Wavelets and the Angle between Past and Future

Treil, Sergei; Volberg, Alexander

doi:10.1006/jfan.1996.2986

Cited by 130 publications

(134 citation statements)

References 8 publications

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“…In general though, it is important to realise that these reducing operators for p = 2 are not averages. Despite this, it is nonetheless very useful to think of them as appropriate averages of W , which is further justified by the following simple but important result proved in [25] when p = 2 and proved in [13] for general 1 < p < ∞.…”

Section: Two Weight Characterization Of Paraproductsmentioning

confidence: 99%

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Boundedness of Commutators and H $${}^1$$ 1 -BMO Duality in the Two Matrix Weighted Setting

Isralowitz

2017

Integr. Equ. Oper. Theory

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Abstract. In this paper we characterize the two matrix weighted boundedness of commutators with any of the Riesz transforms (when both are matrix A p weights) in terms of a natural two matrix weighted BMO space. Furthermore, we identify this BMO space when p = 2 as the dual of a natural two matrix weighted H 1 space, and use our commutator result to provide a converse to Bloom's matrix A 2 theorem, which as a very special case proves Buckley's summation condition for matrix A 2 weights. Finally, we use our results to prove a matrix weighted John-Nirenberg inequality, and we also briefly discuss the challenging question of extending our results to the matrix weighted vector BMO setting.

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Section: Two Weight Characterization Of Paraproductsmentioning

confidence: 99%

“…As we will see later, this translates into a matrix Fefferman-Kenig-Pipher and Buckley condition on matrix A 2 weights W . Note that while the former is well known in the matrix setting (see [6,25]), the latter is to the author's knowledge new.…”

Section: Introductionmentioning

confidence: 99%

Boundedness of Commutators and H $${}^1$$ 1 -BMO Duality in the Two Matrix Weighted Setting

Isralowitz

2017

Integr. Equ. Oper. Theory

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“…See [15] for a similar notion of matrix weights. We will now find a characterization of such functions F in terms of the boundedness of certain averaging operators on the function space L 2 (F * F ).…”

Section: Definition 33mentioning

confidence: 99%

Products of Toeplitz Operators on a Vector Valued Bergman Space

Kerr

2010

Integr. Equ. Oper. Theory

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Abstract. We give a necessary and a sufficient condition for the boundedness of the Toeplitz product T F T G * on the vector valued Bergman space L 2 a (C n ), where F and G are matrix symbols with scalar valued Bergman space entries. The results generalize those in the scalar valued Bergman space case [4]. We also characterize boundedness and invertibility of Toeplitz products T F T G * in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space [8].

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“…An n × n matrix weight almost everywhere positive on the real line will be denoted by w 2 (x), x ∈ R. Further, by M 2 n we denote the class of matrix weights w 2 satisfying Muckenhoupt's condition (A 2 ) [2]: sup…”

Section: To the Centenary Of M G Kreinmentioning

confidence: 99%

“…It is worth mentioning ([3, p. 310]; [6, p. 91]) that for each antidissipative Volterra operator B with trivial kernel and n-dimensional imaginary part there exists a matrix function θ ∈ J n (the characteristic matrix function of B * ) such that B is unitarily equivalent to an operator of the form (2), (3). Further, for each weight V 2 ∈ M 2 n there exists a matrix function V + (respectively, V − ) that is outer [3] in C + (respectively, C − ) and satisfies [2] …”

Section: To the Centenary Of M G Kreinmentioning

confidence: 99%

Perturbations of strongly continuous operator semigroups, and matrix Muckenhoupt weights

Gubreev¹,

Latushkin

2008

Funct Anal Its Appl

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Abstract. Let A and A 0 be linear continuously invertible operators on a Hilbert space H such that A −1 − A −1 0 has finite rank. Assuming that σ(A 0 ) = ∅ and that the operator semigroup V + (t) = exp{iA 0 t}, t 0, is of class C 0 , we state criteria under which the semigroups U ± (t) = exp{±iAt}, t 0, are of class C 0 as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.Key words: nonself-adjoint operator, perturbation of a semigroup, functional model, Muckenhoupt condition. To the centenary of M. G. KreinIn this paper, we apply functional models of nonself-adjoint operators and the technique of matrix Muckenhoupt weights to the theory of one-parameter operator semigroups in Hilbert spaces.1. w-perturbations of linear operators. Let A 0 and A be linear unbounded densely defined and continuously invertible operators on a Hilbert space H such thatwhere f k , g k ∈ H, 1 k n, and the parentheses stand for the inner product in H. We shall assume that the operator A 0 belongs to the class Σ (exp) , that is, and {g k } n 1 such that the operator A related to A 0 ∈ Σ (exp) by (1) generates a C 0 -semigroup U + (t) := exp{iAt} or a C 0 -semigroup U − (t) := exp{−iAt}, t 0. In this paper, we indicate a class of finite-dimensional perturbations A satisfying (1) in which this problem admits a solution.An n × n matrix weight almost everywhere positive on the real line will be denoted by w 2 (x), x ∈ R. Further, by M 2 n we denote the class of matrix weights w 2 satisfying Muckenhoupt's conditionwhere M (w ±2 ) := |∆| −1 ∆ w ±2 (x) dx, ∆ is an arbitrary interval in R, and |∆| is its length. Using the operator A 0 ∈ Σ (exp) and the vectors {g k } n 1 , we construct the n-dimensional rowand introduce the following definition. Definition 1. Suppose that the operators A and A 0 are related by (1).We say that A is a w-perturbation of rank n of the operator A 0 ∈ Σ (exp) if there exists a weight w 2 (x) of the class M 2 n such that the following conditions hold: 1. For each h ∈ H, the function A 0 (x, h)w −1 (x) belongs to L 2 (R).

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Wavelets and the Angle between Past and Future

Cited by 130 publications

References 8 publications

Boundedness of Commutators and H $${}^1$$ 1 -BMO Duality in the Two Matrix Weighted Setting

Boundedness of Commutators and H $${}^1$$ 1 -BMO Duality in the Two Matrix Weighted Setting

Products of Toeplitz Operators on a Vector Valued Bergman Space

Perturbations of strongly continuous operator semigroups, and matrix Muckenhoupt weights

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