Optimal off-diagonal upper estimates for Bergman reproducing kernels

Abstract

In this paper, we establish a sharp off-diagonal pointwise upper estimate for the Bergman reproducing kernel associated with a radial weight on the unit disc. Our proof is self-contained and assumes the weight satisfies a natural one-sided doubling condition on its moments. The standard kernels demonstrate that this estimate is sharp, up to a multiplicative constant, at every point. We also show that the estimate in fact characterizes the class of radial doubling weights under consideration. As applications, we first obtain optimal $L^p$-mean estimates for certain modified Bergman kernels. This approach recovers the key estimates in [Pel\'aez et al., J. Math. Pures Appl. 105(2016), 102--130] and [Pel\'aez et al., arxiv.org/pdf/2407.04645] via a novel and more direct proof. Second, we establish novel connections between the non-tangential maximal function of the Berezin transform, the H\"ormander maximal function, and Carleson measures. Notably, these connections are new even in the setting of the standard weighted Bergman spaces. Finally, we extend our main results to higher dimensions, harmonic Bergman kernels, and two-weight fractional derivatives of kernels, the latter of which yields sharp estimates for Dirichlet reproducing kernels.