| Title:
|
On the regularity of the one-sided Hardy-Littlewood maximal functions (English) |
| Author:
|
Liu, Feng |
| Author:
|
Mao, Suzhen |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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67 |
| Issue:
|
1 |
| Year:
|
2017 |
| Pages:
|
219-234 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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<p<\infty $, 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 $. (English) |
| Keyword:
|
one-sided maximal operator |
| Keyword:
|
Sobolev space |
| Keyword:
|
bounded variation |
| Keyword:
|
continuity |
| MSC:
|
42B25 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 06738514 |
| idMR:
|
MR3633008 |
| DOI:
|
10.21136/CMJ.2017.0475-15 |
| . |
| Date available:
|
2017-03-13T12:10:09Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146050 |
| . |
| Reference:
|
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