Hörmander’s $$L^2$$-Method, $${\overline{\partial }}$$-Problem and Polyanalytic Function Theory in One Complex Variable

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