Weighted maximal inequalities on hyperbolic spaces

Abstract

In this work we study the singularity of the (centered) maximal operator in the hyperbolic spaces. With this aim, we changed the density of the underlying measure to avoid possible compensations due to the symmetries of the hyperbolic measure. 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 & ouml;mberg (1981) [17] who answered a question posed by Stein and Wainger (1978) [16]. 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 (2010) [11]. 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.