In this talk I would like to give an overview of the regularity theory for maximal operators when acting on Sobolev and BV functions, and discuss some partial progress in two conjectures regarding general maximal operators of convolution type. This is based on joint work with Julian Weigt (Univ. of Oulu - Finland). The talk should be very accessible to a broad audience.
There exists a unique minimizer of the L^1-norm among entire functions of exponential type pi satisfying f(0)=1. In 1993 Hörmander and Bernhardsson investigated the properties of this function, proving that all of its zeros are real and simple and finding a remarkably accurate approximation of its L^1-norm. However, they were unable to identify the extremal function explicitly, despite having what they believed to be a very strong numerical approximation of its Fourier transform. In this talk I will give a comprehensive description of this function, which possesses a surprising dual characterization via both differential and functional equations, among other remarkable properties.
On arbitrary Carnot groups, the only hypoelliptic Hodge-Laplacians on forms that have been introduced are 0-order pseudodifferential operators constructed using the Rumin complex. However, to address questions where one needs sharp estimates, this 0-order operator is not suitable. This is a rather difficult problem to tackle in full generality, the main issue being that the Rumin exterior differential is not homogeneous on arbitrary Carnot groups. In this talk, I will focus on the specific example of the free Carnot group of step 3 with 2 generators, where it is possible to introduce different hypoelliptic Hodge-Laplacians on forms. Such Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups.