Duality theorems for certain analytic spaces on the minimal ball
Let \(\mathbb{B}_*= \{z\in\mathbb{C}^n: |z|^2 +|z\cdot z |< 1\}\), \({\mathcal A}^p (\mathbb{B}_*)= {\mathcal O}(\mathbb{B}_*)\cap L^p(\mathbb{B}_*, |z\cdot z|^{(p-2)/2} d\nu)\) where \(z\cdot z=\sum^n_1 (z_j)^2\), \(p>1\) and \(d\nu\) is the Lebesgue measure in \(\mathbb{C}^n\). The main result: Theorem. The dual of the space \({\mathcal A}^p(\mathbb{B}_*)\) can be identified with the space \({\mathcal A}^q (\mathbb{B}_*)\) for \(p>1\) and \({1\over p}+{1\over q}=1\). More precisely, there is a bilinear complex form \(\Lambda\) on \({\mathcal A}^p(\mathbb{B}_*) \times{\mathcal A}^q (\mathbb{B}_*)\) such that every bounded linear functional on \({\mathcal A}^p (\mathbb{B}_*)\) is of the form \[ f\mapsto \Lambda(f,g) \] for some unique \(g\in{\mathcal A}^q (\mathbb{B}_*)\). Furthermore the norm of the linear functional on \({\mathcal A}^p (\mathbb{B}_*)\) is equivalent to the norm of \(g\) in \({\mathcal A}^q(\mathbb{B}_*)\) i.e. there is a positive constant \(\tau\) such that \[ \tau\|g\|_{{\mathcal A}^q(\mathbb{B}_*)} \leq\bigl\|\Lambda(.,g)\bigr \|\leq\tau^{-1} \|g\|_{{\mathcal A}^q (\mathbb{B}_*)}, \] for all \(g\in{\mathcal A}^q(\mathbb{B}_*)\). The proof is based on results from \textit{G. Mengotti} and \textit{E. H. Youssfi} [Bull. Sci. Math. 123, 501-525 (1999; Zbl 0956.32006)].
scientific article; zbMATH DE number 1556592
scientific article