The one-weight inequality for the $\mathcal{H}$-harmonic Bergman projection
Studia Mathematica
MSC: Primary 42B20
DOI: 10.4064/sm250513-2-10
Published online: 3 July 2026
Abstract
Let $n\geqslant 3$ 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 \lt p \lt \infty $ and $-1 \lt \alpha \lt \infty $, $$\|P_\alpha \|_{ L^p(\omega \, d\nu _\alpha )\rightarrow L^p(\omega \, d\nu _\alpha )}\leqslant C [\omega ]_{p,\alpha }^{\max \,\{1,\frac {1}{p-1}\}},$$ where $[\omega ]_{p,\alpha }$ is the Bekollé–Bonami constant. Our proof is inspired by dyadic harmonic analysis, and the key ingredient involves the discretization of the Bergman kernel for the $\mathcal {H}$-harmonic Bergman spaces.
