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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

Citations:

Zbl 0596.42004
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