Fredholmness of Nonlocal Singular Integral Operators with Slowly Oscillating Data

Fernández-Torres, Gustavo; Karlovich, Yuri I.

doi:10.1007/978-3-319-59384-5_9

Cited by 2 publications

(11 citation statements)

References 15 publications

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“…Corollary for

α = β

and

γ = 0

is the main result of . For

α \neq β

and

γ \in C ∖ {0}

satisfying , this result is new.…”

Section: Introductionmentioning

confidence: 79%

“…This paper is a continuation of our recent works . Let

α, β

belong to

S O S ({double-struckR}_{+})

and

a_{k}, b_{k} \in S O false(R_{+} false)

for all

k \in Z

.…”

Section: Introductionmentioning

confidence: 88%

“…Let

α, β

belong to

S O S ({double-struckR}_{+})

and

a_{k}, b_{k} \in S O false(R_{+} false)

for all

k \in Z

. In we found criteria for the Fredholmness and a formula allowing to calculate the index of the weighted singular integral operator of the form

M : = false(a_{0} I + a_{1} U_{α} false) P_{γ}^{+} + false(b_{0} I + b_{1} U_{β} false) P_{γ}^{-} .

Following , we assume in this paper that

0true A_{+} : = \sum_{k \in double-struckZ} a_{k} U_{α}^{k} \in W_{α, p}^{S O}, A_{-} : = \sum_{k \in double-struckZ} b_{k} U_{β}^{k} \in W_{β, p}^{S O}

and consider the weighted singular integral operator of the form

N : = A_{+} P_{γ}^{+} + A_{-} P_{γ}^{-} .

Fernández‐Torres and the second author obtained Fredholm criteria for the operator

N

in the particular case of

α = β

and

γ = 0

. Further, in we obtained sufficient conditions for the left and right Fredholmness of the operator

N

.…”

Section: Introductionmentioning

confidence: 99%

“…The present paper is devoted to the proof of the implication (a)

\Rightarrow

(c). We would like to emphasize that the proof of the necessity of each of the conditions (c‐i) and (c‐ii) for the case

N \in {normalΦ}_{+} false(L^{p} (R) false) \cup {normalΦ}_{-} false(L^{p} (R) false)

is essentially more difficult than the proof for the case

N \in Φ (L^{p} false(double-struckR false))

(see ).…”

Section: Introductionmentioning

confidence: 99%

“…Thus, we complete the proof of the implication (a)

\Rightarrow

(c‐i) in Theorem . The proof of the implication

N \in Φ (L^{p} false(R_{+} false)) \Rightarrow

(c‐ii) of Corollary given in was based on the method of limit operators (see, for example, ). However, it seems that it is not suitable for the study of operators in

false(Φ_{+} (L^{p} false(R_{+} false)) \cup Φ_{-} (L^{p} false(R_{+} false)) false) ∖ Φ false(L^{p} ({double-struckR}_{+}) false)

without any additional hypotheses.…”

Section: Introductionmentioning

confidence: 99%

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Criteria for n(d)-normality of weighted singular integral operators with shifts and slowly oscillating data

Karlovich

Lebre

2018

Proc. London Math. Soc.

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“…Corollary for

α = β

and

γ = 0

is the main result of . For

α \neq β

and

γ \in C ∖ {0}

satisfying , this result is new.…”

Section: Introductionmentioning

confidence: 79%

“…This paper is a continuation of our recent works . Let

α, β

belong to

S O S ({double-struckR}_{+})

and

a_{k}, b_{k} \in S O false(R_{+} false)

for all

k \in Z

.…”

Section: Introductionmentioning

confidence: 88%

“…Let

α, β

belong to

S O S ({double-struckR}_{+})

and

a_{k}, b_{k} \in S O false(R_{+} false)

for all

k \in Z

. In we found criteria for the Fredholmness and a formula allowing to calculate the index of the weighted singular integral operator of the form

M : = false(a_{0} I + a_{1} U_{α} false) P_{γ}^{+} + false(b_{0} I + b_{1} U_{β} false) P_{γ}^{-} .

Following , we assume in this paper that

0true A_{+} : = \sum_{k \in double-struckZ} a_{k} U_{α}^{k} \in W_{α, p}^{S O}, A_{-} : = \sum_{k \in double-struckZ} b_{k} U_{β}^{k} \in W_{β, p}^{S O}

and consider the weighted singular integral operator of the form

N : = A_{+} P_{γ}^{+} + A_{-} P_{γ}^{-} .

Fernández‐Torres and the second author obtained Fredholm criteria for the operator

N

in the particular case of

α = β

and

γ = 0

. Further, in we obtained sufficient conditions for the left and right Fredholmness of the operator

N

.…”

Section: Introductionmentioning

confidence: 99%

“…The present paper is devoted to the proof of the implication (a)

\Rightarrow

(c). We would like to emphasize that the proof of the necessity of each of the conditions (c‐i) and (c‐ii) for the case

N \in {normalΦ}_{+} false(L^{p} (R) false) \cup {normalΦ}_{-} false(L^{p} (R) false)

is essentially more difficult than the proof for the case

N \in Φ (L^{p} false(double-struckR false))

(see ).…”

Section: Introductionmentioning

confidence: 99%

“…Thus, we complete the proof of the implication (a)

\Rightarrow

(c‐i) in Theorem . The proof of the implication

N \in Φ (L^{p} false(R_{+} false)) \Rightarrow

(c‐ii) of Corollary given in was based on the method of limit operators (see, for example, ). However, it seems that it is not suitable for the study of operators in

false(Φ_{+} (L^{p} false(R_{+} false)) \cup Φ_{-} (L^{p} false(R_{+} false)) false) ∖ Φ false(L^{p} ({double-struckR}_{+}) false)

without any additional hypotheses.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Criteria for n(d)-normality of weighted singular integral operators with shifts and slowly oscillating data

Karlovich

Lebre

2018

Proc. London Math. Soc.

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Semi-Fredholmness of Weighted Singular Integral Operators with Shifts and Slowly Oscillating Data

Karlovich

Lebre

2018

Operator Theory, Operator Algebras, and Matrix Theory

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Let α, β be orientation-preserving homeomorphisms of [0, ∞] onto itself, which have only two fixed points at 0 and ∞, and whose restrictions to R+ = (0, ∞) are diffeomorphisms, and let Uα, U β be the corresponding isometric shift operators on the space L p (R+) given byWe prove sufficient conditions for the right and left Fredholmness on L p (R+) of singular integral operators of the form A+P + γ + A−P − γ , where P ± γ = (I ± Sγ )/2, Sγ is a weighted Cauchy singular integral operator, A+ = k∈Z a k U k α and A− = k∈Z b k U k β are operators in the Wiener algebras of functional operators with shifts. We assume that the coefficients a k , b k for k ∈ Z and the derivatives of the shifts α ′ , β ′ are bounded continuous functions on R+ which may have slowly oscillating discontinuities at 0 and ∞.Mathematics Subject Classification (2010). 45E05, 47A53, 47G10, 47G30.

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Fredholmness of Nonlocal Singular Integral Operators with Slowly Oscillating Data

Cited by 2 publications

References 15 publications

Criteria for n(d)-normality of weighted singular integral operators with shifts and slowly oscillating data

Criteria for n(d)-normality of weighted singular integral operators with shifts and slowly oscillating data

Semi-Fredholmness of Weighted Singular Integral Operators with Shifts and Slowly Oscillating Data

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