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Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces. (English) Zbl 1228.42017

Suppose that $T$ is a linear or a sublinear operator which satisfies for any $f\in {L}_{1}\left({ℝ}^{n}\right)$ with compact support and $x\in {\left(suppf\right)}^{c}$

$|Tf\left(x\right)|\le {c}_{0}{\int }_{{ℝ}^{n}}\frac{|f\left(y\right)|}{{|x-y|}^{n}}\phantom{\rule{0.166667em}{0ex}}dy,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where ${c}_{0}$ is independent of $f$ and $x$. For a function $a$, suppose that ${T}_{a}$ is a commutator generated by $T$ and $a$ satisfies for any $f\in {L}_{1}\left({ℝ}^{n}\right)$ with compact support and $x\in {\left(suppf\right)}^{c}$

$|Tf\left(x\right)|\le {c}_{0}{\int }_{{ℝ}^{n}}\frac{|f\left(y\right)||a\left(x\right)-a\left(y\right)|}{{|x-y|}^{n}}\phantom{\rule{0.166667em}{0ex}}dy,\phantom{\rule{2.em}{0ex}}\left(2\right)$

where ${c}_{0}$ is independent of $f$ and $x$.

Let $\varphi \left(x,r\right)$ be a positive measurable function on ${ℝ}^{n}×\left(0,\infty \right)$ and $1\le p<\infty$. We denote by ${M}_{p,\varphi }$ the generalized Morrey space of all functions $f\in {L}_{p}^{\text{loc}}\left({ℝ}^{n}\right)$ with finite quasinorm

${\parallel f\parallel }_{{M}_{p,\varphi }}=\underset{x\in {ℝ}^{n},\phantom{\rule{0.166667em}{0ex}}r>0}{sup}\varphi {\left(x,r\right)}^{-1}{|B\left(x,r\right)|}^{-\frac{1}{p}}{\parallel f\parallel }_{{L}_{p}\left(B\left(x,r\right)\right)}·$

Also, by $W{M}_{p,\varphi }$ we denote the weak generalized Morrey space of all functions $f\in W{L}_{p}^{\text{loc}}\left({ℝ}^{n}\right)$ for which

${\parallel f\parallel }_{W{M}_{p,\varphi }}=\underset{x\in {ℝ}^{n},\phantom{\rule{0.166667em}{0ex}}r>0}{sup}\varphi {\left(x,r\right)}^{-1}{|B\left(x,r\right)|}^{-\frac{1}{p}}{\parallel f\parallel }_{W{L}_{p}\left(B\left(x,r\right)\right)}<\infty ·$

The authors prove the boundedness of the sublinear operator $T$ satisfying condition (1) generated by the Calderón-Zygmund operator from one generalized Morrey space ${M}_{p,{\varphi }_{1}}$ to another space ${M}_{p,{\varphi }_{2}}$ for $1 and from ${M}_{1,{\varphi }_{1}}$ to the weak space $W{M}_{1,{\varphi }_{2}}$. When $a\in \text{BMO}$, they find a sufficient condition on the pair $\left({\varphi }_{1},{\varphi }_{2}\right)$ which ensures the boundedness of the commutator ${T}_{a}$ from ${M}_{p,{\varphi }_{1}}$ to ${M}_{p,{\varphi }_{2}}$ for $1. Finally, they apply their results to several particular operators such as pseudodifferential operators, the Littlewood-Paley operator, the Marcinkiewicz operator, and the Bochner-Riesz operator.

Reviewer: Yu Liu (Beijing)
##### MSC:
 42B20 Singular and oscillatory integrals, several variables 42B35 Function spaces arising in harmonic analysis
##### Keywords:
Calderón-Zygmund operator; Morrey spaces; boundedness