Bergman and Reinhardt weighted spaces of holomorphic functions (Q2567497)

This paper continues very interesting, nontrivial work of the authors [Ann.\ Acad.\ Sci.\ Fenn.\ Math.\ 30, No.~2, 337--352 (2005; Zbl 1090.46018); ``Complete weights and \(v\)-peak points of spaces of weighted holomorphic functions'', Isr.\ J.\ Math.\ (to appear); ``Isometries of spaces of weighted holomorphic functions'' (Preprint)]. Let \(v:U \to \mathbb R\) be a continuous strictly positive weight on the open subset \(U \subset \mathbb C^n.\) The authors' interest is centered on the structure of \({\mathcal H}_v(U)\) (and \({\mathcal H}_{v_0}(U)),\) the Banach spaces of holomorphic functions \(f:U \to \mathbb C\) such that \(\| f\| _v = \sup_{z \in U} v(z)| f(z)| < \infty\) (resp., with the additional requirement that \(v(z)f(z) \to 0\) as \(z\) approaches the boundary of \(U.\)) The \(v\)-boundary of \(U,\) \({\mathcal B}_v(U),\) is \(\{z \in U ~| ~ v(z)\delta_z\) is an extreme point of the unit ball of \({\mathcal H}_{v_0}(U)^\prime\}.\) If the weight \(v\) is such that \({\mathcal B}_v(U) = U,\) then \(v\) is said to be complete. The isometry group of \(v\) is defined as the set of all homeomorphisms of \({\mathcal B}_v(U)\) that induce an isometry of \({\mathcal H}_{v_0}(U),\) and is denoted by \(\Lambda_v(U).\) Let \(D \subset \mathbb C^n\) be a bounded domain, let \(k\) be the usual Bergman kernel on \(D,\) and let \(v_B(z) \equiv k(z,z)^{-1}\) denote the Bergman weight on \(D.\) Among other things, the authors show that if \(D\) is the unit ball of a finite-dimensional JB\(^\ast\)-triple, then \(\Lambda_{v_B}(D) = \text{Aut}(D),\) and as a consequence \(v_B\) is complete. Conditions are given that ensure that a weight on a balanced bounded Reinhardt domain in \(\mathbb C^2\) is complete and, in particular, the authors show that \(v_{p,q}(z) \equiv (1 - | z_1| ^{2p} - | z_2| ^{2q})\) on \(D_{pq} = \{(z_1,z_2) ~| ~ | z_1| ^{2p} + | z_2| ^{2q} < 1\}\) is complete. Finally, using the Monge--Ampère operator \((dd^c)^n,\) the authors investigate isometries of weighted holomorphic functions in higher dimensions. Among other things, they show that for \(m > 1/2,\) if \(E_m = D_{1m}\) and the weight \(v_{\alpha,m}\) is given by \(v_{\alpha,m}(z) \equiv (1 - | z_1| ^2 - | z_2| ^{2m})^\alpha,\) then \(\Lambda_{v_\alpha,m} = \text{Aut}(E_m).\)