Holomorphic functions and the Itô chaos
The author clearly describes results he obtained with L. Gross and L. Saloff-Coste, about the characterization of those holomorphic functions which are square integrable with respect to a given smooth measure \(\lambda\), in terms of the sequence of their derivatives at some base point. Examples are detailed, in particular \(G = \mathbb C^d\). The reproducing kernel properties of square integrable holomorphic functions are emphasized and proved. Moreover some parts of the purely analytical original proof are provided with new proofs, based on Itô's stochastic calculus and the Fock space defined by the Itô chaos. More precisely, the author mainly considers a connected, simply connected complex Lie group \(G\), together with the Laplace operator \(\Delta_q := \sum^m_ {j=1} X^2_j\) associated to linearly independent left-invariant vector fields \(X_j\) generating a subspace \(H_q\) of the Lie algebra \(\mathcal G\), which is supposed to be stable under conjugation, and with the non-negative Hermitian quadratic form \(q := \frac12 \sum^m_ { j=1}\langle X_j, \cdot\rangle \overline{\langle X_j, \cdot\rangle}\) on \(\mathcal G^{\ast}\). The Hörmander condition is assumed: the Lie algebra generated by \(H_q\) is the whole \(\mathcal G\). Then the hypo-elliptic heat kernel measure \(\delta_t\) associated to \(e^{t\Delta_q /4}\) admits a smooth density. On the other hand, consider the Taylor map \(f \mapsto T F := (D^n f (e))_{ n\in N}\) of the successive derivatives \((D^n f (e) \in (\mathcal G^{ \otimes n})^{\ast} )\) at the unit \(e\) of \(G\), and extend \(q\) to \((\mathcal G^{\otimes n} )^{\ast}\) by \[ q^{\otimes n} := 2^{-n} \sum_{1\leq j_1 ,\dots ,j_n \leq m} \langle X_j{}_1 \otimes \cdots \otimes X_j{}_n , \cdot\rangle \overline{\langle X_j{}_1 \otimes\cdots \otimes X_j{}_n , \cdot\rangle}. \] Set \(q_t (f ) := \sum_{n\geq 0}\frac{t^n}{n!} q^{\otimes n} (D^n f (e), D^n f (e))\), for any positive t and any holomorphic function \(f\) on \(G\). Then the main result asserts that an holomorphic function \(f\) belongs to \(L^2 (G, \lambda_t )\) if and only if \(q_t (f )\) is finite, and moreover, that the Taylor map \(T\) is unitary and onto.
square integrable holomorphic functions, Itô chaos
scientific article; zbMATH DE number 6529325
scientific article