Invertibility in the flag kernels algebra on the Heisenberg group

TL;DR

This paper proves that the algebra of convolution operators with flag kernels on the Heisenberg group is inverse-closed, meaning invertible operators within this algebra have inverses that also belong to the same algebra.

Contribution

It establishes the inverse-closed property of the flag kernels algebra specifically on the Heisenberg group, extending understanding of its algebraic structure.

Findings

The algebra of flag kernels is closed under composition.

Invertible operators in this algebra have inverses within the same algebra.

The result applies specifically to the Heisenberg group.

Abstract

Flag kernels are tempered distributions which generalize these of Calderon-Zygmund type. For any homogeneous group $\mathbb{G}$ the class of operators which acts on $L^{2}(\mathbb{G})$ by convolution with a flag kernel is closed under composition. In the case of the Heisenberg group we prove the inverse-closed property for this algebra. It means that if an operator from this algebra is invertible on $L^{2}(\mathbb{G})$, then its inversion remains in the class.