Weighted Sobolev theorem in Lebesgue spaces with variable exponent

Abstract

For the Riesz potential operator I α there are proved weighted estimates

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].…”

Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

“…Note that in the cited papers the integral inequalities were studied as r → 0. Such inequalities are also of interest when they allow to impose different conditions as r → 0 and r → ∞; such a case was dealt with in [35], [23].…”