Commutators of the fractional maximal function on variable exponent Lebesgue spaces

@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 -