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

Karlovich, Yuri I.; Lebre, Amarino B.

doi:10.1007/978-3-319-72449-2_11

Cited by 1 publication

(5 citation statements)

References 26 publications

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

N

. The aim of the present paper is to show that the left (respectively, right) Fredholmness of the operator

N

is equivalent to its

n

‐normality (respectively,

d

‐normality) and that sufficient conditions found in are also necessary.…”

Section: Introductionmentioning

confidence: 79%

“…For

α \neq β

and

γ \in C ∖ {0}

satisfying , this result is new. The implication (c)

\Rightarrow

(b) in Theorem was recently proved in [, Theorem 1.1]. The implication (b)

\Rightarrow

(a) follows from a general fact [, Chapter 4, Theorems 16.1 and 16.2].…”

Section: Introductionmentioning

confidence: 83%

“…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%

“…Proof (a) Assume that the operator

N

n

‐normal on the space

L^{p} false(R_{+} false)

. It follows from [, Lemma 7.1] and Theorem that in order to obtain the left invertibility of the operators

A_{+}

and

A_{-}

from the

n

‐normality of the operator

N

, we have to show that the pairs

(A_{+}, P_{0}^{+})

and

(A_{-}, P_{0}^{-})

satisfy condition

(L)

. To show that the pair

(A_{\pm}, P_{0}^{\pm})

satisfies condition

(L)

, assume that the operator

A_{\pm}

is not left invertible on the space

L^{p} false(R_{+} false)

.…”

Section: First Necessary Condition For the N(d)‐normality Of The Opermentioning

confidence: 99%

“…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%

See 4 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.

Self Cite

View full text Add to dashboard Cite

show abstract

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

N

. The aim of the present paper is to show that the left (respectively, right) Fredholmness of the operator

N

is equivalent to its

n

‐normality (respectively,

d

‐normality) and that sufficient conditions found in are also necessary.…”

Section: Introductionmentioning

confidence: 79%

“…For

α \neq β

and

γ \in C ∖ {0}

satisfying , this result is new. The implication (c)

\Rightarrow

(b) in Theorem was recently proved in [, Theorem 1.1]. The implication (b)

\Rightarrow

(a) follows from a general fact [, Chapter 4, Theorems 16.1 and 16.2].…”

Section: Introductionmentioning

confidence: 83%

“…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%

“…Proof (a) Assume that the operator

N

n

‐normal on the space

L^{p} false(R_{+} false)

. It follows from [, Lemma 7.1] and Theorem that in order to obtain the left invertibility of the operators

A_{+}

and

A_{-}

from the

n

‐normality of the operator

N

, we have to show that the pairs

(A_{+}, P_{0}^{+})

and

(A_{-}, P_{0}^{-})

satisfy condition

(L)

. To show that the pair

(A_{\pm}, P_{0}^{\pm})

satisfies condition

(L)

, assume that the operator

A_{\pm}

is not left invertible on the space

L^{p} false(R_{+} false)

.…”

Section: First Necessary Condition For the N(d)‐normality Of The Opermentioning

confidence: 99%

“…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%

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.

Self Cite

View full text Add to dashboard Cite

show abstract

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

Cited by 1 publication

References 26 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

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