Hyperbolic spaces and weighted estimates

In 1981 Strömberg proved that the center Hardy-Littlewood maximal operator $M$ is of the weak-type $(1,1)$ in the context of Hyperbolic spaces. In fact his result actually is in a more general context (non-compact symmetric spaces). For $p>1$ boundedness results were previously obtained by Stein and Clerk. The main difficulty in the Hyperbolic setting is the exponential growth of the measure of a ball in terms of its radio.This difficulty has generated that, to the best of our knowledge, no general theory of weights has been developed in this context. In this talk using ideas of the discrete setting ($k$-trees) due to Naor and Tao we will show that it is possible to obtain a Fefferman-Stein endpoint weighted estimate generalizing the result of Strömberg. Moreover, we also obtain (sharp) sufficient (geometric) conditions in a weight $w$ for the weak and strong estimates of $M$ in the spaces $L^p(w)$ for $p>1$. The talk is based on a joint work with J. Antezana.