The fractional maximal operator and fractional integrals on variable $L^p$ spaces

Abstract

We prove that if the exponent function p(·) satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maxi-We also prove a weak-type inequality corresponding to the weak (1, n/(n − α)) inequality for M α . We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [3]. As a consequence of these results for M α , we show that the fractional integral operator I α satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable L p spaces.

One‐sided operators inLp(x)spaces

“…The third author is also grateful to Professor Alberto Fiorenza for giving him the details of the proof of Lemma 2.3 from [5] and sending him the manuscript [3].…”

“…We have the following one-sided version of Theorem 4.1 of [3] (see also Lemmas 2.3 and 2.5 of [5] for the two-sided case). …”

Generalized maximal functions and related operators on weighted Musielak-Orlicz spaces

“…This type of inequality is an extension of Hedberg's inequality (see [25]). Another extensions can be found in [5], [21] and [22].…”

“…The boundedness of many operators in harmonic analysis that appear in connection with the study of regularity properties of the solutions of partial differential equations were widely considered in the variable context by different authors, see for instance [9], [11], [14], [15], [16], [30], [31], [32], [38], [39] and [40] for the HardyLittlewood maximal function M, [5], [21], [22] and [28] for the fractional maximal function M α , [18] and [33] for Calderón-Zygmund operators and their commutators, and [1], [10], [24] and [28] for potential type operators (see [13] for other classical operators).…”

Pointwise Inequalities in Variable Sobolev Spaces and Applications

“…We only refer to the papers [27], [38], where the basics of such spaces were developed, to the papers [12], [33], where the denseness of nice functions in variable Sobolev spaces was considered, and to the papers [7], [8], [10], [11], [25], [26], [29], [34], [35] and the recent preprints [5], [6] and references therein, where several results on maximal, potential and singular operators in variable Lebesgue spaces were obtained. We also mention the survey [37].…”