Jose Ramon Madrid Padilla

(Mathematics and system analysis, Aalto University)

On the regularity of classical and fractional maximal operators

Date Thursday, October 27, 2016

We will talk about the regularity properties of maximal operators acting on $BV$ functions and $W^{1,1}$ functions. We will present recent results about new bounds for the derivative and the continuity of maximal operators, both in the continuous and in the discrete settings.

The questions in this area begin with the observation that a nice averaging operator smooths functions. Then, heuristically, the maximal function for a nice sequence of averaging operators should not destroy any smoothness. With this in mind, Hajlasz-Onninen conjectured that the variation of the Hardy-Littlewood maximal function (that is, the HL maximal operator applied to a fixed function) is no worse than the variation of the original function.

Several people studied this question with the $L^p$ theory for $p>1$ becoming quickly well established. This left open the question about the $L^1$ theory because of the main difficulty that the Hardy-Littlewood maximal function is unbounded on $L^1$. Nevertheless, one can ask the question:

For a function $f$, is the total variation of its maximal function $Mf$ bounded by the total variation of $f$?

Tanaka/Aldaz and Perez Lazaro showed that this is true for the uncentered Hardy-Littlewood maximal function, but the case of centered maximal function remained open.

In this talk we will focus our attention on the following questions:

  1. What is the analogous to the Tanaka's result for the fractional uncentered maximal operator?
  2. The Tanaka's result translates as the boundedness of a map. Is this a continuous map?
  3. What about the discrete analogues?