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A note on the multilinear singular integral operators

Authors:

Jiao Yulan

Institution:

(1) College of Science, University of Information Engineering, P. O. Box 2201-160, 450001 Zhengzhou, P. R. China

Abstract:

L ^p(Rⁿ) boundedness is considered for the multilinear singular integral operator defined by

$T_A f(x) = \smallint _{\mathbb{R}^n } \frac{{\Omega (x - y)}}{{|x - y|^{n + 1} }}(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy$

where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives of order one in BMO(Rⁿ). We give a smoothness condition which is fairly weaker than that Ω∈Lip _α(Sⁿ⁻¹) (0<α⪯1) and implies theL ^p(Rⁿ) (1<p<∞) boundedness for the operator T_A. Some endpoint estimates are also established.

Keywords:

multilinear singular integral operator homogeneous of degree zero smoothness condition SINGULAR INTEGRAL OPERATORS estimates established singular integral operator smoothness condition derivatives integrable the unit sphere moment order homogeneous of degree zero boundedness defined

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