Sharp Inequalities for the Variation Of the discrete Maximal Function

Madrid, José

doi:10.1017/s0004972716000903

Cited by 43 publications

(30 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

Section: 2mentioning

confidence: 99%

Regularity and continuity of the multilinear strong maximal operators

Liu

Xue

Yabuta

2020

Journal de Mathématiques Pures et Appliquées

View full text Add to dashboard Cite

show abstract

“…Kinnunen's original result was later extended to a local setting in [14], to a fractional setting in [15] and to a multilinear setting in [9]. Other works on the regularity of maximal operators in Sobolev spaces and other function spaces include [1,16,18,23,25,28].The understanding of the action of M on the endpoint spacewas raised by Haj lasz and Onninen in [12] and remains unsolved in its full generality. A complete solution was reached only in dimension d = 1 in [2, 17, 29] and partial progress on the general case d > 1 was obtained by Haj lasz and Malý [11] and, more recently, by Saari [26] and Luiro [22].Let us elaborate on the achievements in dimension d = 1 since these will be quite important for our purposes here.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Endpoint Sobolev and BV continuity for maximal operators

Carneiro

Madrid

Pierce

2017

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we investigate some questions related to the continuity of maximal operators in W 1,1 and BV spaces, complementing some well-known boundedness results. Letting M be the onedimensional uncentered Hardy-Littlewood maximal operator, we prove that the map f → Mf ′ is continu-In the discrete setting, we prove that M : BV (Z) → BV (Z) is also continuous.For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexamples that the corresponding continuity statements do not hold, both in the continuous and discrete settings, and for the centered and uncentered versions.

show abstract

“…Subsequently, Temur [30] proved (13) for (with constant = 294, 912, 004) following Kurka's breakthrough [12]. Inequality (14) is not optimal, and it was asked in [29] whether the sharp constant for inequality (14) is in fact = 2; this question was resolved in the affirmative by Madrid in [31]. Later on, the above results were extended to a fractional case in [17,32,33], to a one-sided case in [7], and to a high dimensional case in [34].…”

Section: Theorem 1 M + 2 Is Bounded and Continuous Frommentioning

confidence: 99%

“…In Section 2 we shall prove Theorem 2. We remark that the proof of the boundedness part in Theorem 2 is motivated by the method in [31], but our proof is simpler and more direct than that of [31]. The proof of the continuity part in Theorem 2 relies on the previous boundedness result and an useful application of the Brezis-Lieb lemma in [38].…”

Section: Theorem 2 the Operatormentioning

confidence: 99%

A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

Liu

2018

Journal of Function Spaces

View full text Add to dashboard Cite

We investigate the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator. We point out that the above operator is bounded and continuous on the Sobolev spaces , (R 2 ) for 0 ≤ ≤ 1 and 1 < < ∞. More importantly, we establish the sharp boundedness and continuity for the discrete two-dimensional one-sided Hardy-Littlewood maximal operator from ℓ 1 (Z 2 ) to BV(Z 2 ). Here BV(Z 2 ) denotes the set of all functions of bounded variation on Z 2 .

show abstract

Sharp Inequalities for the Variation Of the discrete Maximal Function

Abstract: Abstract. In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, both in the centered and uncentered versions. In particular, we solve a question originally posed by Bober, Carneiro, Hughes and Pierce in [3].

Cited by 43 publications

References 14 publications

Regularity and continuity of the multilinear strong maximal operators

Regularity and continuity of the multilinear strong maximal operators

Endpoint Sobolev and BV continuity for maximal operators

A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

Contact Info

Product

Resources

About