Hypercontractivity over complex manifolds
Consider the complex space \(\mathbb{C}^m\) and the Ornstein-Uhlenbeck operator \(A\); it is known that holomorphic functions satisfy a better hypercontractivity property than more general \(C^\infty\) functions; if \({\mathcal H}^p\) denotes the space of holomorphic functions in \(L^p\), then, for \(0<q\leq p<\infty\), the semigroup \(e^{-tA}\) is a contraction from \({\mathcal H}^q\) into \({\mathcal H}^p\) for \(t\geq t_J(p,q)\); notice that \(q\) and \(p\) can be below 1, and \(t_J(p,q)\) is smaller than the corresponding time for the hypercontractivity on \(C^\infty\) functions. A new proof is given for this result, which enables the extension to operators \(A=d^\star d\) on a complex manifold \(M\); in this case however, \({\mathcal H}^p\) is not always equal to \({\mathcal H}\cap L^p\), but is defined as some closure. The main theorem says that under some assumptions on \(d^\star d\) (in particular it should be holomorphic) and if a logarithmic Sobolev inequality holds, then \(e^{-tA}\) is bounded from \({\mathcal H}^q\) into \({\mathcal H}^p\) for \(t\geq t_J(p,q)\). The proof uses some preliminary results about the structure of the Hilbert space \({\mathcal H}^2\). The article also contains some examples and counterexamples.
scientific article; zbMATH DE number 1541206
scientific article