Mixed Fourier norm spaces of analytic functions on the upper half-plane and Toeplitz operators

TL;DR

This paper introduces weighted mixed Fourier norm spaces of analytic functions on the upper half-plane, characterizes these functions, and explores Toeplitz operators with vertical symbols within these spaces.

Contribution

It defines new function spaces based on Fourier transforms, characterizes analytic functions in these spaces, and analyzes Toeplitz operators with vertical symbols in this context.

Findings

Characterization of analytic functions in mixed Fourier norm spaces

An analog of the Paley-Wiener theorem for these spaces

Analysis of Toeplitz operators with vertical symbols

Abstract

We introduce and study weighted spaces of functions with mixed norm on the upper half-plane, defined in terms of Fourier transform. We give a characterization of analytic functions within these spaces, and in particular, we provide an analog of the Paley-Wiener theorem in this setting. As an application, we consider Toeplitz operators with vertical symbols in these new spaces.