On Witten Laplacians and Brascamp–Lieb’s Inequality on Manifolds with Boundary

Dorian Le Peutrec (1, 2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 MATHERIALS - MATHematics for MatERIALS

In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp–Lieb’s type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the Neumann and Dirichlet self-adjoint realizations of the Witten Laplacian. They moreover imply the usual Brascamp–Lieb’s inequality and its generalization to compact Riemannian manifolds without boundary.

Mots clés: Witten Laplacian; Bakry-\'Emery tensor.; Riemannian manifolds with boundary; Brascamp–Lieb’s inequality; Supersymmetry

Domaines: Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Géométrie différentielle [math.DG]

Dorian Le Peutrec. On Witten Laplacians and Brascamp–Lieb’s Inequality on Manifolds with Boundary. Integral Equations and Operator Theory, 2017, 87 (3), pp.411 - 434. ⟨10.1007/s00020-017-2354-1⟩. ⟨hal-01349786v2⟩