伯格曼再生核的最优非对角线上界估计
Optimal off-diagonal upper estimates for Bergman reproducing kernels
AI总结：
研究单位圆盘上伯格曼再生核的非对角线上界估计，基于权重矩的单边加倍条件证明其精确性及刻画性。通过新方法得到修正伯格曼核的\(L^p\)均值估计，建立相关函数与测度新联系，还将结果推广到高维等情况。
AI中文摘要：
本文为与单位圆盘上径向权重相关的伯格曼再生核建立了一个精确的非对角逐点上界估计。证明基于权重矩满足自然单边加倍条件，且标准核表明该估计在各点至多相差一个乘法常数时是精确的。还证明此估计刻画了所考虑的径向加倍权重类。作为应用得到了某些修正伯格曼核的最优\(L^p\)均值估计，通过新的直接证明恢复了相关关键估计；建立了贝雷辛变换的非切向极大函数、霍尔曼德极大函数和卡尔松测度之间的新联系；并将主要结果推广到高维、调和伯格曼核及核的双权重分数阶导数，后者为狄利克雷再生核给出精确估计。
英文摘要：
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 [Peláez et al., J. Math. Pures Appl. 105(2016), 102--130] and [Peláez 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 Hörmander 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.
