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

By S. Samko, B. Vakulov. 2005

91 citations recorded by the source corpus.

Abstract 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 α in the weighted Lebesgue generalized spaces L p ( ⋅ ) ( R n , ρ ) 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 , ρ ) on the unit sphere S n in R n + 1 are also improved in the same way.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8365ca0a05179f9e856fff444f3759f4966dc856

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Citations in the corpus

Sobolev's inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space Lp(⋅)(logL)q(⋅) (2008, unknown): “8 below) is given in Section 2, which is an extension of Almeida-Samko [3], Diening [6], Futamura-Mizuta [10], Futamura-Mizuta-Shimomura [11, 12], Harjulehto-HästöPere [18], Kokilashvili-Samko [21], Mizuta-Shimomura [27] and Samko-Vakulov [31].”
On the asymptotic behavior of variable exponent power–law functionals and applications (2010, methodology): “A comprehensive introduction to the subject may be found in [8]. See also [4], and [ 5 ].”
VARIABLE BESOV AND TRIEBEL-LIZORKIN SPACES (2008, methodology): “have attracted more attention and have been applied to partial dierential equations and the calculus of variations, see [1]‐[16], [18], [ 19 ], [26], [27].”
Operators of harmonic analysis in weighted spaces with non-standard growth (2008, result): “For potential operators in the case Ω = R we refer to [74] and [68], where for power weights of the class Vp(·)(R,Π) and for radial oscillating weights of the class V osc p(·)(Rn,Π), respectively, there were obtained estimates (4.”
On a nonhomogeneous quasilinear eigenvalue problem in sobolev spaces with variable exponent (2006, unknown): “MR1810360 (2002a:76004) [24] S. Samko and B. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math.”
Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions (2010, background): “For proofs, details and further results on variable exponent Lebesgue and Sobolev spaces we refer to the book of Musielak [9] and the papers of Kovác̆ik and Rákosnı́k [10], Edmunds et al. [11], Edmunds and Rakosnik [12,13], and Samko and Vakulov [14], while for applications of such kind of spaces ”
Fractional integration and differentiation of variable order: an overview (2012, background): “For further results on the integrals Iα(·) Ω f (x) in variable exponent Lebesgue spaces, including the case of unbounded domains and weighted versions we refer for instance to [26, 42, 43, 54, 55], see also references in [7, 24, 28, 51].”
An atomic decomposition of variable Besov and Triebel-Lizorkin spaces (2009, background): “Moreover, these spaces have been applied to partial differential equations and the calculus of variation, see [1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 16, 17, 18, 19, 21, 22, 29] and references therein.”
Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent (2011, unknown): “Our study may be considered at the intersection of two directions in PDEs: the anisotropic Sobolev spaces theory developed by [1–5], and the variable exponent Sobolev spaces theory developed by [6–13].”
Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent (2008, unknown): “The spaces Lp(·)(Ω) and Wm,p(·)(Ω) were thoroughly studied in the monograph by Musielak [25] and the papers by Edmunds et al. [10–12], Kovacik and Rákosník [21], Mihăilescu and Rădulescu [22–24], and Samko and Vakulov [34].”
On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting (2016, unknown): “For more details we refer to the book by Musielak [24] and the papers by Edmunds et al. [8–10], Kováčik and Rákosník [16], Mihăilescu and Rădulescu [17], and Samko and Vakulov [26].”
The maximal operator on weighted variable Lebesgue spaces (2011, result): “) The first results of this kind we proved by Kokilashvili, Samko and their collaborators [25, 40, 41, 42, 43]; these results were more recently extended to classes of weights who oscillate between powers: see [1, 21, 22, 23, 24, 26, 27, 37, 38].”
Sobolev inequalities with variable exponent attaining the values $1$ and $n$ (2008, unknown): “, in [1], [2], [4], [5], [8], [10], [11], [12], [13], [15], [16], [20], [24], [28] in the variable exponent setting.”