| Title:
|
Commutators of the fractional maximal function on variable exponent Lebesgue spaces (English) |
| Author:
|
Zhang, Pu |
| Author:
|
Wu, Jianglong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
183-197 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $M_{\beta }$ be the fractional maximal function. The commutator generated by $M_{\beta }$ and a suitable function $b$ is defined by $[M_{\beta },b]f = M_{\beta }(bf)-bM_{\beta }(f)$. Denote by $\mathscr {P}(\mathbb R^{n})$ the set of all measurable functions $p(\cdot )\colon \mathbb R^{n}\to [1,\infty )$ such that $$ 1< p_{-}:=\mathop {\rm ess inf}_{x\in \mathbb R^n}p(x) \quad \text {and}\quad p_{+}:=\mathop {\rm ess sup}_{x\in \mathbb R^n}p(x)<\infty , $$ and by $\mathscr {B}(\mathbb R^{n})$ the set of all $p(\cdot ) \in \mathscr {P}(\mathbb R^{n})$ such that the Hardy-Littlewood maximal function $M$ is bounded on $L^{p(\cdot )}(\mathbb R^{n})$. In this paper, the authors give some characterizations of $b$ for which $[M_{\beta },b]$ is bounded from $L^{p(\cdot )}(\mathbb R ^{n})$ into $L^{q(\cdot )}(\mathbb R^{n})$, when $p(\cdot )\in \mathscr {P}(\mathbb R^{n})$, $0<{\beta }<n/p_{+}$ and $1/q(\cdot )=1/p(\cdot )-\beta /n$ with $q(\cdot )(n-\beta )/n \in \mathscr {B}(\mathbb R^{n})$. (English) |
| Keyword:
|
commutator |
| Keyword:
|
BMO |
| Keyword:
|
fractional maximal function |
| Keyword:
|
variable exponent Lebesgue space |
| MSC:
|
42B25 |
| MSC:
|
42B30 |
| MSC:
|
46E30 |
| MSC:
|
47B47 |
| idZBL:
|
Zbl 06391486 |
| idMR:
|
MR3247454 |
| DOI:
|
10.1007/s10587-014-0093-x |
| . |
| Date available:
|
2014-09-29T09:51:39Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143959 |
| . |
| Reference:
|
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