A note on parameterized Marcinkiewicz integrals with variable kernels |
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| Authors: | Hui Wang Chun-jie Zhang |
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| Affiliation: | [1]Department of Mathematics, Zhejiang University, Hangzhou 310027, China [2]Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China |
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| Abstract: | In this paper, the parameterized Marcinkiewicz integrals with variable kernels defined by
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$
\mu _\Omega ^\rho (f)(x) = \left( {\int_0^\infty {\left| {\int\limits_{|x - y| \leqslant t} {\frac{{\Omega (x,x - y)}}
{{|x - y|^{n - \rho } }}f(y)dy} } \right|^2 \frac{{dt}}
{{t^{1 + 2\rho } }}} } \right)^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} .
$
\mu _\Omega ^\rho (f)(x) = \left( {\int_0^\infty {\left| {\int\limits_{|x - y| \leqslant t} {\frac{{\Omega (x,x - y)}}
{{|x - y|^{n - \rho } }}f(y)dy} } \right|^2 \frac{{dt}}
{{t^{1 + 2\rho } }}} } \right)^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} . |
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| Keywords: | parameterized Marcinkiewicz integral variable kernel rotation method |
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