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- 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).