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- Boundary value problems for analytic functions in the class of Cauchy-type integrals with density inPublication . Kokilashvili, V.; Paatashvili, V.; G. Samko, StefanWe study the Riemann boundary value problem , for analytic functions in the class of analytic functions represented by the Cauchy-type integrals with density in the spaces with variable exponent. We consider both the case when the coefficient is piecewise continuous and it may be of a more general nature, admitting its oscillation. The explicit formulas for solutions in the variable exponent setting are given. The related singular integral equations in the same setting are also investigated. As an application there is derived some extension of the Szegö-Helson theorem to the case of variable exponents.
- Embeddings of variable Hajlasz-Sobolev spaces into holder spaces of variable orderPublication . Almeida, Alexandre; Samko, StefanPointwise estimates in variable exponent Sobolev spaces on quasi-metric measure spaces are investigated. Based on such estimates, Sobolev embeddings into Holder spaces with variable order are obtained. This extends some known results to the variable exponent setting. (C) 2008 Elsevier Inc. All rights reserved.
- 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.
- Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, IIPublication . Samko, S.; Shargorodsky, E.; Vakulov, B.In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229-246], Sobolev-type p((.)) -> q((.))-theorems were proved for the Riesz potential operator I-alpha in the weighted Lebesgue generalized spaces L-p(.)(R-n, p) with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x(0) and to infinity, under an additional condition relating the weight exponents at x(0) and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces L-p(.) (S-n, p) on the unit sphere S-n in Rn+1 are also improved in the same way. (c) 2006 Elsevier Inc. All rights reserved.
- Weighted Sobolev theorem with variable exponent for spatial and spherical potential operatorsPublication . Samko, Stefan; Vakulov, B.We prove Sobolev-type p((.)) -> q ((.))-theorems for the Riesz potential operator I-alpha in the weighted Lebesgue generalized spaces L-p(.)(R-n, p) with the variable exponent p (x) and a two-parametrical power weight fixed to an arbitrary finite point and to infinity, as well as similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces L-p(.)(S-n, p) on the unit sphere S-n in Rn+1. (c) 2005 Elsevier Inc. All rights reserved.
- Hardy type inequalityin variable lebesgue spacesPublication . Rafeiro, Humberto; Samko, StefanWe prove that in variable exponent spaces where L-p(.)(Omega), where p(.) satisfies the log-condition and Omega is a bounded domain in R-n with the property that R-n\(Omega) over bar has the cone property, the validity of the Hardy type inequality parallel to 1/delta(x)(alpha)integral(Omega)phi(y)/vertical bar x-y vertical bar(n-alpha)dy parallel to(p(.)) <= C parallel to phi parallel to(p(.)), 0 < alpha < min (1, n/p(+)), where delta(x) is approximately equal to dist(x, partial derivative Omega), is equivalent to a certain property of the domain Omega expressed in terms of alpha and chi(Omega).