Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T A , $$ T_{A} f{\left( x \right)} = {\int_{\mathbb{R}^{n} } {\frac{{\Omega {\left( {x - y} \right)}}} {{{\left| {x - y} \right|}^{{n + m - 1}} }}} }R_{m} {\left( {A;x,y} \right)}f{\left( y \right)}dy $$ where R m (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m − 1 in $$ \ifmmode\expandafter\dot\else\expandafter\.\fi{\Lambda }_{\beta } {\left( {0 < \beta < 1} \right)},\Omega {\left( x \right)} \in L^{s} {\left( {S^{{n - 1}} } \right)}{\left( {s \geqslant \frac{n} {{n - \beta }}} \right)}, $$ is homogeneous of degree zero, the authors prove that T A is bounded from L p (ℝ n ) to $$ L^{q} {\left( {\mathbb{R}^{n} } \right)}{\left( {\frac{1} {p} - \frac{1} {q} = \frac{\beta } {n},1 < p < \frac{n} {\beta }} \right)} $$ and from L 1(ℝ n ) to L n/(n−β),∞(ℝ n ) with the bound $$ C{\sum\nolimits_{{\left| \gamma \right|} = m - 1} {{\left\| {D^{\gamma } A} \right\|}} }_{{\ifmmode\expandafter\dot\else\expandafter\.\fi{\Lambda }_{\beta } }} . $$ And if Ω has vanishing moments of order m − 1 and satisfies some kinds of Dini regularity otherwise, then T A is also bounded from L p (ℝ n ) to $$ \ifmmode\expandafter\dot\else\expandafter\.\fi{F}^{{\beta ,\infty }}_{p} {\left( {\mathbb{R}^{n} } \right)}{\left( {1 < {s}\ifmmode{'}\else$'$\fi < p < \infty } \right)} $$ with the bound $$ C{\sum\nolimits_{{\left| \gamma \right|} = m - 1} {{\left\| {D^{\gamma } A} \right\|}} }_{{\ifmmode\expandafter\dot\else\expandafter\.\fi{\Lambda }_{\beta } }} . $$