Invariant subspaces in Bergman spaces and Hedenmalm's boundary value problem

Abstract

A function G in a Bergman space A^p, 0< p<∞, in the unit disk D is called A^p-inner if |G|^p−1 annihilates all bounded harmonic functions in D. Extending a recent result by Hedenmalm for p=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $\Delta \Phi = \chi _D (|G|^p - 1),$ where χ_D denotes the characteristic function of D and G is an arbitrary A^p-inner function, is continuous in C, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman space A² to a general A^p-setting.