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. [ABSTRACT FROM AUTHOR]