Global analytic-hypoellipticity of the $\bar \partial $-Neumann problem

Abstract

Introduction.The (real-)analytic behavior (near the boundary) of solutions of the so-called 9-Neumann problem seems to have been unknown.In this paper we show that the global analytic-hypoellipticity (up to the boundary) holds on certain domains in C n with analytic boundaries.A systematic study of the 9-Neumann problem was made by Kohn [3], and the most difficult part of his work was the proof of the C°°h ypoellipticity (up to the boundary).Soon after, Kohn and Nirenberg [5] gave an elegant proof of the C°° hypoellipticity by establishing the so-called subelliptic estimate.Their method is today used for various problems as the standard technique.However, it seems difficult, even if possible, to deduce the analytic-hypoellipticity of the 3-Neumann problem from the subelliptic estimate.Under these circumstances we introduce in Lemma 2 a certain special vector field tangential along the boundary, which can be constructed in the case the Levi form is non-degenerate.It possesses the properties nice enough to carry out the commutator estimates (Lemmas 4 and 5), and these estimates together with the a priori estimate (Lemma 1) lead us in the usual way (see, e.g., Morrey and Nirenberg [6]) to our result.Our a priori estimate is suggested by a paper of Kohn [4].It should be mentioned that the local problem still remains unsolved, and our method may not be applicable.