Boundary Behaviour of Polyanalytic Functions in Certain Banach Spaces

Abstract

This paper investigates the boundary behaviour of polyanalytic functions of order m in various Banach spaces. We establish structural decomposition theorems in weighted Bergman-type spaces A_m L_P (D,α) and study their implications for boundary regularity. Poly-Hardy spaces H^(n,P) (D), and meta-Hardy spaces H_A^(n,P) (D) are introduced to capture boundary values through radial maximal functions. We solve Dirichlet, Hilbert, and Haseman boundary value problems in polyanalytic settings, establishing Fredholm properties via singular integral operators. Lipschitz regularity conditions are characterized in terms of holomorphic components. Approximation results in L^P spaces on the unit circle are obtained through density criteria for subspaces of the form H^P+∑_(k=1)^m▒〖w_k H^P.〗 The main results include solvability conditions, index formulas, and explicit boundary value representations. The interplay between boundary behaviour and the underlying Banach space structure is emphasized throughout this paper.