The Bargmann transform on a broad family of Banach spaces, with applications to Toeplitz and pseudo-differential operators

TL;DR

This paper explores the properties of the Bargmann transform on broad classes of modulation spaces with rapidly growing weights, demonstrating its isometric bijection to Lebesgue spaces and applying these results to analyze Toeplitz and pseudo-differential operators.

Contribution

It extends the understanding of the Bargmann transform's mapping properties to more general modulation spaces with fast-growing weights, and applies these findings to operator continuity.

Findings

Bargmann transform is isometric and bijective on these modulation spaces.

Modulation spaces with fast-growing weights satisfy key continuity properties.

Established continuity results for Toeplitz and pseudo-differential operators in this context.

Abstract

We investigate mapping properties for the Bargmann transform on modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are well-known when the weights are moderated. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators in the context of these modulation spaces.