Boundedness and unboundedness results for some maximal operators on functions of bounded variation

Aldaz, J. M.; Lázaro, J. Pérez

doi:10.1016/j.jmaa.2007.03.097

Cited by 16 publications

(14 citation statements)

References 15 publications

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“…It follows from [2,39] that if f ∈ W 1,1 (R), then Mf is absolutely continuous on R and it holds that ( Mf ) ′ L 1 (R) ≤ f ′ L 1 (R) . For d ≥ 1, Aldaz and Pérez Lázaro [3] considered a class of local strong maximal operator and proved that it maps BV(U ) into L 1 (U ), where U is an open set of R d and BV(U ) is a subclass of L 1 (U ) functions. See [19,Definition 1.3] and [4,Definition 3.4] for instance.…”

Section: 2mentioning

confidence: 99%

Regularity and continuity of the multilinear strong maximal operators

Liu

Xue

Yabuta

2020

Journal de Mathématiques Pures et Appliquées

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Let m ≥ 1, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operatorwhere x ∈ R d and R denotes the family of all rectangles in R d with sides parallel to the axes. When m = 1, denote MR by MR. Then, MR coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that MR is bounded and continuous from the Sobolev spacesAs a consequence, we further showed that MR is bounded and continuous from the fractional Sobolev spaces W s,p 1 (R d ) × · · · × W s,pm (R d ) to W s,p (R d ) for 0 < s ≤ 1 and 1 < p < ∞. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the p-quasicontinuity of MR. In addition, we proved that MR( f ) is approximately differentiable a.e. if f = (f1, . . . , fm) with each fj ∈ L 1 (R d ) being approximately differentiable a.e. The discrete type of the strong maximal operators has also been considered. We showed that this discrete type of the maximal operators enjoys somewhat unexpected regularity properties.Key words and phrases. Multilinear strong maximal operators, discrete multilinear strong maximal operators, Sobolev spaces, regularity, Tribel-Lizorkin spaces and Besov spaces, mixed Lebesgue spaces and Sobolev spaces, Sobolev capacity.

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Section: 2mentioning

confidence: 99%

Regularity and continuity of the multilinear strong maximal operators

Liu

Xue

Yabuta

2020

Journal de Mathématiques Pures et Appliquées

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“…To the best of our knowledge there are no known higher dimensional results in the case p = 1, and even in the one dimensional case it is still not known whether the Hardy-Littlewood maximal function (i.e. the centered one) of f ∈ W 1,1 (R) belongs locally to W 1,1 (R); see, however, [2], [3]. The results proved in the paper are clearly motivated by this challenging problem.…”

Section: B(xr)mentioning

confidence: 99%

“…Let us also mention that the result of Kinnunen [10] has been applied and generalized by many authors ( [2], [3], [6], [7], [8], [11], [12], [14], [15], [16], [17], [18], [20], [21], [25]). …”

Section: B(xr)mentioning

confidence: 99%

On approximate differentiability of the maximal function

Hajłasz

Malý

2010

Proc. Amer. Math. Soc.

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Abstract. We prove that if f ∈ L 1 (R n ) is approximately differentiable a.e., then the Hardy-Littlewood maximal function Mf is also approximately differentiable a.e. Moreover, if we only assume that f ∈ L 1 (R n ), then any open set of R n contains a subset of positive measure such that Mf is approximately differentiable on that set. On the other hand we present an example of f ∈ L 1 (R) such that Mf is not approximately differentiable a.e.

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“…For other related results, see also e.g. [2,4,5,6,12]. For results considering other concepts than the weak differentiability, see [7,14].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%