×

A note on the endpoint regularity of the Hardy-Littlewood maximal functions. (English) Zbl 1347.42034

Let \(M\) be the uncentered Hardy-Littlewood maximal function on \({\mathbb R}^1\): \[ Mf(x) = \sup_{s,t>0} \frac{1}{s+t}\int_{x-s}^{x+t} | f(y) | \, dy. \]
J. Kinnunen [Isr. J. Math. 100, 117–124 (1997; Zbl 0882.43003)] proved the regularity of \(M\) on Sobolev spaces \(W^{1,p}\) where \(p>1\). H. Tanaka [Bull. Aust. Math. Soc. 65, No. 2, 253–258 (2002; Zbl 0999.42013)] proved that \(\| (Mf)' \|_{L^1} \leq 2 \| f' \|_{L^1}\), and J. M. Aldaz and J. Pérez Lázaro [Trans. Am. Math. Soc. 359, No. 5, 2443–2461 (2007; Zbl 1143.42021)] proved that \[ \| (Mf)' \|_{L^1} \leq \| f' \|_{L^1}.\tag{1} \]
The authors give a simple proof for (1). For the proof they show the following lemma, which is interesting in itself.
Let \(f: {\mathbb R}^1 \to {\mathbb R}^1 \) be continuous and integrable. If \(x_0\) is a local maximum of \(Mf\), then \(Mf(x_0) = | f(x_0) |\).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hajlasz, Ann. Acad. Sci. Fenn. Math. 29 pp 167– (2004)
[2] DOI: 10.1007/978-3-642-61798-0 · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0
[3] DOI: 10.1090/S0002-9939-08-09515-4 · Zbl 1157.42003 · doi:10.1090/S0002-9939-08-09515-4
[4] DOI: 10.1090/S0002-9939-2011-11008-6 · Zbl 1245.42017 · doi:10.1090/S0002-9939-2011-11008-6
[5] DOI: 10.1090/S0002-9947-06-04347-9 · Zbl 1143.42021 · doi:10.1090/S0002-9947-06-04347-9
[6] DOI: 10.1017/S0004972700020293 · Zbl 0999.42013 · doi:10.1017/S0004972700020293
[7] DOI: 10.1007/BF02773636 · Zbl 0882.43003 · doi:10.1007/BF02773636
[8] DOI: 10.1090/S0002-9939-06-08455-3 · Zbl 1136.42018 · doi:10.1090/S0002-9939-06-08455-3
[9] DOI: 10.4153/CMB-2014-070-7 · Zbl 1330.42014 · doi:10.4153/CMB-2014-070-7
[10] DOI: 10.5186/aasfm.2015.4003 · Zbl 1332.42012 · doi:10.5186/aasfm.2015.4003
[11] DOI: 10.1112/S0024609303002017 · Zbl 1021.42009 · doi:10.1112/S0024609303002017
[12] Kinnunen, J. reine angew. Math. 503 pp 161– (1998)
[13] DOI: 10.1017/S0013091507000867 · Zbl 1183.42025 · doi:10.1017/S0013091507000867
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.