國立清華大學 數學系

Abstract: Spherical maximal functions are a classical topic in harmonic analysis arising from the study of differentiability properties of rough functions, going back to works of Stein and Bourgain. In this talk we are concerned with spherical maximal functions with a supremum taken over a fractal set of radii. Our discussion will focus on optimal Lp improving properties, i.e. the sharp range of (1/p, 1/q) such that Lp to Lq boundedness holds. It turns out that this range depends on various fractal properties of the set of radii, such as Minkowski and Assouad dimensions and the Assouad spectrum. We characterize all convex sets that can arise as Lp improving region of such a spherical maximal operator, up to endpoints. Surprisingly it turns out that a critical segment of the boundary of such a set is given by an essentially arbitrary convex curve, which leads to non-polygonal Lp improving regions. An application of our Lp improving properties are new weighted Lp estimates for an associated global spherical maximal operator. Based on joint works with A. Seeger and with T. Anderson, K. Hughes, A. Seeger.