General fractional derivatives and the Bergman projection

TL;DR

This paper explores properties of general fractional derivatives linked to weighted Bergman kernels and introduces a method for finding pre-images under weighted Bergman projections, aiding in understanding their surjectivity.

Contribution

It presents a novel approach for generating pre-images of analytic functions under weighted Bergman projections, especially in complex target spaces.

Findings

Demonstrates a method for pre-image generation under weighted Bergman projections.

Establishes surjectivity of weighted Bergman projections in non-subspace target spaces.

Discusses a fractional Littlewood-Paley formula.

Abstract

In this note we study some basic properties of general fractional derivatives induced by weighted Bergman kernels. As an application we demonstrate a method for generating pre-images of analytic functions under weighted Bergman projections. This approach is useful for proving the surjectivity of weighted Bergman projections in cases when the target space is not a subspace of the domain space (such situations arise often when dealing with Bloch and Besov spaces). We also discuss a fractional Littlewood-Paley formula.