scholarly journals Two-Weight Norm Estimates for Multilinear Fractional Integrals in Classical Lebesgue Spaces

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Mieczysław Mastyło ◽  
Alexander Meskhi
Keyword(s):  
Riesz Potential ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Fractional Integrals ◽  
Potential Operator ◽  
Norm Estimates ◽  
Riesz Potential Operator ◽  
Weight Problems ◽  
Multilinear Fractional Integrals ◽  
Necessary And Sufficient

AbstractWe derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.

scholarly journals Weighted Hardy and Potential Operators in Morrey Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Natasha Samko
Keyword(s):  
Riesz Potential ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Potential Operator ◽  
Potential Operators ◽  
Riesz Potential Operator ◽  
Power Weights ◽  
The One ◽  
Necessary And Sufficient ◽  
Hardy Type Operators

We study the weightedp→q-boundedness of Hardy-type operators in Morrey spacesℒp,λ(ℝn) (orℒp,λ(ℝ+1) in the one-dimensional case) for a class of almost monotonic weights. The obtained results are applied to a similar weightedp→q-boundedness of the Riesz potential operator. The conditions on weights, both for the Hardy and potential operators, are necessary and sufficient in the case of power weights. In the case of more general weights, we provide separately necessary and sufficient conditions in terms of Matuszewska-Orlicz indices of weights.

scholarly journals Necessary and sufficient conditions for the boundedness of the anisotropic Riesz potential in anisotropic modified Morrey spaces

2013 ◽  
Vol 21 (2) ◽  
pp. 111-130
Author(s):  
Malik S. Dzhabrailov ◽  
Sevinc Z. Khaligova
Keyword(s):  
Maximal Operator ◽  
Morrey Space ◽  
Riesz Potential ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Potential Operator ◽  
Fractional Maximal Operator ◽  
Riesz Potential Operator ◽  
Necessary And Sufficient ◽  
Limiting Case

Abstract We prove that the anisotropic fractional maximal operator Mα,σ and the anisotropic Riesz potential operator Iα,σ, 0 < α < ∣σ∣ are bounded from the anisotropic modified Morrey space L̃1,b,σ(Rn) to the weak anisotropic modified Morrey space WL̃q,b,σ(Rn) if and only if, α/|σ|≤1-1/q≤α/(|σ|(1-b)) and from L̃p,b,σ(Rn) to L̃q,b,σ(Rn) if and only if, α/|σ| ≤ 1/p-1/q≤α ((1-b) |σ|). In the limiting case we prove that the operator Mα,σ is bounded from L̃p,b,σ(Rn) to L∞ (Rn) and the modified anisotropic Riesz potential operator Ĩα,σ is bounded from L̃p,b,σ(Rn) to BMOσ(Rn).

scholarly journals Boundedness of homogeneous fractional integrals on Lp for N/α ≤ P ≤ ∞

2002 ◽  
Vol 167 ◽  
pp. 17-33 ◽  
Author(s):  
Yong Ding ◽  
Shanzhen Lu
Keyword(s):  
Integral Operator ◽  
Hardy Spaces ◽  
Fractional Integral ◽  
Riesz Potential ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Potential Operator ◽  
Riesz Potential Operator ◽  
Homogeneous Fractional Integral Operator

AbstractIn this paper we study the map properties of the homogeneous fractional integral operator TΩ, α on Lp(ℝn) for n/α ≤ p ≤ ∞.We prove that if Ω satisfies some smoothness conditions on Sn−1 then TΩ, α is bounded from Ln/α(ℝn) to BMO(ℝn), and from Lp(ℝn) (n/α < p ≤ ∞) to a class of the Campanato spaces l, λ (ℝn), respectively. As the corollary of the results above, we show that when Ω satisfies some smoothness conditions on Sn−1 the homogeneous fractional integral operator TΩ, α is also bounded from Hp(ℝn) (n/(n + α) ≤ p ≤ 1) to Lq(ℝn) for 1/q = 1/p-α/n. The results are the extensions of Stein-Weiss (for p = 1) and Taibleson-Weiss’s (for n/(n + α) ≤ p < 1) results on the boundedness of the Riesz potential operator Iα on the Hardy spaces Hp(ℝn).

Essential norms of weighted differentiation composition operators between Zygmund type spaces and Bloch type spaces

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2877-2889 ◽  
Author(s):  
Amir Sanatpour ◽  
Mostafa Hassanlou
Keyword(s):  
Composition Operators ◽  
Sufficient Conditions ◽  
Essential Norm ◽  
Necessary And Sufficient Conditions ◽  
Norm Estimates ◽  
Type Spaces ◽  
Necessary And Sufficient

We study boundedness of weighted differentiation composition operators Dk?,u between Zygmund type spaces Z? and Bloch type spaces ?. We also give essential norm estimates of such operators in different cases of k ? N and 0 < ?,? < ?. Applying our essential norm estimates, we get necessary and sufficient conditions for the compactness of these operators.

Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces

2020 ◽  
Vol 23 (5) ◽  
pp. 1452-1471
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Riesz Potential ◽  
Fractional Integrals ◽  
Mixed Norm ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Grand Lebesgue Spaces ◽  
Measure Spaces ◽  
Necessary And Sufficient ◽  
Bounded Sets ◽  
Trace Inequalities

Abstract D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.

scholarly journals Riesz potential on the Heisenberg group and modified Morrey spaces

2012 ◽  
Vol 20 (1) ◽  
pp. 189-212
Author(s):  
Vagif S. Guliyev ◽  
Yagub Y. Mammadov
Keyword(s):  
Heisenberg Group ◽  
Fractional Integral ◽  
Morrey Space ◽  
Riesz Potential ◽  
Morrey Spaces ◽  
Potential Operator ◽  
Group Setting ◽  
Riesz Potential Operator ◽  
Homogeneous Dimension ◽  
Limiting Case

Abstract In this paper we study the fractional maximal operator Mα, 0 ≤ α < Q and the Riesz potential operator ℑα, 0 < α < Q on the Heisenberg group in the modified Morrey spaces L͂p,λ(ℍn), where Q = 2n + 2 is the homogeneous dimension on ℍn. We prove that the operators Mα and ℑα are bounded from the modified Morrey space L͂1,λ(ℍn) to the weak modified Morrey space WL͂q,λ(ℍn) if and only if, α/Q ≤ 1 - 1/q ≤ α/(Q - λ) and from L͂p,λ(ℍn) to L͂q,λ(ℍn) if and only if, α/Q ≤ 1/p - 1/q ≤ α/(Q - λ).In the limiting case we prove that the operator Mα is bounded from L͂p,λ(ℍn) to L∞(ℍn) and the modified fractional integral operator Ĩα is bounded from L͂p,λ(ℍn) to BMO(ℍn).As applications of the properties of the fundamental solution of sub-Laplacian Ը on ℍn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of ℑα from the Besov-modified Morrey spaces BL͂spθ,λ(ℍn) to BL͂spθ,λ(ℍn).

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