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Zbl 0895.42005
Lu, Shanzhen; Yang, Dachun; Zhou, Zusheng
Sublinear operators with rough kernel on generalized Morrey spaces.
(English)
[J] Hokkaido Math. J. 27, No.1, 219-232 (1998). ISSN 0385-4035

Let $\phi$ be a positive and increasing function on $(0, \infty)$, satisfying $\phi(2r)\le D \phi(r)$ $(r>0)$, where $D\ge 1$ is a constant independent of $r$. For $1\le p<\infty$, one denotes by $L^{p,\phi}(\Bbb R^n)$ the space of locally integrable functions $f$ for which $\int_{B_r(x_0)}| f(x)| ^p dx \le C^p \phi(r)$ for all $x_0\in \Bbb R^n$ and every $r>0$, where $B_r(x_0)=\{x\in \Bbb R^n; | x-x_0| \le r\}$. These spaces are called the generalized Morrey spaces. The authors give the following: Let $1\le p<\infty$, $1\le D<2^n$ and $\gamma=\log 2^n/\log D$. If a sublinear operator $T$ is bounded on $L^p(\Bbb R^n)$ and for any $f\in L^1(\Bbb R^n)$ with compact support and satisfying $$| Tf(x)| \le C\int_{\Bbb R^n} | \Omega(x-y)| | x-y| ^{-n}| f(y)| dy\tag *$$ for $x\notin \text{supp} f$, where $\Omega$ is homogeneous of degree zero and $\Omega\in L^q(S^{n-1})$ for some $q\ge p/(p-1)$ or some $q>\min\{p, \gamma/(\gamma-1)\}$, then $T$ is also bounded on $L^{p,\phi}(\Bbb R^n)$. Similarly $L^{p,\phi}(\Bbb R^n)$ boundedness is discussed for the commutators $[a,T]$ with $a\in \text{BMO}(\Bbb R^n)$ and a linear $T$ satisfying $(\ast)$. Like as in Riesz operators, they also discuss similar results for the operators $T$ satisfying $$| Tf(x)| \le C\int_{\Bbb R^n}| \Omega(x-y)| | x-y| ^{\alpha-n}| f(y)| dy, \tag**$$ concerning $L^{p,\phi}$-$L^{p,\phi^{q/p}}$ boundedness, where $0<\alpha<n$, $1<p<n/\alpha$ and $1/q=1/p-\alpha/n$. These results are extensions of the corresponding ones by {\it T. Mizuhara} (singular integral operator case) [Harmonic analysis, Proc. Conf., Sendai/Jap. 1990, ICM-90 Satell Conf. Proc., 183-189 (1991; Zbl 0771.42007)] and {\it E. Nakai} (Riesz operator case) [Math. Nachr. 166, 95-103 (1994; Zbl 0837.42008)].
[K.Yabuta (Nara)]
MSC 2000:
*42B20 Singular integrals, several variables
42B25 Maximal functions
42B30 Hp-spaces (Fourier analysis)

Keywords: sublinear operator; CalderÃ³n-Zygmund kernel; Morrey space; commutator; BMO

Citations: Zbl 0837.42008; Zbl 0771.42007

Cited in: Zbl 0969.42009

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