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Holomorphic functions and subelliptic heat kernels over Lie groups. (English) Zbl 1193.32023

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 \({\mathcal U}'\).

MSC:

32W30 Heat kernels in several complex variables
35H20 Subelliptic equations
32C15 Complex spaces
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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