A proof of a Melrose’s trace formula

The goal of this note is to give a proof of the wave trace formula proved by Richard Melrose in the impressive paper [Me-84]. This trace formula is an extension of the Chazarain-Duistermaat-Guillemin trace formula (denoted “CDG trace formula” in this paper) to the case of a sub-Riemannian (“sR”) Laplacian on a 3D contact closed manifold. The proof uses a normal form constructed in the papers [CHT-18, CHT-21], following the pioneering work [Me-84], in order to reduce to the case of the invariant Laplacian on the 3DHeisenberg group. We need also the propagation of singularities results of Victor Ivrii, Bernard Lascar and Richard Melrose [Iv-76, La-82, Me-86]. Aknowledgments: many thanks to Cyril for very useful comments!

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