INTRINSIC MICROLOCAL ANALYSIS AND INVERSION FORMULAS FOR THE HEAT-EQUATION ON COMPACT REAL-ANALYTIC RIEMANNIAN-MANIFOLDS

This paper is devoted to a new intrinsic description of microlocal ana lytic singularities on a connected compact C-omega Riemannian manifold (X, g). In this approach, the microlocal singularities of a distribut ion u on X are described in terms of the growth, as t --> 0(+), of the analytic extension of epsilon(-t Delta)u to a suitable complexificati on X' of X, identified with a tubular neighborhood of the zero section in TX. First we show that the analytic extension of the heat kernel of (X, g) to X' is an F.B.I. transform in the sense of Sjostrand. Then we establish various inversion formulae for the heat semigroup e(-t D elta) analogous to Lebeau's inversion formula for the Euclidean Fourie r-Bros-Iagolnitzer transform.