We will discuss some recent developments in dimensional-free bounds for the Hardy--Littlewood averaging operators defined over convex symmetric bodies in R^d. Specifically we will be interested in r-variational bounds.

We also prove the dimension-free bounds on L^p(Z^d) with p>3/2 for the discrete maximal function associated with cubes in Z^d. Using similar methods we also give a new simplified proof for the dimension-free bounds on L^p(R^d) with p>3/2 for maximal functions corresponding to symmetric convex bodies in R^d. If the time permits we will discuss problems for discrete Euclidean balls in Z^d as well.

This is joint project with J. Bourgain, E.M. Stein and B. Wrobel.

### Analysis Seminar

Thursday, September 21, 2017 - 3:00pm

#### Mariusz Mirek

UCL