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| ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR (I) |
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Citation： |
Chen Wengu,Hu Guoen,Lu Shanzhen.ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR (I)[J].Chinese Annals of Mathematics B,1997,18(2):181~190 |
| Page view： 1120
Net amount： 653 |
Authors： |
Chen Wengu; Hu Guoen;Lu Shanzhen |
Foundation： |
Project supported by the National Natural Science
Foundation of China |
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| Abstract： |
The authors consider the multilinear oscillatory
singular integral operator defined by
$$ T_{A_1,\,A_2,\,\cdots,\,A_k}f(x)=\int_{{\Bbb R}^n} e^{iP(x,y)}\prod_{j=1}^k
R_{m_j}(A_j;\,x,y)\frac {\Omega(x-y)}{|x-y|^{n+M}}f(y)\,dy,$$
where $P(x,y)$ is a real-valued polynomial on ${\Bbb R}^n\times {\Bbb R}^n$,
$\Omega$ is homogeneous of degree zero,
$R_{m_j}(A_j;\,x,y)$ denotes the $m_j$-th order Taylor series remainder
of $A_j$ at $x$ expanded about $y$, $M=\sum^k_{j=1}\limits m_j$.
It is shown that
if $\Omega$ belongs to the space $L\log^+L(S^{n-1})$ and has vanishing moment
up to order $M$, then
$$ \|T_{A_1,\,A_2,\,\cdots,\,A_k}f\|_q \le C \prod^k_{j=1}\Big(\sum_{|\alpha|=m_j}
\|D^\alpha A_j\|_{r_j}\Big)\|f\|_p,$$
provided that $1 |
Keywords： |
Multilinear operator, Oscillatory singular
integral, Maximal operator |
Classification： |
42B20 |
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