Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Zhang, Pu, and Wu, Jianglong. "Commutators of the fractional maximal function on variable exponent Lebesgue spaces." Czechoslovak Mathematical Journal 64.1 (2014): 183-197. <http://eudml.org/doc/262009>.

@article{Zhang2014,
abstract = {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 $\mathcal \{P\}(\mathbb \{R\}^\{n\})$ the set of all measurable functions $p(\cdot )\colon \mathbb \{R\}^\{n\}\rightarrow [1,\infty )$ 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 $\mathcal \{B\}(\mathbb \{R\}^\{n\})$ the set of all $p(\cdot ) \in \mathcal \{P\}(\mathbb \{R\}^\{n\})$ 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 $p(\cdot )\in \mathcal \{P\}(\mathbb \{R\}^\{n\})$, $0<\{\beta \}<n/p_\{+\}$ and $1/q(\cdot )=1/p(\cdot )-\beta /n$ with $q(\cdot )(n-\beta )/n \in \mathcal \{B\}(\mathbb \{R\}^\{n\})$.},
author = {Zhang, Pu, Wu, Jianglong},
journal = {Czechoslovak Mathematical Journal},
keywords = {commutator; BMO; fractional maximal function; variable exponent Lebesgue space; fractional maximal function; variable exponent Lebesgue space; BMO; commutator},
language = {eng},
number = {1},
pages = {183-197},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Commutators of the fractional maximal function on variable exponent Lebesgue spaces},
url = {http://eudml.org/doc/262009},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Zhang, Pu
AU - Wu, Jianglong
TI - Commutators of the fractional maximal function on variable exponent Lebesgue spaces
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 183
EP - 197
AB - 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 $\mathcal {P}(\mathbb {R}^{n})$ the set of all measurable functions $p(\cdot )\colon \mathbb {R}^{n}\rightarrow [1,\infty )$ 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 $\mathcal {B}(\mathbb {R}^{n})$ the set of all $p(\cdot ) \in \mathcal {P}(\mathbb {R}^{n})$ 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 $p(\cdot )\in \mathcal {P}(\mathbb {R}^{n})$, $0<{\beta }<n/p_{+}$ and $1/q(\cdot )=1/p(\cdot )-\beta /n$ with $q(\cdot )(n-\beta )/n \in \mathcal {B}(\mathbb {R}^{n})$.
LA - eng
KW - commutator; BMO; fractional maximal function; variable exponent Lebesgue space; fractional maximal function; variable exponent Lebesgue space; BMO; commutator
UR - http://eudml.org/doc/262009
ER -