Weighted Sobolev theorem in Lebesgue spaces with variable exponent

For the Riesz potential operator IαIα there are proved weighted estimates‖Iαf‖Lq(⋅)(Ω,wqp)⩽C‖f‖Lp(⋅)(Ω,w),Ω⊆Rn,1q(x)≡1p(x)−αn within the framework of weighted Lebesgue spaces Lp(⋅)(Ω,w)Lp(⋅)(Ω,w) with variable exponent. In case Ω   is a bounded domain, the order α=α(x)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⊂RnSn⊂Rn.