The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces (original) (raw)
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Acta Applicandae Mathematicae, 2013
In this paper, the authors establish several general theorems for the boundedness of sublinear operators (B sublinear operators) satisfies the condition (1.2), generated by B singular integrals on a weighted Lebesgue spaces L p,ω,γ (R n k,+), where B = k i=1 (∂ 2 ∂x 2 k + γ i x i ∂ ∂x i). The condition (1.2) are satisfied by many important operators in analysis, including B maximal operator and B singular integral operators. Sufficient conditions on weighted functions ω and ω 1 are given so that B sublinear operators satisfies the condition (1.2) are bounded from L p,ω,γ (R n k,+) to L p,ω 1 ,γ (R n k,+). Keywords Weighted Lebesgue space • B sublinear operator • B maximal operator • B singular integral operator • Two-weighted inequality Mathematics Subject Classification (2000) 42B20 • 42B25 • 42B35
Weighted boundedness of multilinear singular integral operators
Journal of Inequalities and Applications, 2014
In this paper, we establish the weighted sharp maximal function inequalities for the multilinear singular integral operators. As an application, we obtain the boundedness of the multilinear operators on weighted Lebesgue and Morrey spaces. MSC: 42B20; 42B25
TWO-WEIGHTED INEQUALITY FOR p-ADMISSIBLE Bk,n–SINGULAR OPERATORS IN WEIGHTED LEBESGUE SPACES
2014
In this paper, we study the boundedness of p-admissible singular operators, associated with the Laplace-Bessel differential operator Bk,n = n ∑ i=1 ∂ ∂xi + k ∑ j=1 γj xj ∂ ∂xj (p-admissible Bk,n–singular operators) on a weighted Lebesgue spaces Lp,ω,γ(Rk,+) including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Bk,n–maximal operator, Bk,n–singular integral operators and so on. Sufficient conditions on weighted functions ω and ω1 are given so that p-admissible Bk,n–singular operators are bounded from Lp,ω,γ(Rk,+) to Lp,ω1,γ(Rk,+) for 1 < p <∞ and weak p-admissible Bk,n–singular operators are bounded from Lp,ω,γ(Rk,+) to Lp,ω1,γ(Rk,+) for 1 ≤ p <∞.
The Cauchy Singular Integral Operator on Weighted Variable Lebesgue Spaces
Operator Theory: Advances and Applications, 2013
Let p : R → (1, ∞) be a globally log-Hölder continuous variable exponent and w : R → [0, ∞] be a weight. We prove that the Cauchy singular integral operator S is bounded on the weighted variable Lebesgue space L p(•) (R, w) = {f : f w ∈ L p(•) (R)} if and only if the weight w satisfies sup −∞<a<b<∞ 1 b − a wχ (a,b) p(•) w −1 χ (a,b) p ′ (•) < ∞ (1/p(x) + 1/p ′ (x) = 1).
On integral operators in weighted grand Lebesgue spaces of Banach-valued functions
Authorea
The paper deals with boundedness problems of integral operators in weighted grand Bochner-Lebesgue spaces. We will treat both cases: when a weight function appears as a multiplier in the definition of the norm, or when it defines the absolute continuous measure of integration. Along with the diagonal case, we deal with the off-diagonal case. To get the appropriate result for the Hardy-Littlewood maximal operator, we rely on the reasonable bound of the sharp constant in the Buckley-type theorem, which is also derived in the paper.
Two-Weight Norm Inequalities for Certain Singular Integrals
Taiwanese Journal of Mathematics, 2012
In this paper we prove the boundedness of certain convolution operator in a weighted Lebesgue space with kernel satisfying the generalized Hörmander's condition. The sufficient conditions for the pair of weights ensuring the validity of two-weight inequalities of a strong type and of a weak type for singular integral with kernel satisfying the generalized Hörmander's condition are found.
Bulletin des Sciences Mathématiques, 2013
Let T be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of R n. We assume that the associated kernel of T satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. We then show the boundedness for T and the boundedness of the commutator of T with BMO functions on products of weighted Lebesgue spaces of R n. As an application, we obtain the weighted norm inequalities of multilinear Fourier multipliers and of their commutators with BMO functions on the products of weighted Lebesgue spaces when the number of derivatives of the symbols is the same as the best known result for the multilinear Fourier multipliers to be bounded on the products of unweighted Lebesgue spaces. Contents