Hormander's L-2-Method, <(?)over bar >-Problem and Polyanalytic Function Theory in One Complex Variable

In this paper we consider the classical (?) over bar -problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of poly-analytic functions in order to construct particular solutions of this problem and obtain new Hormander type estimates using suitable powers of the Cauchy-Riemann oper-ator. We also compute particular solutions of the (?) over bar -problem for specific polyanalytic data such as the Ito complex Hermite polynomials and polyanalytic Fock kernels.