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

Let $n⩾3$ be an integer. For the Bekollé-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 $1<p<\mathrm{\infty }$ and $-1<\alpha <\mathrm{\infty }$, $$\mathrm{\mid }\mathrm{\mid }{P}_{\alpha }\mathrm{\mid }{\mathrm{\mid }}_{L^p(\omega d{\nu }_{\alpha })⟶L^p(\omega d{\nu }_{\alpha })}⩽C[\omega {]}_{p,\alpha }^{\max \{1,\frac{1}{p-1}\}},$$ where $[\omega {]}_{p,\alpha }$ is the Bekollé-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.