The author establishes an -boundedness criterion for a class of multilinear oscillatory singular integrals with standard Calderón-Zygmund kernels. Let
where ; is a real-valued polynomial in and ; is a function that satisfies for all multi-indices with ; has derivatives of order in , , and is a standard Calderón-Zygmund kernel. The truncated operator is defined by
The author proved that if satisfies certain conditions, then, for , , ,
if and only if
where , are constants depending only on , , and .
This result is also true if , has derivatives of order in and is replaced by .