Jonathan Hickman

Title: The circular maximal function for Heisenberg radial functions

Abstract: We consider a variant of Bourgain's circular maximal function defined over the Heisenberg group $H^1$. Determining the $L^p$ mapping properties of this operator is a challenging open problem and involves analysing a number of interesting singularities which are not present in the euclidean case. In recent joint work with David Beltran, Shaoming Guo and Andreas Seeger, the $L^p$ mapping properties were determined under a radial assumption on the input function. Even under this assumption considerable difficulties arise, and the proof involves the analysis of a maximal function on the euclidean plane associated to a non-smooth curve distribution which fails both the rotational curvature and cinematic curvature conditions.