Lacunary elliptic maximal operator on the Heisenberg group

TL;DR

This paper establishes conditions for the boundedness of lacunary elliptic maximal operators on the Heisenberg group, extending previous results from skew-symmetric matrices to arbitrary matrices and analyzing the role of curvature.

Contribution

It provides a comprehensive characterization of when lacunary elliptic maximal operators are bounded on L^p spaces based on matrix properties, extending prior work.

Findings

L^p boundedness results for lacunary elliptic maximal operators on the Heisenberg group.

Extension of L^p estimates from skew-symmetric to arbitrary matrices.

Necessary and sufficient conditions on matrices A for operator boundedness.

Abstract

In this paper, we prove \( L^p \) boundedness results for lacunary elliptic maximal operators on the Heisenberg group. Furthermore, we extend these \( L^p \) estimates from skew-symmetric matrices, which naturally arise in Heisenberg group operations, to arbitrary matrices \( A \), investigating how the curvature induced by \( A \) governs the \( L^p \) boundedness of lacunary circular and elliptic maximal operators. Specifically, we provide necessary and sufficient conditions on \( A \) that determine whether these operators are bounded or unbounded on \( L^p \).