Littlewood-type theorems for random Dirichlet multipliers

Abstract

In this paper we present a systematic study of random Dirichlet functions. In 1993, Cochran-Shapiro-Ullrich proved the following elegant result on random Dirichlet multipliers [21]: For any $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n\in D$, the Dirichlet space over the unit disk, almost all of its randomizations$\left(Rf\right)\left(z\right)=\sum _{n=0}^{\infty }\pm a_n{z}^n$ are multipliers of $D$. The purpose of this paper is to exploit this result and to extend it in three directions, inspired by the 1930 theorem of Littlewood on random Hardy functions:

• (A)
  We introduce a symbol space $M_{\star }$ for random multipliers on $D$ and reformulate (and strengthen) the problem as the characterization of $M_{\star }$. We then characterize ${\left(M_{\alpha }^p\right)}_{\star }$ for all $p\in (0,\infty ),\alpha \in (-1,\infty )$ when $\alpha \ne p-1$ (Theorem 55). The case $p=2,\alpha =0$ recovers the 1993 result.
• (B)
  We obtain a two-parameter version by formulating and solving a Littlewood-type problem for all $(p,q)\in {(0,\infty )}^2$ (Theorem 69). The case $p=q=2$ recovers the 1993 result.
• (C)
  We consider $D_{p-1}^p$ and $D_{p-2}^p$ and solve completely the two-parameter Littlewood-type problem for these two spaces (Theorem 74 and Theorem 77). Moreover, the case $p=2$ in Lemma 78 recovers the 1993 result.

Technical challenges in this paper include two aspects. First, this paper is the first systematic study of random Dirichlet functions; consequently, many basic questions need to be answered. In particular, we encounter several fundamental topics, clearly of general interests for random analytic functions yet seldom treated in literature, including:

• (i)
  random Carleson measures;
• (ii)
  the cotype of Banach spaces of analytic functions;
• (iii)
  the norm convergence of random Taylor polynomials; and
• (iv)
  the Marcus-Pisier space.

We establish properties for these topics as prompted by our study. The second challenge we face is that many of our results are satisfactory “if and only if” type; to achieve this, we need to construct many examples to show the optimality of various embedding problems.