Weighted Sobolev theorem in Lebesgue spaces with variable exponent

For the Riesz potential operator I α there are proved weighted estimates I αf Lq(·)(Ω,w qp ) C f Lp(·)(Ω,w), Ω⊆ Rn, 1 q(x) ≡ 1 p(x) − α n within the framework of weighted Lebesgue spaces Lp(·)(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 Sn ⊂ Rn. © 2007 Elsevier Inc. All rights reserved.