Maximal operators: scales, curvature and the fractal dimension

We establish L bounds for the Bourgain-Stein spherical maximal operator in the setting of compactly supported Borel measures μ, ν satisfying natural local size assumptions μ(B(x, r)) ≤ Crμ , ν(B(x, r)) ≤ Crν . Taking the supremum over all t > 0 is not in general possible for reasons that are fundamental to the fractal setting, but we are able to obtain single scale (t ∈ [1, 2]) results. The range of possible L exponents is, in general, a bounded open interval where the upper endpoint is closely tied with the local smoothing estimates for Fourier Integral Operators. In the process, we establish L(μ) → L(ν) bounds for the convolution operator Tλf(x) = λ∗(fμ), where λ is a tempered distribution satisfying a suitable Fourier decay condition. More generally we establish a transference mechanism which yields L(μ) → L(ν) bounds for a large class of operators satisfying suitable L-Sobolev bounds. This allows us to effectively study the dimension of a blowup set ({x : Tf(x) = ∞}) for a wide class of operators, including the solution operator for the classical wave equation. Some of the results established in this paper have already been used to study a variety of Falconer type problems in geometric measure theory.