Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators

Böttcher, Albrecht; Karlovich, Yuri I.

doi:10.1007/978-3-0348-8922-3

Cited by 212 publications

(305 citation statements)

References 0 publications

Supporting

Mentioning

303

Contrasting

Unclassified

Order By: Relevance

“…Karlovich [2], see also [3]. For a variable Lebesgue space with a power weight, that is, in the case when Im γ = 0, this result (and the converse) is due to V. Kokilashvili, V. Paatashvili, and S. Samko [22].…”

Section: 3mentioning

confidence: 88%

“…In this setting, the circular arc depends on the value of the exponent p(t) at a discontinuity point t ∈ Γ. Later on, the author gradually extended results known for singular integral operators with piecewise continuous coefficients on weighted standard Lebesgue spaces (see [3,13]) to the case of weighted variable Lebesgue spaces (see [17,19] for power weights and Lyapunov curves; [18] for power weights and so-called logarithmic Carleson curves; [20] for radial oscillating weights and logarithmic Carleson curves).…”

Section: Introductionmentioning

confidence: 99%

“…The Fredholm theory for the Banach algebra generated by matrix singular integral operators on L p (Γ, w) under the most general conditions on the curve Γ and the weight w is constructed by A. Böttcher and Yu. Karlovich and is presented in the monograph [3] (although, we advise to start the study of this theory from the nice survey [4]). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights

Karlovich

2009

Operator Algebras, Operator Theory and Applications

View full text Add to dashboard Cite

Abstract. In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces L p (Γ) over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces L p(·) (Γ) where p : Γ → (1, ∞) satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.

show abstract

Section: 3mentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights

Karlovich

2009

Operator Algebras, Operator Theory and Applications

View full text Add to dashboard Cite

show abstract

“…We refer to the well-known books by Gakhov [74] and Muskhelishvili [75] for the 'classical' period and to the books by Gohberg and Krupnik [76,77] for the later 'operator theory' period; many aspects of the modern theory of singular integral operators may be found in Böttcher and Karlovich [78,79].…”

Section: Fredholmness Of Singular Integral Equations In the Spaces L mentioning

confidence: 99%

On a progress in the theory of lebesgue spaces with variable exponent: maximal and singular operators

Samko¹

2005

Integral Transforms and Special Functions

246

108

View full text Add to dashboard Cite

This paper represents a broadened version of the plenary lecture presented by the author at the conference Analytic Methods of Analysis and Differential Equations (AMADE-2003), September 4-9, 2003, Minsk, Belarus. We give a survey of investigations on 'the variable exponent business', concentrating mainly on recent advances in the operator theory and harmonic analysis in the generalized Lebesgue and Sobolev spaces L p (·) and W m,p(·) .

show abstract

“…For the convenience of the reader, we quote here one of the main results of this theory, which will be basic for our further considerations. For its proof and also for the background on Carleson curves and Muckenhoupt weights used below, we refer the reader to [6,Sections 4 and 5] and [6, the restrictions of functions in C ∞ 0 (R 2 ) to . For > 0, u ∈ and g ∈ C ∞ 0 ( ), let S , g(u) denotes a truncated singular integral of g at the point u, i.e.,…”

mentioning

confidence: 99%

Resolvent characterisation of generators of cosine functions and C 0-groups

Król

2013

J. Evol. Equ.

View full text Add to dashboard Cite

Abstract. The paper provides new characterisations of generators of cosine functions and C 0 -groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C 0 -groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini's result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh's characterisation of the boundedness of the H ∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L 2 space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C 0 -semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.

show abstract

Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators

Cited by 212 publications

References 0 publications

Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights

Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights

On a progress in the theory of lebesgue spaces with variable exponent: maximal and singular operators

Resolvent characterisation of generators of cosine functions and C 0-groups

Contact Info

Product

Resources

About