In this talk, we will consider the problem of obtaining sharp (up to endpoints) $L^p\to L^q$ estimates for the local maximal operator associated with averaging over dilates of the Korányi sphere on Heisenberg groups. This is a codimension one surface compatible with the non-isotropic Heisenberg dilation structure. I will describe the main features of the problem, some of which are helpful while others are obstructive. These include the non-Euclidean group structure (the extra "twist" due to the Heisenberg group law), the geometry of the Korányi sphere (in particular, the flatness at the poles) and an "imbalanced" scaling argument encapsulated by a new type of Knapp example.
The Korányi Spherical Maximal Function on Heisenberg groups
Rajula Srivastava, University of Wisconsin-MadisonAuthors: Rajula Srivastava
2022 AWM Research Symposium
Women in Analysis Research Network - Special Session for Graduate Students and Postdoctoral Fellows