Sharp off-diagonal weighted norm estimates for the Bergman projection

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