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.