## On multilinear singular integrals in $$\mathbf R^n$$.(English)Zbl 1006.42019

The authors treat the multilinear operator defined by $T^Af(x)= \text{p.v.}\int_{\mathbb R^n}\frac{A(x)-A(y)-\nabla A(y)\cdot (x-y)} {|x-y|^{n+1}}\Omega(x-y)f(y) dy,$ where $$\nabla A$$ belongs to the Lipschitz space $$\text{Lip}_\beta(\mathbb R^n)$$ $$(0<\beta<1)$$ and $$\Omega$$ satisfies (i) $$\Omega$$ is homogeneous of degree zero, (ii) $$|\Omega(x)-\Omega(y)|\leq C|x-y|$$ for $$|x|=|y|=1$$, and (iii) $$\int_{|x|=1}x_j\Omega(x)dx=0$$, $$j=1,2,\ldots, n$$. They show the following: If $$1<p<\infty$$ and $$0<\beta<1$$, then there exits $$C>0$$ such that $\|T^Af\|_ {\dot F_p^{\beta,\infty}}\leq C\|\nabla A\|_{\text{Lip}_\beta}\|f\|_p,$ where $$\dot F_p^{\beta,\infty}$$ is the homogeneous Triebel-Lizorkin space. This result extends the result $\|T^Af\|_p\leq C\|\nabla A\|_{\text{BMO}}\|f\|_p$ by J. Cohen [“A sharp estimate for a multilinear singular integral in $$\mathbb R^n$$”, Indiana Univ. Math. J. 30, 693-702 (1981; Zbl 0596.42004)].

### MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Zbl 0596.42004