The one-weight inequality for $\mathcal{H}$-harmonic Bergman projection

Abstract

Let $n\geqslant 3$ be an integer. For the Bekoll\'e-Bonami weight $\omega$ on the real unit ball $\mathbb{B}_n$, we obtain the following sharp one-weight estimate for the $\mathcal{H}$-harmonic Bergman projection: for $1<p<\infty$ and $-1<\alpha<\infty$, \[||P_\alpha||_{ L^p(\omega d\nu_\alpha)\longrightarrow L^p(\omega d\nu_\alpha)}\leqslant C [\omega]_{p,\alpha}^{\max\left\{1,\frac{1}{p-1}\right\}}, \] where $[\omega]_{p,\alpha}$ is the Bekoll\'e-Bonami constant. Our proof is inspired by the dyadic harmonic analysis, and the key ingredient involves the discretization of the Bergman kernel for the $\mathcal{H}$-harmonic Bergman spaces.