Weighted inequality of integral operators induced by Hardy kernels
Zipeng Wang, Kenan Zhang
TL;DR
The paper addresses the problem of characterizing the boundedness of the Hardy-kernel-induced operator $K_ obreak alpha$ on weighted spaces $L^p( obreak oop) omega$ over the unit disk. It shows that, under doubling weights, $K_ obreak alpha$ is bounded on $L^p( obreak oomega)$ if and only if the Carleson-box quantity $ obreak rac{|Q_I|_ obreak omega |Q_I|_ obreak sigma^{p-1}}{|Q_I|^{peta/2}}$ is uniformly finite for all Carleson boxes $Q_I$, where $ obreak sigma= omega^{1-p'}$ and $eta= obreak alpha/2$, with the necessary part holding without doubling and the sufficiency part under reverse-doubling. The authors combine Sawyer duality, dyadic Carleson-box decompositions using two grids $ D^0$ and $ D^{1/3}$, and Carleson-embedding theorems to establish the result, thereby confirming Guo-Wang's conjecture for $eta=1$ in the doubling setting and providing a full characterization for all $ obreaketa>1$.
Abstract
For doubling weights, we obtain a necessary and sufficient condition such that the one weighted inequality of the integral operator induced by Hardy kernels on the unit disk holds. This confirms a conjecture by Guo and Wang in such situations.