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A sharp estimate for a multilinear singular integral in \({\mathbb{R}}^ n\). (English) Zbl 0596.42004

Summary: The purpose of this paper is to prove a good \(\lambda\) inequality for certain multilinear singular integrals in \({\mathbb{R}}^ n\). The operators considered here are the maximal and limiting operators of the singular integral \[ C_{\epsilon}(\nabla A,f)(x)=\int_{| x-y| >\epsilon}P_ 2(A;x,y)\frac{\Omega (x-y)}{| x-y/^{n+1}}f(y)dy \] where \(P_ 2(A;x,y)=A(x)-A(y)-\nabla A(y)\cdot (x-y)\) is the second order Taylor series remainder of A, \(\Omega\) satisfies certain homogeneity, smoothness and symmetry conditions, \(\nabla A\in BMO\) and \(f\in L^ p({\mathbb{R}}^ n)\). The maximal function \(C_*(\nabla A,f)(x)=\sup_{\epsilon >0}| C_{\epsilon}(\nabla A,f)(x)|\) is shown to be controlled by a ”sharp function” of \(\nabla A\) and a Hardy- Littlewood maximal function of f.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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