On Identical Vanishing of Holomorphic Functions in Several Complex Variables.

We will make some comment on the following recent result of J. J. Kohn. In the complex space Cn:(z1, . . . , zn) if fl, . . . , fn are holomorphic in the unit ball and have continuous boundary values on the boundary, and if on the boundary the linear combination 1fi+ + j,,f is 0, then the functions fl, . .. , f,, are identically 0. 1. We denote by D any bounded circular domain in C., by B its boundary, and by Bo an open neighborhood in B. We do not assume that D contains the origin in C,, but only that Bo has a positive distance from it. For any integer m, m 5 n, we take functions fl, . . , fm which are holomorphic in D and have continuous boundary values on Bo. We also take m polynomials Pi, . . . , Pm in n symbols Ai ... . . ,n homogeneous and of a common degree and we denote the degree by r. Finally, we introduce the function m D(r; z) = E P~()f"(z) (1) A=1