Dimension-Free Maximal Operators over Convex Bodies
We study the Hardy--Littlewood maximal operator attached to symmetric convex bodies in $\mathbb{R}^n$ and its dimension-free $L^p$ bounds. We begin with Stein's dimension-independent $L^p$ theorem for Euclidean balls via semigroup methods. Building on this, we present Bourgain's $L^2$ estimate for arbitrary symmetric convex bodies and Carbery's extension to $L^p$ for $p>3/2$, obtained through dyadic reduction, Littlewood--Paley estimates, and multipliers for fractional derivatives.. We then discuss Müller's projection-parameter criterion $q(C)$, which yields dimension-free bounds for all $p>1$ in structured families, including the $\ell^{q}$-balls. The course concludes with Bourgain's result for cubes---dimension-free boundedness for every $p>1$---and the sharp limitations at $p=1$: the weak-type $(1,1)$ constants for cubes grow unboundedly with $n$, with quantitative lower bounds due to Aubrun and to Iakovlev--Strömberg. Throughout we emphasize Poisson and heat semigroup techniques, complex interpolation on strips, Fourier multiplier bounds, and log-concavity.
Lecturer
Date
24th October, 2025 ~ 9th January, 2026
Location
Weekday | Time | Venue | Online | ID | Password |
---|---|---|---|---|---|
Friday | 15:20 - 16:55 | A7-301 | ZOOM 05 | 293 812 9202 | BIMSA |
Audience
Graduate
, Postdoc
, Researcher
Video Public
No
Notes Public
No
Language
English