On multilinear singular integrals in \(\mathbf R^n\) (Q2767384)

The authors treat the multilinear operator defined by NEWLINE\[NEWLINET^Af(x)= \text{p.v.}\int_{\mathbb R^n}\frac{A(x)-A(y)-\nabla A(y)\cdot (x-y)} {|x-y|^{n+1}}\Omega(x-y)f(y) dy,NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE\|T^Af\|_ {\dot F_p^{\beta,\infty}}\leq C\|\nabla A\|_{\text{Lip}_\beta}\|f\|_p,NEWLINE\]NEWLINE where \(\dot F_p^{\beta,\infty}\) is the homogeneous Triebel-Lizorkin space. This result extends the result NEWLINE\[NEWLINE\|T^Af\|_p\leq C\|\nabla A\|_{\text{BMO}}\|f\|_pNEWLINE\]NEWLINE by \textit{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)].