Inverse-Closedness of a Banach Algebra of Integral Operators on the Heisenberg Group

Abstract

Let $\mathbb{H}$ be the general, reduced Heisenberg group. Our main result establishes the inverse-closedness of a class of integral operators acting on $L^p(\mathbb{H})$, given by the off-diagonal decay of the kernel. As a consequence of this result, we show that if ${\alpha }_{1}I+S_f$, where $S_f$ is the operator given by convolution with $f$, $f\in L_v^1(\mathbb{H})$, is invertible in $\B(L^{p}(\mathbb{H}))$, then (α_{1}I+S_{f})^{-1}=α_{2}I+S_{g}$,and$g\in L^{1}_{v}(\mathbb{H})$. We prove analogous results for twisted convolution operators and apply the latter results to a class of Weyl pseudodifferential operators. We briefly discuss relevance to mobile communications.

arXiv commentThis version corrects two mistakes and recognizes the work of other authors related to a corollary of our main theorem

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