Local Maximum Modulus Property for Polyanalytic Functions

Let be an open subset and let be a space of functions defined on . is said to have the local maximum modulus property if: for every and for every sufficiently small domain with it holds true that where denotes the set of points at which attains strict local maximum. This property fails for We verify it however for the set of complex-valued functions whose real and imaginary parts are real analytic. We show by example that the property cannot be improved upon whenever is the set of n-analytic functions on , in the sense that locality cannot be removed as a condition and independently cannot be removed from the conclusion.