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

TL;DR

This work extends sharp one-weight inequalities to the $\\mathcal{H}$-harmonic Bergman projection $P_\alpha$ on the unit ball for $n\\ge 3$. By building a dyadic Carleson-box discretization and dominating the Bergman kernel with positive dyadic kernels, the authors prove that $P_\alpha$ is bounded on $L^p(\omega d\nu_\alpha)$ for Bekollé-Bonami weights with a norm bound $||P_\alpha||_{L^p(\omega d\nu_\alpha)\to L^p(\omega d\nu_\alpha)} \le C [\omega]_{p,\alpha}^{\max\{1,1/(p-1)\}}$, and they show this exponent is sharp. The proof combines discretization techniques, extrapolation, and a dyadic modeling of the kernel, enriching the harmonic analysis toolkit for Bergman-type projections beyond the holomorphic setting. A weight-specific lower-bound construction confirms linear growth in $[\omega]_{2,\alpha}$, establishing optimality. The results provide a precise, scalable framework for weighted inequalities of $\\mathcal{H}$-harmonic Bergman projections with potential applications to function spaces on the unit ball.

Abstract

Let $n\geqslant 3$ be an integer. For the Bekollé-Bonami weight $ω$ 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 $-1<α<\infty$, \[||P_α||_{ L^p(ωdν_α)\longrightarrow L^p(ωdν_α)}\leqslant C [ω]_{p,α}^{\max\left\{1,\frac{1}{p-1}\right\}}, \] where $[ω]_{p,α}$ 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.

Theorems & Definitions (29)

• Theorem 1.1
• Remark 1.1
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• Remark 2.1
• Lemma 2.4
• proof