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 .
Highlights
Last years harmonic analysis in variable exponent spaces attracts enormous interest of researchers due to both mathematical curiosity caused by the difficulties of investigation in variable exponent spaces and by various applications
A generalization of the Stein–Weiss inequality [26], that is, Sobolev embedding with power weight on unbounded domains for variable exponents was considered in [24], where this generalization was obtained with a certain additional restriction on the parameters involved
The main results of the paper are given in Theorems A–C, see Section 3
Summary
Last years harmonic analysis in variable exponent spaces attracts enormous interest of researchers due to both mathematical curiosity caused by the difficulties of investigation in variable exponent spaces and by various applications. A generalization of the Stein–Weiss inequality [26], that is, Sobolev embedding with power weight on unbounded domains for variable exponents was considered in [24], where this generalization was obtained with a certain additional restriction on the parameters involved. This restriction was withdrawn in [25]. The main novelty of the results obtained in this paper is admission of a certain class of general weights w(|x − x0|), x0 ∈ Ω, of radial-type (in the case of unbounded domains we admit radial type weights “fixed” to infinity). W , see (2.10); Φγβ , see Definition 2.2; Ψγβ , see Definition 2.8; by c or C we denote various positive absolute constants
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