Norms of embeddings between quadratically weighted spaces of holomorphic functions

TL;DR

This paper investigates the operator norms of embeddings between spaces of holomorphic functions with Gaussian weights, linking these embeddings to Fourier integral operators with complex phase, relevant in the context of FBI-Bargmann transforms.

Contribution

It provides a novel characterization of embedding norms between weighted holomorphic function spaces using Fourier integral operators with complex phase.

Findings

Explicit formulas for embedding operator norms.

Connection established between embeddings and Fourier integral operators.

Applications to FBI-Bargmann transform theory.

Abstract

We consider spaces of holomorphic functions which are square-integrable against a Gaussian weight, which appear in the theory of metaplectic FBI--Bargmann transforms. We identify the operator norm of embeddings between two such spaces, by relating these embeddings to Fourier integral operators with complex phase.