Abstract
We prove Sobolev-type p ( ⋅ ) → q ( ⋅ ) -theorems 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-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 , ρ ) on the unit sphere S n in R n + 1 .
Highlights
An obvious interest to the operator theory in the generalized Lebesgue spaces with variable exponent p(x) could be observed in a variety of papers, the main objects being the maximal operator, Hardy operators, singular operators and potential type operators, we refer, in particular to surveys [13,24].S
In this paper we prove a weighted Sobolev-type theorem for the Riesz potential operator
We prove a similar theorem for the spherical analogue (Kαf )(x) =
Summary
An obvious interest to the operator theory in the generalized Lebesgue spaces with variable exponent p(x) could be observed in a variety of papers, the main objects being the maximal operator, Hardy operators, singular operators and potential type operators, we refer, in particular to surveys [13,24]. Samko [17] for weighted boundedness on bounded domains. Sobolev p(·) → q(·)-theorem for potential operators on bounded domains was considered in S.G. Samko [25] and L. Diening [6], in [6] there being treated the case of unbounded domains under the assumption that the maximal operator is bounded. A weighted statement on p(·) → p(·)-boundedness for the Riesz potential operators on bounded domains was obtained in V. Kokilashvili and S.G. Samko [17], limiting inequalities for bounded domains having been recently proved in S. In this paper we prove a weighted Sobolev-type theorem for the Riesz potential operator. Weighted p(·) → q(·)-estimates for the operator I α in the case of bounded domains were proved in [28].
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