Carleson embeddings
type: Item
instance of: scholarly article
title: Carleson embeddings
zbMATH Open document ID: 0937.47024
DOI: 10.1155/S1085337596000097
author: Helmut J. Heiming
published in: Abstract and Applied Analysis
publication date: 2000-06-13T00:00:00Z
full work available at URL: https://eudml.org/doc/48414
full work available at URL: issue-2.html
review text: This is a paper on special spaces of functions \(\mathbb{D}\to\mathbb{C}\) where \(\mathbb{D}\) is the open unit disk in \(\mathbb{C}\). Let \(\mathbb{T}\) be the boundary of \(\mathbb{D}\), and let \(m\) be normalized Lebesgue measure on \(\mathbb{T}\). Put \(B(0)= \mathbb{T}\), and if \(0\neq z\in\mathbb{D}\), let \(B(z)\) be the arc of length \(1-|z|^2\) centered at \(z/|z|\). The Stoltz domain for \(\zeta\in\mathbb{T}\) is \(\Gamma(\zeta)= \{z\in\mathbb{D}: \zeta\in B(z)\}\), and the tent over an open set \(\Omega\subset\mathbb{T}\) is \(\theta(\Omega)= \Omega\cup \{z\in\mathbb{D}: B(z)\subset \Omega\}\). Following \textit{R. R. Coifman}, \textit{Y. Meyer} and \textit{E. M. Stein} [J. Funct. Anal. 62, 304-335 (1985; Zbl 0569.42016)] the non-tangential supremum \(Nf(\zeta):= \sup_{z\in\Gamma(\zeta)}|f(z)|\) of a function \(f:D\to \mathbb{C}\) at \(\zeta\in\mathbb{T}\) is used to define, for \(0<q\leq\infty\), the `tent space' \(T^q(D)\) as the closure of \({\mathcal C}(\overline D)\) in the (\(q\)-)Banach space of all \(f:\mathbb{D}\to \mathbb{C}\) such that \(\|Nf\|_{L^q(m)}\) is finite. For \(1<q<\infty\) (resp. \(0<q<\infty\)), the usual harmonic space \(h^q(\mathbb{D})\) (Hardy space \(H^q(\mathbb{D}))\) just consists of all harmonic (analytic) members of \(T^q(\mathbb{D})\). Given \(\beta>0\), say that a (regular) Borel measure \(\mu\) on \(\overline{\mathbb{D}}\) is a \(\beta\)-Carleson measure if there is a constant \(C\) such that \(|\mu|(\theta(\Omega))\leq C\cdot m(\Omega)^\beta\) for every open set \(\Omega\subset\mathbb{T}\). Let \(\|\mu\|_\beta\) be the smallest of such constants. The \(\beta\)-Carleson measures form a Banach space \(M^\beta\) with norm \(\|\cdot\|_\beta\); it is isometrically isomorphic to the dual of the tent space \(T^{1/\beta}(\mathbb{D})\). Moreover, the formal identity \(I_\mu: T^q(\mathbb{D})\hookrightarrow L^{\beta q}(\mu)\) exists and is continuous if and only if \(\mu\) is in \(M^\beta\). In the light of the fact that the i…
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 1370620
zbMATH Keywords: tent space
zbMATH Keywords: Carleson measure
zbMATH Keywords: Stoltz domain
zbMATH Keywords: non-tangential supremum
zbMATH Keywords: Borel measure
zbMATH Keywords: composition operator
zbMATH Keywords: weakly compact
zbMATH Keywords: \(p\)-integral
zbMATH Keywords: absolutely \(p\)-summing
reviewed by: Hans Jarchow
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1370620