Weighted Sobolev theorem in Lebesgue spaces with variable exponent - Samko, N. G. (2007)
For the Riesz potential operator I-alpha there are proved weighted estimates [GRAPHICS] within the framework of weighted Lebesgue spaces L (P(center dot)) (Omega, omega) with variable exponent. In case Omega is a bounded domain, the order alpha = alpha (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 subset of R-n. (c) 2007 Elsevier Inc. All rights reserved.