Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II

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.info:eu-repo/semantics/publishedVersio

Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators

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.info:eu-repo/semantics/publishedVersio

Weighted Sobolev theorem in Lebesgue spaces with variable exponent: corrigendum

We fill in a gap discovered in the proof of Theorem A, on weighted Sobolev type boundedness for potential operators in variable exponent Lebesgue spaces, in the paper of the authors "Weighted Sobolev theorem in Lebesgue spaces with variable exponent", J. Math. Anal. and Applic., 2007, vol. 335, No 1, 560-583. The proof remains the same in the case where the Matuszewska-Orlich indices $m(w)$ and $M(w)$ of the weight $w$ are both positive or negative, but in the case where they have different signs, the proof needs some additional arguments and requires a slightly different formulation of the result

On Some Classical Operators of Variable Order in Variable Exponent Spaces

We give a survey of a selection of recent results on weighted and non-weighted estimations of classical operators of Harmonic Analysis in variable exponent Lebesgue, Morrey and Hölder spaces, based on the talk presented at International Conference Analysis, PDEs and Applications on the occasion of the 70th birthday of Vladimir Maz’ya, Rome, June 30–July 3, 2008. We touch both the Euclidean case and the general setting within the frameworks of quasimetric measure spaces. Some of the presented results are new