Maximal Operators: Scales, Curvature and the Fractal Dimension

Abstract

The spherical maximal operator

$$Af(x) = \mathop {sup}\limits_{t > 0} \left| {{A_t}f(x)} \right| = \mathop {sup}\limits_{t > 0} \left| \int{f(x - ty)d\sigma (y)} \right|$$

where σ is the surface measure on the unit sphere, is a classical object that appears in a variety of contexts in harmonic analysis, geometric measure theory, partial differential equation and geometric combinatorics.

We establish L_p bounds for the Stein spherical maximal operator in the setting of compactly supported Borel measures μ, ν satisfying natural local size assumptions $\mu (B(x,r)) \leqslant C{r {{s_\mu }}},v(B(x,r)) \leqslant C{r {{s_v}}}$.

Taking the supremum over all t > 0 is not in general possible for reasons that are fundamental to the fractal setting, but we can obtain single scale (t ∈ [a, b] ⊂ (0,∞)) results. The range of possible L_p 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 L²(μ) → L²(ν) bounds for the convolution operator T_λf(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^p(μ) → L^p(ν) bounds for a large class of operators satisfying suitable L^p-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.