Carleson Embeddings Research Articles

Generalized Carleson embeddings of Müntz spaces

This paper establishes Carleson embeddings of Müntz spaces MΛq into weighted Lebesgue spaces Lp(dμ), where μ is a Borel regular measure on [0,1] satisfying μ([1−ε])≲εβ. In the case β⩾1 we show that such measures are exactly the ones for which Carleson embeddings Lpβ↪Lp(dμ) hold. The case β∈(0,1) is more intricate, but we characterize such measures μ in terms of a summability condition on their moments. Our proof relies on a generalization of Lp estimates à la Gurariy-Macaev in the weighted Lp spaces setting, which we think can be of interest in other contexts.

Generalized Carleson embeddings into weighted outer measure spaces

We prove generalized Carleson embeddings for the continuous wave packet transform from Lp(R,w) into an outer Lp space over R×R×(0,∞) for 2<p<∞ and weight w∈Ap/2. This work is a weighted extension of the corresponding Lebesgue result in [13] and generalizes a similar result in [10]. The proof in this article relies on L2 restriction estimates for the wave packet transform which are geometric and may be of independent interest.

Carleson embeddings for Hardy-Orlicz and Bergman-Orlicz spaces of the upper-half plane

In this paper we characterize off-diagonal Carleson embeddings for both Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper-half plane. We use these results to obtain embedding relations and pointwise multipliers between these spaces.

Carleson embeddings with loss for Bergman–Orlicz spaces of the unit ball

ABSTRACT We prove Carleson embeddings for Bergman–Orlicz spaces of the unit ball that extend the lower triangle estimates for the usual Bergman spaces.

A Toeplitz-type operator on Hardy spaces in the unit ball

We study a Toeplitz type operator $Q_\mu$ between the holomorphic Hardy spaces $H^p$ and $H^q$ of the unit ball. Here the generating symbol $\mu$ is assumed to a positive Borel measure. This kind of operator is related to many classical mappings acting on Hardy spaces, such as composition operators, the Volterra type integration operators and Carleson embeddings. We completely characterize the boundedness and compactness of $Q_\mu:H^p\to H^q$ for the full range $1<p,q<\infty$; and also describe the membership in the Schatten classes of $H^2$. In the last section of the paper, we demonstrate the usefulness of $Q_\mu$ through applications.

On Laplace–Carleson embeddings, and $L^p$-mapping properties of the Fourier transform

We investigate so-called Laplace–Carleson embeddings for large exponents. In particular, we extend some results by Jacob, Partington, and Pott. We also discuss some related results for Sobolev-and Besov spaces, and mapping properties of the Fourier transform. These variants of the Hausdorff–Young theorem appear difficult to find in the literature. We conclude the paper with an example related to an open problem.

The reproducing kernel thesis for lower bounds of weighted composition operators

It is shown that the property of being bounded below (having closed range) of weighted composition operators on Hardy and Bergman spaces can be tested by their action on a set of simple test functions, including reproducing kernels. The methods used in the analysis are based on the theory of reverse Carleson embeddings.

Absolutely summing Carleson embeddings on Hardy spaces

We consider the Carleson embeddings of the classical Hardy spaces (on the disk) into a Lp(μ) space, where μ is a Carleson measure on the unit disk. This includes the case of composition operators. We characterize such operators which are r-summing on Hp, where p>1 and r≥1. This completely extends the former results on the subject and answers a question posed on the early seventies.

Hilbert spaces of analytic functions with a contractive backward shift

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift f(z)↦f(z)−f(0)z is a contraction on the space. We present a model for this operator and use it to prove the surprising result that functions which extend continuously to the closure of the disk are dense in the space. This has several applications, for example we can answer a question regarding reverse Carleson embeddings for these spaces. We also identify a large class of spaces which are similar to the de Branges–Rovnyak spaces and prove some results which are new even in the classical case.

A modulation invariant Carleson embedding theorem outside local L2

The article arXiv:1309.0945 by Do and Thiele develops a theory of Carleson embeddings in outer $L^p$ spaces for the wave packet transform of functions in $ L^p(\mathbb R)$, in the $2\leq p\leq \infty$ range referred to as local $L^2$. In this article, we formulate a suitable extension of this theory to exponents $1<p<2$, answering a question posed in arXiv:1309.0945. The proof of our main embedding theorem involves a refined multi-frequency Calder\'on-Zygmund decomposition. We apply our embedding theorem to recover the full known range of $L^p$ estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation.

Vectorial Hankel operators, Carleson embeddings, and notions of BMOA

Let {textrm{BMOA}_{mathcal{N}mathcal{P}}left(mathcal{L}right)} denote the space of {mathcal{L}}-valued analytic functions {phi} for which the Hankel operator {Gamma_phi} is {H^2left(mathcal{H}right)}-bounded. Obtaining concrete characterizations of {textrm{BMOA}_{mathcal{N}mathcal{P}}left(mathcal{L}right)} has proven to be notoriously hard. Let {D^alpha} denote fractional differentiation. Motivated originally by control theory, we characterize {H^2left(mathcal{H}right)}-boundedness of {D^alphaGamma_phi}, where {alpha > 0}, in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that {textrm{BMOA}_{mathcal{N}mathcal{P}}left(mathcal{L}right)} is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of {Gamma_phi}. The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.

Plongements de Carleson absolument sommants

We study the Carleson embeddings on the Hardy spaces Hp with p>1. We characterize their membership in the class of r-summing operators for any r≥1. This generalizes in a strong way former results and solves a problem open since the 1970s. This characterization is different in nature according to the value of the couple (p,r). As particular cases, this settles the case of composition and weighted composition operators.

Carleson Embeddings and Two Operators on Bergman Spaces of Tube Domains over Symmetric Cones

We prove Carleson embeddings for Bergman spaces of tube domains over symmetric cones, we apply them to characterize symbols of bounded Cesaro-type operators from weighted Bergman spaces to weighted Besov spaces. We also obtain Schatten class criteria of Toeplitz operators and Cesaro-type operators on weighted Hilbert-Bergman spaces.

Applications of Laplace--Carleson Embeddings to Admissibility and Controllability

It is shown how results on Carleson embeddings induced by the Laplace transform can be used to derive new and more general results concerning the weighted (infinite-time) admissibility of control and observation operators for linear semigroup systems with $q$-Riesz bases of eigenvectors. As an example, the heat equation is considered. Next, a new Carleson embedding result is proved, which gives further results on weighted admissibility for analytic semigroups. Finally, controllability by smoother inputs is characterized by means of a new result about weighted interpolation.

On Laplace–Carleson embedding theorems

This paper gives embedding theorems for a very general class of weighted Bergman spaces: the results include a number of classical Carleson embedding theorems as special cases. The little Hankel operators on these Bergman spaces are also considered. Next, a study is made of Carleson embeddings in the right half-plane induced by taking the Laplace transform of functions defined on the positive half-line (these embeddings have applications in control theory): particular attention is given to the case of a sectorial measure or a measure supported on a strip, and complete necessary and sufficient conditions for a bounded embedding are given in many cases.

Choquet integrals, weighted Hausdorff content and maximal operators

Abstract The boundedness of maximal operators on the weighted Choquet space and the Choquet–Morrey space is established. These results are used to study Carleson embeddings for weighted Sobolev spaces.

Boundedness, Compactness and Schatten-class Membership of Weighted Composition Operators

The boundedness and compactness of weighted composition operators on the Hardy space \({{\mathcal H}^2}\) of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces \({\mathcal{A}^2_\alpha(\mathbb{D})}\) are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.

Carleson embeddings and some classes of operators on weighted Bergman spaces

We prove Carleson-type embedding theorems for weighted Bergman spaces with Békollé weights. We use this to study properties of Toeplitz-type operators, integration operators and composition operators acting on such spaces. In particular, we investigate the membership of these operators to Schatten class ideals.

Applications of the Discrete Weiss Conjecture in Operator Theory

In this paper, we study a discrete version of the Weiss Conjecture. In Section 1 we discuss the Reproducing Kernel Thesis and in Section 2 we introduce the operators which concern us. Section 3 shows how to relate these operators to Carleson embeddings and weighted composition operators, so that we can apply the Carleson measure theorem to obtain conditions for boundedness and compactness of many weighted composition operators. Section 4 contains Theorem 4.4 which is a discrete version of the Weiss Conjecture for contraction semigroups, and finally Section 5 shows how the usual (continuous time) Weiss Conjecture is related to the discrete version studied here; in fact they are equivalent (for scalar valued observation operators). The main advantage of the discrete version is that it is technically simpler – the observation operators are automatically bounded and the functional calculus can be achieved using power series.

Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings

We study weighted norm inequalities for the derivatives (Bernstein-type inequalities) in the shift-coinvariant subspaces K Θ p of the Hardy class H p in the upper half-plane. It is shown that the differentiation operator acts from K Θ p to certain spaces of the form L p ( w ) , where the weight w ( x ) depends on the density of the spectrum of Θ near the point x of the real line. We discuss an application of the Bernstein-type inequalities to the problems of the description of measures μ , for which K Θ p ⊂ L p ( μ ) , and of compactness of such embeddings. New versions of Carleson-type embedding theorems are obtained generalizing the theorems due to W.S. Cohn and A.L. Volberg–S.R. Treil.