Geodesic
On the regularity of the one-sided Hardy-Littlewood maximal functions
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 219-234 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators $\mathcal {M}^+$ and $\mathcal {M}^-$. More precisely, we prove that $\mathcal {M}^+$ and $\mathcal {M}^-$ map $W^{1,p}(\mathbb {R})\rightarrow W^{1,p}(\mathbb {R})$ with $1
In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators $\mathcal {M}^+$ and $\mathcal {M}^-$. More precisely, we prove that $\mathcal {M}^+$ and $\mathcal {M}^-$ map $W^{1,p}(\mathbb {R})\rightarrow W^{1,p}(\mathbb {R})$ with $1$, boundedly and continuously. In addition, we show that the discrete versions $M^+$ and $M^-$ map ${\rm BV}(\mathbb {Z})\rightarrow {\rm BV}(\mathbb {Z})$ boundedly and map $l^1(\mathbb {Z})\rightarrow {\rm BV}(\mathbb {Z})$ continuously. Specially, we obtain the sharp variation inequalities of $M^+$ and $M^-$, that is, $${\rm Var}(M^{+}(f))\leq {\rm Var}(f)\quad \text {and}\quad {\rm Var}(M^{-}(f))\leq {\rm Var}(f)$$ if $f\in {\rm BV}(\mathbb {Z})$, where ${\rm Var}(f)$ is the total variation of $f$ on $\mathbb {Z}$ and ${\rm BV}(\mathbb {Z})$ is the set of all functions $f\colon \mathbb {Z}\rightarrow \mathbb {R}$ satisfying ${\rm Var}(f)\infty $.
DOI : 10.21136/CMJ.2017.0475-15
Classification : 42B25, 46E35
Keywords: one-sided maximal operator; Sobolev space; bounded variation; continuity 
@article{10_21136_CMJ_2017_0475_15,
     author = {Liu, Feng and Mao, Suzhen},
     title = {On the regularity of the one-sided {Hardy-Littlewood} maximal functions},
     journal = {Czechoslovak Mathematical Journal},
     pages = {219--234},
     year = {2017},
     volume = {67},
     number = {1},
     doi = {10.21136/CMJ.2017.0475-15},
     mrnumber = {3633008},
     zbl = {06738514},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0475-15/}
}
TY  - JOUR
AU  - Liu, Feng
AU  - Mao, Suzhen
TI  - On the regularity of the one-sided Hardy-Littlewood maximal functions
JO  - Czechoslovak Mathematical Journal
PY  - 2017
SP  - 219
EP  - 234
VL  - 67
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0475-15/
DO  - 10.21136/CMJ.2017.0475-15
LA  - en
ID  - 10_21136_CMJ_2017_0475_15
ER  - 
%0 Journal Article
%A Liu, Feng
%A Mao, Suzhen
%T On the regularity of the one-sided Hardy-Littlewood maximal functions
%J Czechoslovak Mathematical Journal
%D 2017
%P 219-234
%V 67
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0475-15/
%R 10.21136/CMJ.2017.0475-15
%G en
%F 10_21136_CMJ_2017_0475_15
Liu, Feng; Mao, Suzhen. On the regularity of the one-sided Hardy-Littlewood maximal functions. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 219-234. doi: 10.21136/CMJ.2017.0475-15

[1] Aldaz, J. M., Lázaro, J. Pérez: Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities. Trans. Am. Math. Soc. 359 (2007), 2443-2461. | DOI | MR | JFM

[2] Bober, J., Carneiro, E., Hughes, K., Pierce, L. B.: On a discrete version of Tanaka's theorem for maximal functions. Proc. Am. Math. Soc. 140 (2012), 1669-1680. | DOI | MR | JFM

[3] Calderón, A. P.: Ergodic theory and translation invariant operators. Proc. Natl. Acad. Sci. USA 59 (1968), 349-353. | DOI | MR | JFM

[4] Carneiro, E., Hughes, K.: On the endpoint regularity of discrete maximal operators. Math. Res. Lett. 19 (2012), 1245-1262. | DOI | MR | JFM

[5] Carneiro, E., Moreira, D.: On the regularity of maximal operators. Proc. Am. Math. Soc. 136 (2008), 4395-4404. | DOI | MR | JFM

[6] Dunford, N., Schwartz, J.: Convergence almost everywhere of operator averages. Proc. Natl. Acad. Sci. USA 41 (1955), 229-231. | DOI | MR | JFM

[7] Hajłasz, P., Onninen, J.: On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 29 (2004), 167-176. | MR | JFM

[8] Hardy, G. H., Littlewood, J. E.: A maximal theorem with function-theoretic applications. Acta Math. 54 (1930), 81-116 \99999JFM99999 56.0264.02. | DOI | MR

[9] Kinnunen, J.: The Hardy-Littlewood maximal function of a Sobolev function. Isr. J. Math. 100 (1997), 117-124. | DOI | MR | JFM

[10] Kinnunen, J., Lindqvist, P.: The derivative of the maximal function. J. Reine Angew. Math. 503 (1998), 161-167. | DOI | MR | JFM

[11] Kinnunen, J., Saksman, E.: Regularity of the fractional maximal function. Bull. Lond. Math. Soc. 35 (2003), 529-535. | DOI | MR | JFM

[12] Kurka, O.: On the variation of the Hardy-Littlewood maximal function. Ann. Acad. Sci. Fenn. Math. 40 (2015), 109-133. | DOI | MR | JFM

[13] Liu, F., Chen, T., Wu, H.: A note on the endpoint regularity of the Hardy-Littlewood maximal functions. Bull. Aust. Math. Soc. 94 (2016), 121-130. | DOI | MR | JFM

[14] Liu, F., Wu, H.: On the regularity of the multisublinear maximal functions. Can. Math. Bull. 58 (2015), 808-817. | DOI | MR | JFM

[15] Luiro, H.: Continuity of the maximal operator in Sobolev spaces. Proc. Am. Math. Soc. 135 (2007), 243-251. | DOI | MR | JFM

[16] Luiro, H.: On the regularity of the Hardy-Littlewood maximal operator on subdomains of $\mathbb{R}^n$. Proc. Edinb. Math. Soc. II. 53 (2010), 211-237. | DOI | MR | JFM

[17] Sawyer, E.: Weighted inequalities for the one-sided Hardy-Littlewood maximal function. Trans. Am. Math. Soc. 297 (1986), 53-61. | DOI | MR | JFM

[18] Stein, E. M., Shakarchi, R.: Real Analysis. Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis 3, Princeton University Press, Princeton (2005). | MR | JFM

[19] Tanaka, H.: A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function. Bull. Aust. Math. Soc. 65 (2002), 253-258. | DOI | MR | JFM

[20] Temur, F.: On regularity of the discrete Hardy-Littlewood maximal function. Available at ArXiv:1303.3993v1 [math.CA].

Cité par Sources :