Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds

This paper is devoted to a new intrinsic description of microlocal analytic singularities on a connected compact C-omega Riemannian manifold (X, g). In this approach, the microlocal singularities of a distribution 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 complexification X' of X, identified with a tubular neighborhood of the zero section in T*X. 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 Delta) analogous to Lebeau's inversion formula for the Euclidean Fourier-Bros-Iagolnitzer transform.