Analysis Seminar: "The circular maximal operator on Heisenberg radial functions"

Abstract: An analogue of the Stein spherical maximal function was introduced on the Heisenberg group $\mathbb{H}^n$ by Nevo and Thangavelu in 1997. Whilst the sharp $L^p$ estimates for this object are known for $n \geq 2$ (independently obtained by M\"uller and Seeger and Narayanan and Thangavelu), the case of $\mathbb{H}^1$ remains open and it is currently unknown if the circular maximal function is bounded on $L^p(\mathbb{H}^1)$ for any finite $p$.  

In this talk, we present sharp $L^p$ estimates for this object when restricted to a class of Heisenberg radial functions. Under this assumption, the problem reduces to studying a variable-coefficient version of Bourgain's circular maximal operator on the Euclidean plane which presents a number of interesting singularities: it is associated to a non-smooth curve distribution and, furthermore, fails both the usual rotational curvature and cinematic curvature conditions.

This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.

Host: Francesco Di Plinio