## 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

### Keywords:

multilinear singular integrals; maximal function
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