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- Weighted Sobolev theorem in Lebesgue spaces with variable exponentPublication . Samko, N. G.; Samko, Stefan; Vakulov, B. G.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.
- Operators of harmonic analysis in weighted spaces with non-standard growthPublication . Kokilashvili, V. M.; Samko, StefanLast years there was increasing an interest to the so-called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues. (C) 2008 Elsevier Inc. All rights reserved.