A Littlewood-Type Theorem for Random Bergman Functions

Let f(z)=∑ n=0 ∞ a n zn be a formal power series with complex coefficients. Let (R f)(z)= ∑ n=0 ∞ pm a n zn be the randomization of f by choosing independently a random sign for each coefficient. Let Hp(D) and L p a(D) (p>0) denote the Hardy space and the Bergman space, respectively, over the unit disk in the complex plane. In 1930, Littlewood proved that if f in H2(D), then R f \in Hp(D) for any p \in (0, ∞) almost surely, and if f ϵ H2(D), then R f ϵ Hp(D) for any p \in (0, ∞) almost surely. In this paper, we obtain a characterization of the pairs (p, q) \in (0, ∞) 2 such that R f is almost surely in L q a(D) whenever f in L p a(D), including counterexamples to show the optimality of the embedding. In contrast to Littlewood's theorem, random Bergman functions exhibit no improvement of regularity for any p>0, but the loss of regularity for p<2 is not as drastic as the Hardy case; there is indeed a nontrivial boundary curve given by 1 q- 2p+ 1 2=0. Several other results about random Bergman functions are established along the way. The technical difficulties, especially when p<1, are different from the Hardy space and we devise a different route of proof. The Dirichlet space follows as a corollary. An improvement of the original Littlewood theorem is obtained.