INVARIANT SUBSPACES IN BERGMAN SPACES AND HEDENMALMS BOUNDARY-VALUE PROBLEM

A function G in a Bergman space A(p), 0<p<infinity, in the unit disk D is called AP-inner if \G\(p)-1 annihilates all bounded harmonic funct ions in D. Extending a recent result by Hedenmalm for p=2, we show (Th m. 2) that the unique compactly-supported solution Phi of the problem Delta Phi = chi(D) (\G\(P) - 1), where chi(D), denotes the characteris tic function of D and G is an arbitrary A(P)-inner function, is contin uous 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 A(2) t o a general A(P)-setting.