Two-Weight Estimates for Strong Fractional Maximal Functions and Potentials With Multiple Kernels

Kokilashvili, Vakhtang; Meskhi, Alexander

doi:10.4134/jkms.2009.46.3.523

Cited by 11 publications

(11 citation statements)

References 24 publications

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“…where, as before, 1 p = m i=1 1 p i . Similar results were derived in [12] (see also [13], Ch.4) for sublinear maximal operators. It is natural to study the appropriate potential operator with product kernels:…”

Section: Resultssupporting

confidence: 85%

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

Kokilashvili

Mastyło

Meskhi

2015

FCAA

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We 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.

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Section: Resultssupporting

confidence: 85%

“…Proof of this lemma follows in the same way as in the case of (sub)linear case (see [12], [13], P. ); therefore we omit the details.…”

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confidence: 99%

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

Kokilashvili

Mastyło

Meskhi

2015

FCAA

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“…for any I, J ∈ DR, where I ⊂ J and |I| = 1 2 n |J|. Remark 2.1.The definition, dyadic reverse doubling condition or reverse doubling condition associated with cubes, can be found in [8], [16], [25]. In [8], it was introduced to investigate the boundedness of the dyadic fractional maximal function.…”

Section: Definitions and Main Resultsmentioning

confidence: 99%

On multilinear fractional strong maximal operator associated with rectangles and multiple weights

Cao¹,

Xue²,

Yabuta³

2017

Rev. Mat. Iberoam.

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In this paper, the multilinear fractional strong maximal operator M R,α associated with rectangles and corresponding multiple weights A ( p,q),R are introduced. Under the dyadic reverse doubling condition, a necessary and sufficient condition for two-weight inequalities is given. As consequences, we first obtain a necessary and sufficient condition for one-weight inequalities. Then, we give a new proof for the weighted estimates of multilinear fractional maximal operator M α associated with cubes and multilinear fractional integral operator I α , which is quite different and simple from the proof known before.

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“…Let w(x, y) = w 1 (x)w 2 (y). Then the proposition follows in the same way as the appropriate statements regarding the Hardy operators defined on R 2 + in [14], [12] (see also Theorem 1.1.6 of [13]). If v is a product weight, i.e.…”

Section: The Next Statements Deals With the Double Hardy-type Operatomentioning

confidence: 74%

A characterization of the two-weight inequality for Riesz potentials on cones of radially decreasing functions

Meskhi

Murtaza²,

Sarwar

2014

J Inequal Appl

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We establish necessary and sufficient conditions on a weight pair (v, w) governing the boundedness of the Riesz potential operatoris the Lebesgue space defined for non-negative radially decreasing functions on G. The same problem is also studied for the potential operator with product kernels I α 1 ,α 2 defined on a product of two homogeneous groups G 1 × G 2 . In the latter case weights, in general, are not of product type. The derived results are new even for Euclidean spaces. To get the main results we use Sawyer-type duality theorems (which are also discussed in this paper) and two-weight Hardy-type inequalities on G and G 1 × G 2 , respectively. MSC: 42B20; 42B25

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Two-Weight Estimates for Strong Fractional Maximal Functions and Potentials With Multiple Kernels

Abstract: Abstract. In the paper two-weight inequalities of various type for strong fractional maximal functions and potentials with multiple kernels defined on R 2 are established.

Cited by 11 publications

References 24 publications

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

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

On multilinear fractional strong maximal operator associated with rectangles and multiple weights

A characterization of the two-weight inequality for Riesz potentials on cones of radially decreasing functions

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