Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T A , TAf(x)=Rn(xy)|xy|n*m1Rm(A;x,y)f(y)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\else\expandafter\.\fi(0>>1),(x)Ls(Sn1)(snn), is homogeneous of degree zero, the authors prove that T A is bounded from L p(n) to Lq(Rn)(1p1q=n,1>p>n) and from L 1(n) to L n/(n),(n) with the bound C||=m1DA\ifmmode\expandafter\else\expandafter\.\fi. And if has vanishing moments of order m 1 and satisfies some kinds of Dini regularity otherwise, then T A is also bounded from L p(n) to \ifmmode\expandafter\else\expandafter\.\fiF,p(Rn)(1>s\ifmmode\else$$\fi> p>) with the bound C||=m1DA\ifmmode\expandafter\else\expandafter\.\fi.