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Commutators and Morrey spaces. (English) Zbl 0761.42009
A locally $L\sp p$ function $f$ is said to belong to the Morrey space $L\sp{p,\lambda}(\bbfR\sp n)$ if $$\Vert f\Vert\sb{p,\lambda}\sp p=\sup\sb{x,\rho}\rho\sp{ -\lambda}\int\sb{\vert x-y\vert\le \rho}\vert f(y)\vert\sp p dy<\infty.$$ As is known, the commutators between the Calderón-Zygmund singular integral operators and the multiplication operator by a function $a(x)$ are bounded on $L\sp p (\bbfR\sp n)$, $1<p<\infty$, if and only if $a(x)$ belongs to the John-Nirenberg space $\roman{BMO}$. The authors show that the same result holds for the Morrey space, in place of $L\sp p$. Commutators between a fractional integral operator and $a(x)$ are also dealt with in the Morrey space as in the $L\sp p$ space.
Reviewer: K.Yabuta (Nara)

42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory
42B30$H^p$-spaces (Fourier analysis)
46E99Linear function spaces and their duals