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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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