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

Abstract

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 $1<p<\infty$ and $-1<\alpha <\infty$, \[||P_α||_{ L^p(ωdν_α)\longrightarrow L^p(ωdν_α)}\leqslant C [ω]_{p,α}^{\max\left\{1,\frac{1}{p-1}\right\}}, \] 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.