Endpoint estimates and optimality for the generalized spherical maximal operator on radial functions

Abstract

We find sharp conditions for the maximal operator associated with generalized spherical mean Radon transform on radial functions $ M^{ \alpha, \beta}_t $ to be bounded on power weighted Lebesgue spaces. Moreover, we also obtain the corresponding endpoint results in terms of optimal power weighted weak and restricted weak type estimates. All this complements significantly previous partial results existing in the literature.