$L^p$ regularity for a class of averaging operators on the Heisenberg group

TL;DR

This paper establishes $L^p$ to $L^p_s$ boundedness for certain averaging operators on the Heisenberg group using advanced harmonic analysis techniques, including oscillatory integrals and decoupling inequalities.

Contribution

It introduces a new Sobolev space adapted to the Heisenberg group and proves boundedness of averaging operators within this framework, advancing understanding of harmonic analysis on non-commutative groups.

Findings

Proved $L^p_{comp} o L^p_s$ boundedness for averaging operators.

Constructed a Sobolev space suited to the Heisenberg group.

Applied decoupling inequalities to analyze oscillatory integrals.

Abstract

We prove $L^p_{comp}\to L^p_{s}$ boundedness for averaging operators associated to a class of curves in the Heisenberg group $\mathbb{H}^1$ via $L^2$ estimates for related oscillatory integrals and Bourgain-Demeter decoupling inequalities on the cone. We also construct a Sobolev space adapted to translations on the Heisenberg group to which these averaging operators map all $L^p$ functions boundedly.