Maximal estimates for a generalized spherical mean Radon transform acting on radial functions
Abstract
We study a generalized spherical means operator,
viz.\ generalized spherical mean Radon transform, acting on radial functions.
As the main results, we find conditions for the associated maximal operator and its local variant
to be bounded on power weighted Lebesgue spaces. This translates, in particular, into almost everywhere convergence
to radial initial data results for solutions to certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations.
Moreover, our results shed some new light on the interesting and important question of optimality
of the yet known $L^p$ boundedness results for the maximal operator in the general non-radial case.
It appears that these could still be notably improved, as indicated by our conjecture of the ultimate
sharp result.