Holomorphic functions and subelliptic heat kernels over Lie groups

Bruce K. Driver ; Leonard Gross ; Laurent Saloff-Coste

Journal of the European Mathematical Society, Tome 11 (2009) no. 5, pp. 941-978.

Résumé

A Hermitian form q on the dual space, g∗, of the Lie algebra, g, of a Lie group, G, determines a sub-Laplacian, Δ, on G. It will be shown that Hörmander’s condition for hypoellipticity of the sub-Laplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universal enveloping algebra, U′, is non-degenerate. The subelliptic heat semigroup, etΔ/4, is given by convolution by a C∞ probability density ρt​. When G is complex and u:G→C is a holomorphic function, the collection of derivatives of u at the identity in G gives rise to an element, u^(e)∈U′. We will show that if G is complex, connected, and simply connected then the “Taylor” map, u↦u^(e), defines a unitary map from the space of holomorphic functions in L2(G,ρt​) onto a natural Hilbert space lying in U′.

Détail

DOI : 10.4171/jems/171

Classification : 32-XX, 00-XX
Keywords: Subelliptic, heat kernel, complex groups, universal enveloping algebra, Taylor map Schrödinger equations, L²-critical NLS, pseudo-conformal blow-up