Spherical maximal operators with fractal sets of dilations on radial functions

For a given set of dilations E⊂[1,2], Lebesgue space mapping properties of the spherical maximal operator with dilations restricted to E are studied when acting on radial functions. In higher dimensions, the type set only depends on the upper Minkowski dimension of E, and in this case complete endpoint results are obtained. In two dimensions we determine the closure of the Lp→Lq type set for every given set E in terms of a dimensional spectrum closely related to the upper Assouad spectrum of E.