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 Lp 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 Lp 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 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 Lp(μ) → Lp(ν) bounds for a large class of operators satisfying suitable Lp-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.