Polyharmonic Bergman spaces and Bargmann type transforms
Let \(dA\) denote the area measure on the upper half-plane \(\Pi\). For \(j=1,2,\dots\), the poly-Bergman space \(\mathcal{A}_j^2(\Pi)\) consists of those complex-valued smooth functions \(f\in L^2(\Pi, dA)\) for which \(\overline{\partial}{}^j f =0\). The polyharmonic Bergman space \(\mathcal{H}_j^2(\Pi)\) consists of \(u\in L^2(\Pi, dA)\) such that \(\Delta^j u = 0\), where \(\Delta\) is the Euclidean Laplacian. The authors show that \[ \mathcal{H}_j^2(\Pi) = \mathcal{A}_j^2(\Pi) \oplus \overline{\mathcal{A}_j^2}(\Pi), \quad j=1,2,\dots. \] The proof of the above decomposition is based on certain properties of the two-sided compression of the Beurling-Ahlfors transform to \(L^2(\Pi, dA)\). Also, the authors establish relations between \(\mathcal{H}_j^2(\Pi)\), its true polyharmonic Bergman spaces and true poly-Bergman spaces. They construct isometric isomorphisms from \(L^2(\mathbb{R})\) onto the true polyharmonic Bergman spaces. Finally, using the classical Paley-Wiener theorem, the authors prove that \(\mathcal{H}_j^2(\Pi)\) is isometrically isomorphic to \(2j\) copies of the Hardy space.
scientific article; zbMATH DE number 6675609
scientific article