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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.

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