TY - JOUR

T1 - A Littlewood-Type Theorem for Random Bergman Functions

AU - Cheng, Guozheng

AU - Fang, Xiang

AU - Liu, Chao

N1 - Publisher Copyright:
© 2021 The Author(s). Published by Oxford University Press. All rights reserved.

PY - 2022/7/1

Y1 - 2022/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85134024126&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnab018

DO - 10.1093/imrn/rnab018

M3 - 期刊論文

AN - SCOPUS:85134024126

SN - 1073-7928

VL - 2022

SP - 11056

EP - 11091

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 14

ER -