},y^{\alpha+(2+\alpha)(\frac{q}{p}-1)}dxdy)}\le C_{p,q,\alpha}[\omega]_{B_{p,q,\alpha}}^{(\frac{1}{p'}+\frac{1}{q})\max\{1,\frac{p'}{q}\}}\|\omega f\|_{L^p(\mathcal{H},y^{\alpha}dxdy)}$$ where $P_\alpha$ is the weighted Bergman projection of the upper-half plane $\mathcal{H}$, and $$[\omega]_{B_{p,q,\alpha}}:=\sup_{I\subset \mathbb{R}}\left(\frac{1}{|I|^{2+\alpha}}\int_{Q_I}\omega^{q}dV_\alpha\right)\left(\frac{1}{|I|^{2+\alpha}}\int_{Q_I}\omega^{-p'}dV_\alpha\right)^{\frac{q}{p'}},$$ with $Q_I=\{z=x+iy\in \mathbb{C}: x\in I, 0<y<|I|\}$.