Maximal function and Carleson measures in Békollé-Bonami weights

Abstract

Let $\omega$ be a Békollé-Bonami weight. We give a complete characterization of the positive measures $\mu$ such that $${\int }_{\mathcal{H}}|M_{\omega }f(z){|}^{q}d\mu (z)\le C{\left({\int }_{\mathcal{H}}|f(z){|}^p\omega (z)dV(z)\right)}^{q/p}$$ 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 $1\le p,q<\infty$.