Speaker
Hrit Roy, University of Jyväskylä
Abstract
The Nikodym maximal operator takes a function f and computes its maximal average over all unit length delta-tubes centered at a given point. Cordoba showed that the planar Nikodym maximal operator is “essentially bounded” on L^2, in the sense that its L^2 operator norm grows subpolynomially in delta^{-1}. In fact, 2 is the critical exponent for essential boundedness; that is, it is the smallest exponent for which this is true. How does the critical exponent change if the directions of the tubes are restricted to a prescribed set? I will discuss a joint work with Tuomas Orponen, where we show that the critical exponent for essential boundedness is one plus the quasi-Assouad dimension of the direction-set, provided this dimension is at least one half.