Holomorphic functions and subelliptic heat kernels over Lie groups
Summary: A Hermitian form \(q\) on the dual space \(g^*\) of the Lie algebra \(g\) of a Lie group \(G\) determines a sub-Laplacian \(\Delta \) on \(G\). It is 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 \({\mathcal U}'\) is non-degenerate. The subelliptic heat semigroup \(e^{t\Delta /4}\) is given by convolution by a \(C^{\infty }\) probability density \(\rho _{t}\). When \(G\) is complex and \(u : G \to \mathbb C\) is a holomorphic function, the collection of derivatives of \(u\) at the identity in \(G\) gives rise to an element \(\hat u(e) \in{\mathcal U}'\). We show that, if \(G\) is complex, connected, and simply connected, then the ``Taylor'' map \(u \rightarrowtail \hat u(e)\) defines a unitary map from the space of holomorphic functions in \(L^{2}(G, \rho _{t})\) onto a natural Hilbert space lying in \({\mathcal U}'\).
scientific article; zbMATH DE number 5608950
scientific article