A converse of Littlewood-type theorem on random analytic functions in mixed norm spaces

Source abstract

Abstract This article investigates coefficient randomizations of holomorphic functions and the resulting random analytic functions within the framework of mixed norm spaces. Adopting the perspective of the random symbol space ( X ) ⋆ = { f ∈ H ( B n ) : P ( R f ∈ X ) = 1 } $(\mathcal {X})_{\star }=\{f\in H(\mathbb {B}_n): \mathbb {P}(\mathcal {R} f\in \mathcal {X})=1\}$ left parenthesis script upper X right parenthesis Subscript star Baseline equals StartSet f element of upper H left parenthesis double struck upper B Subscript n Baseline right parenthesis colon double struck upper P left parenthesis script upper R f element of script upper X right parenthesis equals 1 EndSet , we first focus on the unit disk D $\mathbb {D}$ double struck upper D . Specifically, for q ≥ 4 $q\geq 4$ q greater than or equals 4 , we identify the conditions under which the random symbol space of the mixed-norm space H p , q , u ( D ) $H^{p,q,u}(\mathbb {D})$ upper H Superscript p comma q comma u Baseline left parenthesis double struck upper D right parenthesis coincides with H 2 , q , u ( D ) $H^{2,q,u}(\mathbb {D})$ upper H Superscript 2 comma q comma u Baseline left parenthesis double struck upper D right parenthesis . Furthermore, we extend our analysis to the higher-dimensional unit ball B n $\mathbb {B}_n$ double struck upper B Subscript n . We establish sufficient conditions and necessary constraints linking the deterministic membership of a function to the almost sure membership of its randomization in mixed norm spaces. A key finding is the identification of a sharp, dimension-dependent weight loss phenomenon that arises for n > 1 $n>1$ n greater than 1 .