Local Maximum Modulus Property for Polyanalytic Functions

Let (Omega subset {mathbb {C}}) be an open subset and let ({mathcal {F}}) be a space of functions defined on (Omega ). ({mathcal {F}}) is said to have the local maximum modulus property if: for every (fin {mathcal {F}},p_0in Omega ,) and for every sufficiently small domain (Dsubset Omega ,) with (p_0in D,) it holds true that (max _{zin overline{D}}left| f(z)right| = max _{zin Sigma cup partial D}left| f(z)right| ,) where (Sigma subset Omega ) denotes the set of points at which (left| fright| ) attains strict local maximum. This property fails for ({mathcal {F}}=C^{infty }.) 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 ({mathcal {F}}) is the set of n-analytic functions on (Omega ), (nge 2,) in the sense that locality cannot be removed as a condition and independently (Sigma ) cannot be removed from the conclusion.