ON THE SINGULARITIES OF THE SZEGO PROJECTIONS ON LOWER ENERGY FORMS

Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n - 1, n >= 2. Let rectangle((q))(b) be the Gaffney extension of Kohn Laplacian on (0, q)-forms. We show that the spectral function of rectangle((q))(b) admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if X is compact and the Levi form is non-degenerate of constant signature on X, then the spectrum of rectangle((q))(b) in ]0, infinity[ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szego kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szego kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR S-1 actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR S-1 action can be CR embedded into C-N, for some N is an element of N.