Weighted maximal inequalities on hyperbolic spaces

In this work we develop a weight theory in the setting of hyperbolic spaces. Our starting point is a variant of the well-known endpoint Fefferman-Stein inequality for the centered Hardy-Littlewood maximal function. This inequality generalizes, in the hyperbolic setting, the weak $(1,1)$ estimates obtained by Str\"omberg in "Weak type L1 estimates for maximal functions on noncompact symmetric spaces", Ann. of Math. 114 (1981), where Str\"omberg answered a question posed by Stein and Wainger in "Problems in harmonic analysis related to curvature", Bull. Amer. Math. Soc. 84 (1978). Our approach is based on a combination of geometrical arguments and the techniques used in the discrete setting of regular trees by Naor and Tao in "Random martingales and localization of maximal inequalities", J. Funct. Anal. 259 (2010). This variant of the Fefferman-Stein inequality paves the road to weighted estimates for the maximal function for $p>1$. On the one hand, we show that the classical $A_p$ conditions are not the right ones in this setting. On the other hand, we provide sharp sufficient conditions for weighted weak and strong type $(p,p)$ boundedness of the centered maximal function, when $p>1$. The sharpness is in the sense that, given $p>1$, we can construct a weight satisfying our sufficient condition for that $p$, and so it satisfies the weak type $(p,p)$ inequality, but the strong type $(p,p)$ inequality fails. In particular, the weak type $(q,q)$ fails as well for every $q < p$.