On a discrete version of Tanaka’s theorem for maximal functions

Bober, Jonathan; Carneiro, Emanuel; Hughes, Kevin; Pierce, Lillian B.

doi:10.1090/s0002-9939-2011-11008-6

Cited by 57 publications

(58 citation statements)

References 17 publications

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“…When d = 1, the regularity properties of the discrete maximal type operators were studied by Bober et al [6], Temur [52] and Madrid [49], Carneiro and Madrid [10] and Liu [37]. The following sharp inequalities have been established.…”

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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“…Var Mf ≤ C Var f for a universal constant C. Question 1.1 and the validity of Theorem 1.2 were already studied in the discrete setting in [3]. In the present paper, we do not care how small the constant C may be.…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

“…In the present paper, we do not care how small the constant C may be. It is a plausible hypothesis that the inequality holds for C = 1, in the same way as in the non-centered case (see also [3,Question B]). …”

Section: Introduction and Main Resultsmentioning

confidence: 92%

On the variation of the Hardy-Littlewood maximal function

Kurka¹

2015

Ann. Acad. Sci. Fenn. Math.

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“…As M β f j − M β f ∞ → 0 as j → ∞, there exists j 1 (K) such that C K,j > C K /2 for all j > j 1 (K). 4 For each x ∈ K, let B x,j := B(z x,j , r x,j ) ∈ B β x,j . Then…”

Section: 4mentioning

confidence: 99%

Endpoint Sobolev Continuity of the Fractional Maximal Function in Higher Dimensions

Beltran

Madrid

2020

International Mathematics Research Notices

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We establish continuity mapping properties of the non-centered fractional maximal operator M β in the endpoint input space W 1,1 (R d ) for d ≥ 2 in the cases for which its boundedness is known. More precisely, we provef is radial and for 1 ≤ β < d for general f . The results for 1 ≤ β < d extend to the centered counterpart M c β . Moreover, if d = 1, we show that the conjectured boundedness of that map for M c β implies its continuity. Af 1,q ≤ C f 1,p . At the endpoint p = 1, one cannot expect boundedness of M β from W 1,1 to W 1, d d−β to hold, as M β fails to be bounded at the level of Lebesgue spaces. However, one Date: June 4, 2019. 2010 Mathematics Subject Classification. 42B25, 46E35.

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On a discrete version of Tanaka’s theorem for maximal functions

Cited by 57 publications

References 17 publications

Regularity and continuity of the multilinear strong maximal operators

Regularity and continuity of the multilinear strong maximal operators

On the variation of the Hardy-Littlewood maximal function

Endpoint Sobolev Continuity of the Fractional Maximal Function in Higher Dimensions

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