Maximal function and Carleson measures in B\'ekoll\'e-Bonami weights

TL;DR

This paper characterizes measures for which the weighted Hardy-Littlewood maximal function is bounded in Békollé-Bonami weighted spaces, providing a complete description of Carleson measures in this context.

Contribution

It offers a full characterization of measures ensuring boundedness of the maximal function in Békollé-Bonami weighted spaces, extending understanding of Carleson measures in this setting.

Findings

Characterization of measures for maximal function boundedness

Complete description of Carleson measures in Békollé-Bonami weights

Results applicable to upper-half plane analysis

Abstract

Let $\omega$ be a B\'ekoll\'e-Bonami weight. We give a complete characterization of the positive measures $\mu$ such that $$\int_{\mathcal H}|M_\omega f(z)|^qd\mu(z)\le C\left(\int_{\mathcal H}|f(z)|^p\omega(z)dV(z)\right)^{q/p}$$ and $$\mu\left(\{z\in \mathcal H: Mf(z)>\lambda\}\right)\le \frac{C}{\lambda^q}\left(\int_{\mathcal H}|f(z)|^p\omega(z)dV(z)\right)^{q/p}$$ where $M_\omega$ is the weighted Hardy-Littlewood maximal function on the upper-half plane $\mathcal H$, and $1\le p,q<\infty$.