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

Abstract

We prove $L_{comp}^p\to L_s^p$ 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.

32 pages, 2 figures. Added references, added detail to proof of Corollary 1.1, result unchanged. Made additional changes reflecting anonymous referee's comments

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