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Essential norm estimates for multilinear singular and fractional integrals

100%
Meskhi A.
Commentationes Mathematicae
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2019
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tom 59
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nr 1-2
EN
We derive lower two-weight estimates for the essential norm (measure of noncompactness) for multilinear Hilbert and Riesz transforms, and Riesz potential operators in Banach function lattices. As a corollary we have appropriate results in weighted Lebesgue spaces. From these statements we conclude that there is no \((m+1)\)-tuple of weights \((v,w_1, \dots, w_m)\) for which these operators are compact from \(L^{p_1}_{w_1} \times \dots \times L^{p_m}_{w_m}\) to \(L^q_v\).
2
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Trace inequalities for fractional integrals in grand Lebesgue spaces

64%
Kokilashvili V., Meskhi A.
Studia Mathematica
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2012
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tom 210
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nr 2
159-176
EN
rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from $L^{p),θ}(X,μ)$ to $L^{q),qθ/p}(X,ν)$ (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace inequalities for one-sided potentials, strong fractional maximal functions and potentials with product kernels, fractional maximal functions and potentials defined on the half-space are also proved in terms of Adams-type criteria. Finally, we remark that a Fefferman-Stein-type inequality for Hardy-Littlewood maximal functions and Calderón-Zygmund singular integrals holds in grand Lebesgue spaces.
3
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One-weight weak type estimates for fractional and singular integrals in grand Lebesgue spaces

64%
Kokilashvili V., Meskhi A.
Banach Center Publications
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2014
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tom 102
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nr 1
131-142
EN
We investigate weak type estimates for maximal functions, fractional and singular integrals in grand Lebesgue spaces. In particular, we show that for the one-weight weak type inequality it is necessary and sufficient that a weight function belongs to the appropriate Muckenhoupt class. The same problem is discussed for strong maximal functions, potentials and singular integrals with product kernels.
4
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Two-weighted criteria for integral transforms with multiple kernels

64%
Kokilashvili V., Meskhi A.
Banach Center Publications
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2006
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tom 72
|
nr 1
119-140
EN
Necessary and sufficient conditions governing two-weight $L^p$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.
5
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Boundedness of commutators of singular and potential operators in generalized grand Morrey spaces and some applications

51%
Kokilashvili V., Meskhi A., Rafeiro H.
Studia Mathematica
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2013
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tom 217
|
nr 2
159-178
EN
In the setting of spaces of homogeneous type, it is shown that the commutator of Calderón-Zygmund type operators as well as the commutator of a potential operator with a BMO function are bounded in a generalized grand Morrey space. Interior estimates for solutions of elliptic equations are also given in the framework of generalized grand Morrey spaces.
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