Multilinear Singular Integral Forms of Christ-Journé Type

We introduce a class of multilinear singular integral forms Lambda : L-p1 (R-d) x ... x Lpn+2 (R-d) -> C which generalize the Christ-Journe multilinear forms; here Sigma(n+2 )(j=1)p(j)(-1) = 1, p(j) is an element of (1, infinity]. The research is partially motivated by an approach to Bressan's problem on incompressible mixing flows. A key aspect of the theory is that the class of operators is closed under adjoints (i.e. the class of multilinear forms is closed under permutations of the entries). This, together with an interpolation, allows us to reduce the L-p1 x ... x L(p)(n+2 )boundedness to L-infinity x ... x L-infinity x L-p x L-p' boundedness. We obtain estimates of the form |Lambda(f(1), ... ,f(n+2))| <= Cn(2) log(3)(2+n) Pi(n+2)(j=1)parallel to f(j)parallel to(Lpj), where the constant C does not depend on n.