Asymptotic behaviour of doubly orthogonal Bergman systems: An elementary approach
Let \(X\) be a complex space, let \(D\) be an open subset of \(X\), and let \(K\) be a compact subset of \(D\). Assume that \(D\) is hyperconvex with finitely many connected components and that the intersection of \(K\) with each component of \(D\) is non-pluripolar, i.e., the pair \((K,D)\) is a condenser in \(X\). The relative extremal function \(\omega( \cdot ;K;D)\) for \(K\) in \(D\) is defined as the least upper semi-continuous majorant of \(\sup\{u\in \text{PSH}(D)\); \(u\leq 1\), \(u|K\leq 0\}\) and the relative equlibrium measure \(\mu_{K,D}\) is defined as \((dd^c\omega( \cdot ;K;D))^n\), where \((dd^c\cdot)^n\) is the Monge-Ampère operator. Let \(\mu_1\) be a positive Borel measure of the form \(\mu_1=\rho dV\), where \(\rho\) is a continuous positive function on \(D\) and \(dV\) is the volume element associated to a given hermitean metric on \(X\), and assume that the Bergman space \(H_1=L^2_h(D,\mu_1)={\mathcal O}(D)\cap L^2(D,\mu_1)\) is of infinite dimension. Let \(\mu_0\) be a positive Borel measure on \(D\) such that \(\omega( \cdot ;E;D)=\omega( \cdot ;K;D)\) for every Borel set \(E\subset K\) with \(\mu_0(E)=\mu_0(K)\) and let \(H_0\) be the closure of \({\mathcal O}(K)\) in \(L^2(K,\mu_0)\). (The equilibrium measure \(\mu_{K,D}\) satisfies this condition.) The main result of the paper is that there exists a doubly orthogonal system \((B_j)_{j\geq 1}\) in \(H_0\) and \(H_1\) with the property, that if \(\|B_j\|_{H_0}=1\) and \(\|B_j\|_{H_1}=\gamma_j\), then \[ \sum_{j=1}^{+\infty}\gamma_j^{-\varepsilon}<+\infty \] for all \(\varepsilon >0\), and \[ \limsup_{j\to +\infty}\dfrac{\log|B_j(x)|}{\log\gamma_j} \leq \omega(x;K;D). \] for all \(x\in D\). In particular, for every \(\alpha\in ]0,1[\) and every compact subset \(E\) of \(\{x\in D\); \(\omega(x;K;D)<\alpha\}\) there exists a constant \(c(K,\alpha)>0\) such that \(\|B_j\|_E \leq c(E,\alpha)\gamma_j^\alpha\) for all \(j\geq 1\).
scientific article; zbMATH DE number 1885668