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

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.