On the Cauchy transform of functionals on a~Bergman space

V. V. Napalkov ; R. S. Yulmukhametov

Sbornik. Mathematics, Tome 82 (1995) no. 2, pp. 327-336

Résumé

The strong dual space of the Bergman space $$ B_2(G)=\biggl\{f\in H(G):\|f\|_{B_2(G)}^2=\int_G|f(x)|^2\,d\mathrm{v}(z)\infty\biggr\}, $$ is described in terms of the Cauchy transformation, where $\mathrm{v}(z)$ is Lebesgue measure and $G$ is a simply connected domain with boundary of class $C^{1+0}$. As a normed space, $B_2^*(G)$ is isomorphic to the space $$ B_2^1(\mathbb{C}\setminus \overline G) =\biggl\{\gamma (\zeta )\in H(\mathbb{C}\setminus \overline G), \gamma (\infty )=0: \|\gamma \|_{B_2^1(\mathbb C\setminus\overline G)}^2 =\int_{{\mathbb C}\setminus {\overline G}} |\gamma'(\zeta )|^2\,d\mathrm{v}(\zeta)\infty\biggr\}. $$ An example is given of a domain with nonsmooth boundary for which the spaces $B_2^*(G)$ and $B_2^1(\mathbb C\setminus\overline G)$ are not isomorphic.