Sharp off-diagonal weighted norm estimates for the Bergman projection

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 $Q_I=\{z=x+iy\in \mathbb{C}: x\in I, 0<y<|I|\}$.