NTU Math Seminar

演講者：Professor Joris Roos (University of Wisconsin at Madison)

講 題：Spherical maximal functions and fractal dimensions

時 間：2020年1月10日 (星期五) 10:00 - 11:00 a.m.

地 點：臺灣大學天數館

摘 要：Spherical maximal functions are a classical topic in real harmonic analysis arising from

the study of differentiability properties of functions, going back to works of Stein and Bourgain.

In this talk we are concerned with spherical maximal functions in dimensions $d\ge 2$ with a supremum taken over a fractal set of radii. Our discussion will focus on optimal $L^p$ improving properties, i.e. the sharp range of $(1/p, 1/q)$ such that $L^p\to L^q$ boundedness holds. It turns out that this range depends on various fractal dimensions of the set of radii, such as Minkowski and Assouad dimensions and the Assouad spectrum. We characterize all convex sets that can arise as $L^p$ improving region of such a spherical maximal operator, up to endpoints.

Surprisingly, a critical segment of the boundary of such a set is given by an essentially arbitrary convex curve, which leads to non-polygonal $L^p$ improving regions. An application of our

$L^p$ improving properties are new weighted $L^p$ estimates for an associated global spherical maximal operator. Based on joint works with A. Seeger and with T. Anderson, K. Hughes, A.

Seeger.

茶 會：11:00-12:00

NTU Math Seminar

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