On derivatives of holomorphic functions on a complex Wiener space
Let \((X,\mu)\) be any complex Wiener space. The functions \(P(z)=\sum a_{k_1\dots k_n}\langle z,\ell_1\rangle^{k_1}\cdot\dots\cdot \langle z,\ell_n\rangle^{k_n}\), for \(\ell_j\in X^*\) are called holomorphic polynomials on \(X\). Let \(H^p(X,\mu)\) be the \(L^p(X,\mu)\)-closure of those polynomials. As a first result it is shown that the Gâteaux derivative \(D_hf\) of each function \(f\in H^p(X,\mu)\) exists \(\mu\)-a.e. on \(X\), but a counterexample shows that \(D_hf\) need not belong to \(L^p(X,\mu)\). In order to obtain positive results in this direction, the author defines inductively new measures \(\rho_n\) on \(X\), and he shows as a main result that for each function \(f\in H^2(X,\mu)\) the limit \(\widetilde{\mathcal L}f=\lim t^{-1}(T_tf-f)\) exists in \(L^2(X,\rho_2)\), and that \(\int_X|\widetilde{\mathcal L}f|^2 d\rho_2\leq \int_X|f|^2 d\mu\) holds true. In this sense, each \(f\in H^2(X,\mu)\) belongs to the Sobolev space \(W^{2,1}(X,\rho_2)\). Using the measures \(\rho_{2^n}\) the result can be generalized to higher derivatives.
scientific article; zbMATH DE number 746250