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A BMO estimate for the multilinear singular integral operator

Authors:

Institution:

(1) Department of Applied Mathematics, University of Information Engineering, P. O. Box 1001-7410, Zhengzhou, 450002, P. R. China

Abstract:

The behavior on the space L^∞ (ℝ ⁿ) for the multilinear singular integral operator defined by

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

is considered, 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 (ℝ ⁿ). It is proved that if Ω satisfies some minimum size condition and an L¹-Dini type regularity condition, then for f ε L^∞(ℝ ⁿ), T_Af is either infinite almost everywhere or finite almost everywhere, and in the latter case, T_Af ε BMO(ℝ ⁿ).

Keywords:

multilinear singular integral operator L1-Dini type regularity condition SINGULAR INTEGRAL OPERATOR case finite minimum size regularity condition type derivatives integrable the unit sphere moment order homogeneous of degree zero singular integral operator defined behavior space

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