Weighted Sobolev theorem in Lebesgue spaces with variable exponent

By Natasha Samko, S. Samko, B. Vakulov. 2007

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Abstract For the Riesz potential operator I α there are proved weighted estimates ‖ I α f ‖ L q ( ⋅ ) ( Ω , w q p ) ⩽ C ‖ f ‖ L p ( ⋅ ) ( Ω , w ) , Ω ⊆ R n , 1 q ( x ) ≡ 1 p ( x ) − α n within the framework of weighted Lebesgue spaces L p ( ⋅ ) ( Ω , w ) with variable exponent. In case Ω is a bounded domain, the order α = α ( x ) is allowed to be variable as well. The weight functions are radial type functions “fixed” to a finite point and/or to infinity and have a typical feature of Muckenhoupt–Wheeden weights: they may oscillate between two power functions. Conditions on weights are given in terms of their Boyd-type indices. An analogue of such a weighted estimate is also obtained for spherical potential operators on the unit sphere S n ⊂ R n .

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/ef756ebf8cf6ff7f5732e3ffabfddb1f338c5d10

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In Computational Theory and Mathematics

Citations in the corpus

The maximal operator on weighted variable Lebesgue spaces (2011, result): “) The first results of this kind we proved by Kokilashvili, Samko and their collaborators [25, 40, 41, 42, 43]; these results were more recently extended to classes of weights who oscillate between powers: see [1, 21, 22, 23, 24, 26, 27, 37, 38].”
An atomic decomposition of variable Besov and Triebel-Lizorkin spaces (2009, background): “Moreover, these spaces have been applied to partial differential equations and the calculus of variation, see [1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 16, 17, 18, 19, 21, 22, 29] and references therein.”
Fractional integration and differentiation of variable order: an overview (2012, background): “For further results on the integrals Iα(·) Ω f (x) in variable exponent Lebesgue spaces, including the case of unbounded domains and weighted versions we refer for instance to [26, 42, 43, 54, 55], see also references in [7, 24, 28, 51].”
Weighted Hardy operators in the local generalized vanishing Morrey spaces (2012, unknown): “Most of them may be found dispersed in various papers, for instance, in [8,15,25,27,28], but for reader’s convenience we gathered them in Appendix.”
Operators of harmonic analysis in weighted spaces with non-standard growth (2008, background): “For potential operators in the case Ω = R n we refer to [75] and [69], where for power weights of the class V p ( · ) ( R n , Π) and for radial oscillating weights of the class V oscp ( · ) ( R n , Π), respectively, there were obtained estimates (4.9) under assumptions more general than should be…”
Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces (2010, background): “Such inequalities are also of interest when they allow to impose different conditions as r → 0 and r → ∞; such a case was dealt with in [35], [23].”
Commutators of Hardy operators in vanishing Morrey spaces (2012, background): “Following [29], Subsection 2.”
VARIABLE BESOV AND TRIEBEL-LIZORKIN SPACES (2008, methodology): “have attracted more attention and have been applied to partial dierential equations and the calculus of variations, see [1]‐[16], [ 18 ], [19], [26], [27].”