Maximal function and Carleson measures in the theory of Békollé–Bonami weights
Volume 142 / 2016
Colloquium Mathematicum 142 (2016), 211-226
MSC: Primary 42B25; Secondary 42A61.
DOI: 10.4064/cm142-2-4
Abstract
Let $\omega$ be a Békollé–Bonami weight. We give a complete characterization of the positive measures $\mu$ such that $$ \int_{\mathcal H}|M_\omega f(z)|^q\,d\mu(z)\le C\biggl(\int_{\mathcal H}|f(z)|^p\omega(z)\,dV(z)\bigg)^{q/p} $$ and $$ \mu(\{z\in \mathcal H: Mf(z)>\lambda\})\le \frac{C}{\lambda^q}\biggl(\int_{\mathcal H}|f(z)|^p\omega(z)\,dV(z)\bigg)^{q/p}, $$ where $M_\omega$ is the weighted Hardy–Littlewood maximal function on the upper half-plane $\mathcal H$ and $1\le p,q<\infty$.
