Weak-type bounds for the Bergman projection with Bekollé-Bonami weights
Jiale Chen, Zoe Nieraeth, Cody B. Stockdale, Nathan A. Wagner
TL;DR
This work develops a quantitative weighted weak-type theory for Bergman projections on simple domains with Bekollé-Bonami weights. It provides two independent proofs: a sparse-domination approach yielding an endpoint weak-type bound and a logarithmic improvement for $p>1$, and a testing-condition approach that gives a logarithm-free weak-type bound for $p>1$ plus a strong-type range via interpolation; it also presents a mixed two-weight weak-type inequality in the spirit of Sawyer. The results extend to general simple domains, including strongly pseudoconvex domains, via a dyadic boundary framework and dyadic domination of the Bergman kernel, with weights defined on tent analogues of boundary cubes. These findings yield sharp, weight-dependent bounds for the Bergman projection, advancing the understanding of weighted weak-type behavior in complex analysis and several complex variables, with new results even in classical settings like the upper half-plane and unit disk.
Abstract
We establish weighted weak-type bounds for the Bergman projection with respect to Bekollé-Bonami characteristics. We present two proofs of an improved quantitative weak-type $(1,1)$ estimate, as well as sharp weak-type $(p,p)$ bounds for $p>1$ and mixed weighted weak-type $(1,1)$ inequalities. Our results, which hold for a wide class of simple domains in $\mathbb{C}^n$, are new even in the classical settings of the upper half-plane and the unit disk.