Weighted estimates of the fourier transformation

Parfenov, O. G.

doi:10.1007/bf02355829

Cited by 4 publications

(6 citation statements)

References 7 publications

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“…Our conditions, stated in terms of dyadic partition of the disc, are similar to a theorem of Luecking [24] which characterizes Schatten-von Neumann class embeddings of the whole Hardy class, as well as to results of Parfenov [28,29]. For further discussion see Sections 5-6.…”

Section: Introduction and Main Resultsmentioning

confidence: 91%

Compact embeddings of model subspaces of the Hardy space

Baranov

2007

Preprint

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We study embeddings of model (star-invariant) subspaces K p Θ of the Hardy space H p , associated with an inner function Θ. We obtain a criterion for the compactness of the embedding of K p Θ into L p (µ) analogous to the Volberg-Treil theorem on bounded embeddings and answer a question posed by Cima and Matheson. The proof is based on Bernstein inequalities for functions in K p Θ . Also we study measures µ such that the embedding operator belongs to a Schatten-von Neumann ideal.

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Section: Introduction and Main Resultsmentioning

confidence: 91%

Compact embeddings of model subspaces of the Hardy space

Baranov

2007

Preprint

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“…We obtain a condition, which is analogous to (8) and is sufficient for (19) and, therefore, sufficient for the inclusion of ϕ t ( S) − ϕ t (S) into the class S p . Proposition 7.2.…”

Section: Estimates For the Trace Classmentioning

confidence: 96%

“…Let ν = n∈Z δ n be the sum of unit point masses at all integers. Thus, the total mass of ν is infinite, and hence the corresponding measure µ on T does not satisfy condition (8). For ψ t (x) = e itx in the last formula, we obtain the integral…”

Section: Estimates For the Hilbert-schmidt Classmentioning

confidence: 99%

“…We expect that the condition ν(R) < ∞ equivalent to (8), is also necessary for the inclusions ϕ t ( S) − ϕ t (S) ∈ S 2 , that is, the factor 1 − Θ(z) does not change essentially the convergence of the integral in (16). Now we present an example of a measure ν such that the corresponding operator ϕ t ( S) − ϕ t (S) is not in the Hilbert-Schmidt class for any t = 0.…”

Section: Estimates For the Hilbert-schmidt Classmentioning

confidence: 99%

“…We reduce this problem to a question about embeddings of the Paley-Wiener space PW t of all entire functions of exponential type at most t that are square-summable on R. Embedding operators for spaces of analytic functions and their inclusion to ideals S p were studied in detail by O.G. Parfenov [7,8]. In [3] generalizations of some of Parfenov's results to embeddings of coinvariant subspaces K θ are obtained.…”

Section: Estimates For the Trace Classmentioning

confidence: 99%

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On perturbations of the isometric semigroup of shifts on the semiaxis

Amosov

Baranov

Капустин

2011

St. Petersburg Math. J.

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Abstract. Perturbations (r τ t ) t≥0 of the semigroup of shifts (τ t ) t≥0 on L 2 (R + ) are studied under the assumption that r τ t −τ t belongs to a certain Schatten-von Neumann class S p with p ≥ 1. It is shown that, for the unitary component in the WoldKolmogorov decomposition of the cogenerator of the semigroup (r τ t ) t≥0 , any singular spectral type may be achieved by S 1 -perturbations. An explicit construction is provided for a perturbation with a given spectral type, based on the theory of model spaces of the Hardy space H 2 . Also, it is shown that an arbitrary prescribed spectral type may be obtained for the unitary component of the perturbed semigroup by a perturbation of class S p with p > 1. §1. IntroductionConsider the isometric semigroup (τ t ) t≥0 of shifts on the space L 2 (R + ),In this paper we are concerned with perturbations (r τ t ) of the semigroup (τ t ) satisfying the following properties: (r τ t ) t≥0 is a strongly continuous semigroup of isometric operators on L 2 (R + ); the difference r τ t − τ t belongs to a certain Schatten-von Neumann ideal S p for every t > 0.The central problem considered in this paper is to describe all possible spectral types of perturbed isometric semigroups. The spectral type of a semigroup determines the semigroup uniquely up to unitary equivalence; it is defined by the spectral type of the cogenerator of the group (the definition can be found in §2). For p = 1, the stability of the absolutely continuous spectrum of the unitary dilation implies that the absolutely continuous parts of the cogenerators for the unitary dilations of the semigroups (τ t ) and (r τ t ) are unitarily equivalent (see §2 for the details). The cogenerator of the semigroup (τ t ) is unitarily equivalent to the unilateral shift operator on the Hardy space H 2 . Thus, for p = 1, our problem reduces to the description of all possible singular parts, and we show that any singular type may be realized by some semigroup (r τ t ) t≥0 . For p = 2, in [1,2] it was shown that any spectral type of the unitary component may occur; for the case of a singular spectral measure, a model of such perturbations was constructed. Here we show that similar results are valid for all p > 1.

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Embeddings of model subspaces of the Hardy space: compactness and Schatten-von Neumann ideals

Baranov¹

2009

Izv. Math.

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Weighted estimates of the fourier transformation

Abstract: The Fourier transformation is regarded as an operator from L2 (-r, 7r)

Cited by 4 publications

References 7 publications

Compact embeddings of model subspaces of the Hardy space

Compact embeddings of model subspaces of the Hardy space

On perturbations of the isometric semigroup of shifts on the semiaxis

Embeddings of model subspaces of the Hardy space: compactness and Schatten-von Neumann ideals

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