The Commutator Type And The Levi Form Type In C~3

The finite type conditions gave their rise from the investigation of the subellipticity of the ?-Neumann operator.For any boundary point of a smooth pseudoconvex domain in C2,Kohn introduced three types of integer invariants,which are respectively the regular contact type,the commutator type and the Levi form type.Kohn proved that these invariants are equivalent to each other.When they are finite at a boundary point,the domain possesses local sub-elliptic estimates near this point.Bloom generalized Kohn’s type conditions in C2 directly to higher dimensional spaces.D’Angelo introduced D’Angelo finite type and conjectured that these two types equal to each other when the real hypersuface is pseudoconvex.He confirmed the conjecture when one of the type is exactly 4.The present paper is devoted to proving this conjecture when the real hypersurface is in C3.For any fixed（1,0）vector field of a pseudoconvex hypersurface in C3,we prove that its commutator type and Levi form are equivalent to each other.This answers affirmatively a problem of D’Angelo in complex dimension three.