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Holomorphic functions of moderate growth and distribution vectors. (Fonctions holomorphes à croissance modérée et vecteurs distributions.) (French) Zbl 1057.22017
A holomorphic function \(f(w)=\sum_{k=0}^\infty a_kw^k\) in the unit disk \(\mathbb D\subset\mathbb C\) is said to be of moderate growth if there exist constants \(C\) and \(N\) such that \(| f(w)| \leq C(1-| w| )^{-N}\). Using the Cayley transformation, we say that a holomorphic function \(F\) on the upper half plane \(\mathbb T \) is of moderate growth if there exist constants \(C\), \(\alpha\), \(\beta\) such that \[ | F(z)| \leq C(1+| z| ^2)^\alpha(1+y^{-\beta}). \] Let \(\mathcal H_\nu(\mathbb D)\) be the space of holomorphic functions on \(\mathbb D\) such that \[ \int_{\mathbb D}| f(w)| ^2(1-| w| ^2)^{\nu-2}\,d\lambda(w)<\infty, \] where \(\lambda\) is the Lebesgue measure and \(\nu> 1\). Let \(\pi_\nu\) be the representation of \(SU(1,1)\) in \(\mathcal H_\nu(\mathbb D)\) defined by \(\left(\pi_\nu(g)f\right)(w)= (\overline\beta w+\overline\alpha)^{-\nu} f\left(\frac{\alpha w+\beta}{\overline\beta w+\overline\alpha}\right),\) where \(g^{-1}=\left( \begin{matrix} \alpha&\beta\\ \overline\beta&\overline\alpha \end{matrix} \right)\), \(\alpha,\beta\in\mathbb C\), \(| \alpha^2| -| \beta| ^2=1.\)
In this paper, the space of the distribution vectors of the representation \(\pi_\nu\) is identified to the space of holomorphic functions with moderate growth on \(\mathbb D\). This is the main result of the paper. The Cayley transformation permits to obtain the representation \(\pi_\nu\) in the space \(\mathcal H_\nu(\mathbb T)\) of functions holomorphic on \(\mathbb T\). Using the Laplace transformation, this is also achieved in \(L^2(0,\infty; u^{\nu-1}\,du)\). As a consequence, the Hankel transformation is an isomorphism of the Schwartz space \(S([0,\infty[)\). An application to multivariate Laguerre series expansions is also given. These results are obtained in the case of a bounded symmetric domain. The relation that exists between the notion of holomorphic function of moderate growth and the notion of moderate growth introduced by N. R. Wallach is established.

MSC:
22E46 Semisimple Lie groups and their representations
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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