Embedding theorems for spaces of analytic functions via Khinchine's inequality
Let \(D\) denote the open unit disk of the complex plane, and suppose we are given a space of functions analytic on \(D\) with norm \(\| \cdot \|\). Fix an integer \(n \geq 0\) and a positive number \(s\). The problem is then (for various spaces) to classify those positive measures \(\mu\) for which there exists a positive constant \(C_ \mu\) such that \[ \Bigl( \int_ D \bigl | f^{(n)} (z) \bigr |^ sd \mu (z) \Bigr)^{1/s} \leq C_ \mu \| f \| \] for all \(f\) in the space under discussion; such an embedding problem is inspired by \textit{L. Carleson}'s famous theorem for the \(H^ p\) spaces (then \(n=0\) and \(s=p)\) [Ann. of Math., II. Ser. 76, 547-559 (1962; Zbl 0112.297)]. The paper under review is part of a series of papers by the same author on similar problems [see, e.g., Proc. Edinb. Math. Soc., II. Ser. 29, 125-131 (1986; Zbl 0587.30048); Proc. Lond. Math. Soc., III. Ser. 63, 595-619 (1991; Zbl 0774.42011)]. The main purpose of the paper is to introduce a new method for handling such problems. The method consists in combining known embeddings of sequence spaces into spaces of analytic functions with Khinchine's inequality in order to obtain a necessary condition on \(\mu\). In each of the cases discussed in the paper, it is quite easy to prove that this condition is sufficient as well. The method is first illustrated in the case of unweighted Bergman spaces (here the solution is already known); then it is used to obtain embedding theorems for the mixed norm spaces, i.e., \[ \| f \| = \| f \|_{p,q} = \left( \int^ 1_ 0 \left( \int^{2 \pi}_ 0 \bigl | f(re^{i \theta}) \bigr |^ p \right)^{q/p} dr \right)^{1/q}, \] \((p,q\) positive numbers) and certain analytic tent spaces [inspired by the tent spaces introduced by \textit{R. Coifman}, \textit{Y. Meyer} and \textit{E. M. Stein}, J. Funct. Anal. 62, 304-335 (1985; Zbl 0569.42016)]. The two latter situations seem to require the full power of the method, and the author says: ``It is possible to use the same techniques to investigate a nearly unending array of embedding theorems''.
embeddings of sequence spaces into spaces of analytic functions with Khinchine's inequality
scientific article; zbMATH DE number 474974
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