Carleson embeddings

Helmut J Heiming

doi:10.1155/s1085337596000097

Abstract

In this paper we discuss several operator ideal properties for so called Carleson embeddings of tent spaces into specificL q(μ)-spaces, whereμis a Carleson measure on the complex unit disc. Characterizing absolutelyq-summing, absolutely continuous andq-integral Carleson embeddings in terms of the underlying measure is our main topic. The presented results extend and integrate results especially known for composition operators on Hardy spaces as well as embedding theorems for function spaces of similar kind.

Highlights

Introduction and Main ResultsCarleson measures proved to be an effective tool to discuss composition operators on classical Hardy spaces Hq(D) on the complex unit disc D
As restrictions of compact operators are still compact, a compact Carleson embedding is induced by a vanishing Carleson measure
First of all we show that μ is a β-Carleson measure

Summary

Introduction and Main Results

Carleson measures proved to be an effective tool to discuss composition operators on classical Hardy spaces Hq(D) on the complex unit disc D (see Hunziker, Jarchow [6], or Zhu [9, Chapter 8]). Carleson embedding, absolutely summing operator, tent space, Hardy space, composition operator. As restrictions of compact operators are still compact, a compact Carleson embedding is induced by a vanishing Carleson measure. If μ is a vanishing β-Carleson measure the Carleson embedding Iμ : T q(D) → Lβq(μ) can be approximated in operator norm by (βq)-integral operators, dist(Iμ, { u :. If μ is a vanishing β-Carleson measure for all q ≥ β−1 the induced Iμ : T q(D) → Lβq(μ) is absolutely continuous. Μ is a vanishing β-Carleson measure if and only if the induced Carleson embedding Iμ : T q(D) → Lβq(μ) is absolutely continuous. Iμ : T q(D) → Lβq(μ) is absolutely (βq)-summing for some, and all, finite q ≥ β−1

Preliminaries

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3

Concluding Remarks