On the range of weighted planar Cauchy transform

TL;DR

This paper characterizes the range of the weighted planar Cauchy transform and its k-Bergman projection within weighted true poly-Bargmann spaces, expanding understanding of their functional analytic properties.

Contribution

It provides a detailed description of the range of these transforms on specific weighted function spaces, a novel analysis in complex analysis and operator theory.

Findings

Range characterized for weighted Cauchy transform

Results on k-Bergman projection in poly-Bargmann spaces

Enhanced understanding of Gaussian function spaces

Abstract

We describe the range of of weighted Cauchy transform and its $k$-Bergman projection when action on weighted true poly-Bargmann spaces constituting an orthogonal Hilbertian decomposition of the Hilbert space of Gaussian functions on the complex plane.