H\"ormander's $L^2$-method, $\bar{\partial}$-problem and polyanalytic function theory in one complex variable

TL;DR

This paper explores the classical ar-problem in one complex variable, extending Hrmanderb4s method to polyanalytic functions, and provides new estimates and solutions for specific polyanalytic data.

Contribution

It introduces new Hrmander type estimates for the ar-problem with polyanalytic data and constructs particular solutions using the decomposition of polyanalytic functions.

Findings

Derived new ar-estimates for polyanalytic functions.

Constructed particular solutions for specific polyanalytic data such as Hermite polynomials.

Computed solutions for polyanalytic Fock kernels.

Abstract

In this paper we consider the classical $\bar{\partial}$-problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new H\"ormander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the $\bar{\partial}$-problem for specific polyanalytic data such as the It\^o complex Hermite polynomials and polyanalytic Fock kernels.