Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights

Karlovich, Alexei Yu.

doi:10.1007/978-3-0346-0174-0_9

Cited by 8 publications

(3 citation statements)

References 36 publications

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“…A Fredholm criterion for Banach algebras of singular integral operators with piecewise continuous coefficients on variable Lebesgue spaces L p(·) (Γ, w) over Carleson Jordan curves with weights having finite sets of singularities were obtained in [17][18][19] (see also the references therein). The approach of these works is based on further developments of the methods of the monograph [4] based on localization techniques, Wiener-Hopf factorization and heavy use of results and methods from the theory of submultiplicative functions.…”

Section: Introductionmentioning

confidence: 99%

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Spitkovsky

2011

Journal of Mathematical Analysis and Applications

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Let a be a semi-almost periodic matrix function with the almost periodic representatives a l and a r at −∞ and +∞, respectively. Suppose p : R → (1, ∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space L p(·) (R). We prove that if the operator aP + Q with P = (I + S)/2 and Q = (I − S)/2 is Fredholm on the variable Lebesgue space L p(·) N (R), then the operators a l P + Q and a r P + Q are invertible on standard Lebesgue spaces L q l N (R) and L qr N (R) with some exponents q l and q r lying in the segments between the lower and the upper limits of p at −∞ and +∞, respectively.

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Section: Introductionmentioning

confidence: 99%

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Spitkovsky

2011

Journal of Mathematical Analysis and Applications

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“…Similar to the case of the constant p, the Fredholm theory of the mentioned operators in spaces related to L p(·) has also a big interest. With respect to one-dimensional singular integral operators in variable exponent Lebesgue spaces we refer, for instance, to [17][18][19][20][21][22][23][24][25][26]41].…”

Section: Introductionmentioning

confidence: 99%

“…In [18][19][20], some results of the book [1] were transferred to algebras of SIO acting in the Lebesgue spaces with variable exponents.…”

Section: Introductionmentioning

confidence: 99%

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

Rabinovich

2010

Integr. Equ. Oper. Theory

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The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L p(·) (Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L p(·) (R+, dμ) where dμ is an invariant measure on multiplicative group R+ = {r ∈ R : r > 0}.(2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L p(·) (Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on R+ and local invertibility of singular integral operators on R.(3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L p(·) (Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities. Mathematics Subject Classification (2000). Primary 47G30.

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Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves

Karlovich

2009

Mathematische Nachrichten

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We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights ϕt,γ(τ ) = |(τ − t) γ |, where γ is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point t and γ is not real, then ϕt,γ is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.

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Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights

Cited by 8 publications

References 36 publications

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves

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