Circular maximal functions on the Heisenberg group

TL;DR

This paper establishes the boundedness of the circular maximal function on the Heisenberg group for certain p-values, using a novel phase space analysis related to vector fields specific to the Heisenberg structure.

Contribution

It introduces a new approach based on a square sum estimate associated with a 2x2 cone in phase space, differing from traditional Euclidean methods.

Findings

Proves L^p boundedness for 2<p≤∞ on the Heisenberg group.

Uses phase space analysis involving vector fields on the Heisenberg group.

Employs a novel square sum estimate related to the 2x2 cone.

Abstract

We prove the $L^p$ boundedness of the circular maximal function on the Heisenberg group $\mathbb{H}^1$ for $2<p\le \infty$. The proof is based on the square sum estimate associated with the $2\times 2$ cone $|(\xi_1',\xi_2')|= |(\xi_3',\xi_4')| $ of the phase space arising from the vector fields $X_1,X_2,tX_3,\partial/\partial t$ on the Heisenberg group, rather than the $2\times 1$ cone $ |(\xi_1,\xi_2)|= |\xi_3|$ of the frequency space arising from $\partial/\partial x_1, \partial/\partial x_2, \partial/\partial t$ on the Euclidean space.