Two-weight inequalities for convolution operators in Lebesgue space

Bandaliev, Rovshan A.

doi:10.1007/s11006-006-0101-z

Cited by 7 publications

(7 citation statements)

References 7 publications

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“…Obviously, if we take l = 1, B 1 ( x ) = K ( x ), and ϕ 1 ( y ) ≡ 1, then condition (6) is exactly the classical Lipschitz condition (4). We remark that the function K ( x ) = sin x / x satisfies conditions ( K 1 )–( K 4 ), but does not satisfy the Hörmander's condition (1) (see [11] page 5).…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Under the assumption of Theorem 3, several authors have studied two-weight inequalities for the convolution operator T , for example [11–13]. Recently, the authors [14] introduce a variant of the classical L r -Hörmander's condition in the scope of (2) and establish the weighted norm inequalities for singular integral operator with its kernel satisfying such a variant of the classical L r -Hörmander's condition.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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Commutators of Singular Integral Operators Satisfying a Variant of a Lipschitz Condition

Zhang

2014

The Scientific World Journal

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Let T be a singular integral operator with its kernel satisfying |K(x − y) − ∑k=1 ℓ‍B k(x)ϕ k(y)| ≤ C | y|γ/|x − y|n+γ, |x | > 2 | y | > 0, where B k and ϕ k (k = 1,…, ℓ) are appropriate functions and γ and C are positive constants. For trueb→=false(b1,…,bmfalse) with b j ∈ BMO(ℝn), the multilinear commutator Tb→ generated by T and b→ is formally defined by Ttrueb→ffalse(xfalse)=∫ℝn[]|false∏j=1mnormal‍false(bjfalse(xfalse)-bjfalse(yfalse)false)Kfalse(x,yfalse)ffalse(yfalse)dy. In this paper, the weighted L p-boundedness and the weighted weak type Llog⁡L estimate for the multilinear commutator Tb→ are established.

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Section: Introduction and Resultsmentioning

confidence: 99%

Section: Introduction and Resultsmentioning

confidence: 99%

Commutators of Singular Integral Operators Satisfying a Variant of a Lipschitz Condition

Zhang

2014

The Scientific World Journal

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“…However, although this looks like a small difference, such problems often behave quite differently. We mention [3,44], which work more generally on R n .…”

Section: Results (See Theorem 42 and Theorem 43)mentioning

confidence: 99%

Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

Harper

2008

Complex Anal. Oper. Theory

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We ask when convolution operators with scalar-or operator-valued kernel functions map between weighted L 2 spaces of Hilbert space-valued functions. For a certain class of decreasing weights, including negative powers (t + a) −m for example, we solve the one-weight problem completely by using Laplace transforms and Bergman-type spaces of vector-valued analytic functions. For a much more general class of decreasing weights, we solve the one-weight problem for all positive real kernels (also for L p (w) with p > 1), by results on Steklov operators which generalise the weighted Hardy inequality. When the kernel function is a strongly continuous semigroup of bounded linear Hilbert space operators, which arises from input-output maps of certain linear systems, then the most obvious sufficient condition for boundedness, obtained by taking norm signs inside the integrals, is also necessary in many cases, but not in general. Mathematics Subject Classification (2000). Primary 44A35; Secondary 47B32, 47D06.

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“…Note that for the case in which u = u 1 = 1 , Theorem 3 was proved in [20] by using different methods. Further, in the case 1 < p < ∞ Theorems 6 and 8 was proved in [3].…”

Section: Remarkmentioning

confidence: 96%

“…There exist other conditions stronger than condition (2) (see [9,21]). The function K(x) = (sin x)/x satisfies conditions (K1) − (K4) and does not satisfy conditions 1), 2), and Hörmander's condition (2) (see [3]).…”

Section: Remarkmentioning

confidence: 99%

Two‐weight inequalities for singular integral operators satisfying a variant of Hörmander′s condition

Guliyev

2009

Journal of Function Spaces

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Two-weight inequalities for convolution operators in Lebesgue space

Cited by 7 publications

References 7 publications

Commutators of Singular Integral Operators Satisfying a Variant of a Lipschitz Condition

Commutators of Singular Integral Operators Satisfying a Variant of a Lipschitz Condition

Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

Two‐weight inequalities for singular integral operators satisfying a variant of Hörmander′s condition

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