Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Karlovich, Yuri I.; Hernández, Jesús Hernández

doi:10.1007/s00020-009-1685-y

Cited by 13 publications

(3 citation statements)

References 21 publications

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“…(2) Separation of discontinuities: we separate discontinuities of a k ∈ P C by reducing to the case of only one discontinuity point of the functions a k on R. Using the index formula from [17] for Wiener-Hopf type operators and approximation and stability arguments, we obtain ind A t 0 .…”

Section: ])mentioning

confidence: 99%

Index formula for convolution type operators with data functions in alg(SO,PC)

Bastos

Carvalho

Karlovich

2016

Journal of Mathematical Analysis and Applications

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We establish an index formula for the Fredholm convolution type operators A = m k=1 a k W 0 (b k ) acting on the space L 2 (R), where a k , b k belong to the C * -algebra alg(SO, P C) of piecewise continuous functions on R that admit finite sets of discontinuities and slowly oscillate at ±∞, first in the case where all a k or all b k are continuous on R and slowly oscillating at ±∞, and then assuming that a k , b k ∈ alg(SO, P C) satisfy an extra Fredholm type condition. The study is based on a number of reductions to operators of the same form with smaller classes of data functions a k , b k , which include applying a technique of separation of discontinuities and eventually lead to the so-called truncated operators A r = m k=1 a k,r W 0 (b k,r ) for sufficiently large r > 0, where the functions a k,r , b k,r ∈ P C are obtained from a k , b k ∈ alg(SO, P C) by extending their values at ±r to all ±t ≥ r, respectively. We prove that ind A = lim r→∞ ind A r although A = s-lim r→∞ A r only.

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Section: ])mentioning

confidence: 99%

Index formula for convolution type operators with data functions in alg(SO,PC)

Bastos

Carvalho

Karlovich

2016

Journal of Mathematical Analysis and Applications

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“…λ and SO 3 λ of three times continuously differentiable slowly oscillating functions For a point λ ∈ Ṙ, let C 3 (R \ {λ}) be the set of all three times continuously differentiable functions a : R \ {λ} → C. Slightly extending definitions in [22,Section 3] and [21, Section 2.3], consider the commutative Banach algebras…”

Section: Banach Algebra Somentioning

confidence: 99%

Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers

Fernandes

Karlovich

2021

Banach J. Math. Anal.

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Let M X(R) be the Banach algebra of all Fourier multipliers on a Banach function space X(R) such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X (R). For two sets Ψ, Ω ⊂ M X(R) , let Ψ Ω be the set of those c ∈ Ψ for which there exist d ∈ Ω such that the multiplier norm of χ R\[−N,N ] (c − d) tends to zero as N → ∞.In this case we say that the Fourier multiplier c is equivalent at infinity to the Fourier multiplier d. We show that if Ω is a unital Banach subalgebra of M X(R) consisting of nice Fourier multipliers (for instance, continuous or slowly oscillating in certain sense) and Ψ is an arbitrary unital Banach subalgebra of M X(R) , then Ψ Ω is a also a unital Banach subalgebra of M X(R) . KeywordsFourier convolution operator • Fourier multiplier • slowly oscillating function • equivalence at infinity • Banach algebra • C * -algebra. Mathematics Subject Classification (2010) 47G10 • 42A45 • 46E30 This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/MAT/ 00297/2020 (Centro de Matemática e Aplicações) and by the SEP-CONACYT project A1-S-8793 (México).

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“…I, §8.3], [15,). In particular, the density of the linear span of Φ 2 in L p (R + ) for 1 < p < ∞ plays a crucial role in the proof of the fact that the Banach algebra alg(W (C( Ṙ)) generated by Wiener-Hopf operators with continuous symbols contains all compact operators on L p (R + ) (see, [4,Section 9.9] and also [12,).…”

Section: Introductionmentioning

confidence: 99%

On the Density of Laguerre Functions in Some Banach Function Spaces

FERNANDES¹,

KARLOVYCH²,

VALENTE³

2022

JIASF

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Let λ > 0 and Φλ := {ϕ1,λ, ϕ2,λ, . . . } be the system of dilated Laguerre functions. We show that if L1 (R+) ∩ L∞(R+) is embedded into a separable Banach function space X(R+), then the linear span of Φλ is dense in X(R+). This implies that the linear span of Φλ is dense in every separable rearrangement-invariant space X(R+) and in every separable variable Lebesgue space Lp(·) (R+)

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Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Cited by 13 publications

References 21 publications

Index formula for convolution type operators with data functions in alg(SO,PC)

Index formula for convolution type operators with data functions in alg(SO,PC)

Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers

On the Density of Laguerre Functions in Some Banach Function Spaces

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