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

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.

Operators of harmonic analysis in weighted spaces with non-standard growth

“…In the sequel we give results of such a kind for other operators. For potential operators in the case Ω = R n we refer to [74] and [68], where for power weights of the class V p(·) (R n , Π) and for radial oscillating weights of the class V osc p(·) (R n , Π), respectively, there were obtained estimates (4.9) under assumptions more general than should be imposed by the usage of Theorem 2.12.…”

“…Martell and C. Perez [10], D. Cruz-Uribe, A. Fiorenza and C.J. Neugebauer [11], L. Diening [13][14][15], L. Diening and M. Ružička [17], V. Kokilashvili, N. Samko and S. Samko [38], V. Kokilashvili and S. Samko [41][42][43]45], A. Nekvinda [58], S. Samko [70][71][72], S. Samko, E. Shargorodsky and B. Vakulov [74] and references therein.…”

One‐sided operators inLp(x)spaces

“…Weighted inequalities with power-type weights for the Hardy transforms, Hardy-Littlewood maximal functions, singular and fractional integrals were established in [18], [19], [13], [29], [32], [31], [20], [12], [10] and for general-type weights in [8], [17], [12] (see also [28], [16]). …”

The maximal operator on weighted variable Lebesgue spaces

“…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]. Other results in this direction have been proved by a number of authors; see, for example, [4,3,15,20,32,33].…”