A Littlewood-Type Theorem for Random Bergman Functions

Let $f(z)=\sum _{n=0}^{\infty }a_n z^n$ be a formal power series with complex coefficients. Let $({\mathcal{R}} f)(z)= \sum _{n=0}^{\infty }\pm a_n z^n$ be the randomization of $f$ by choosing independently a random sign for each coefficient. Let $H^p({\mathbb{D}})$ and $L^p_a({\mathbb{D}})$  $(p>0)$ denote the Hardy space and the Bergman space, respectively, over the unit disk in the complex plane. In 1930, Littlewood proved that if $f \in H^2({\mathbb{D}})$, then ${\mathcal{R}} f \in H^p({\mathbb{D}})$ for any $p \in (0, \infty )$ almost surely, and if $f \notin H^2({\mathbb{D}})$, then ${\mathcal{R}} f \notin H^p({\mathbb{D}})$ for any $p \in (0, \infty )$ almost surely. In this paper, we obtain a characterization of the pairs $(p, q) \in (0, \infty )^2$ such that ${\mathcal{R}} f$ is almost surely in $L^q_a({\mathbb{D}})$ whenever $f \in L^p_a({\mathbb{D}})$, 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 $\frac{1}{q}-\frac{2}{p}+\frac{1}{2}=0$. 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.