Curious properties of canonical divisors in weighted Bergman spaces (Q2782533)

Let \(D\) be the open unit disk and \(dA\) be area measure on \(D\). Consider a nonnegative Borel measurable function \(\omega\) in \(D\) with the following properties: NEWLINE\[NEWLINEh(0)=\int_Dh(z)\omega(z)dA(z)\tag{1}NEWLINE\]NEWLINE for all bounded harmonic functions \(h\) in \(D\).NEWLINENEWLINENEWLINE(2) The weighted Bergman spaces \(L^2_a (\omega)\) and \(L^2_h(\omega)\), consisting of analytic and harmonic functions in \(L^2(D,\omega dA)\), respectively, are closed in \(L^2(D,\omega dA)\).NEWLINENEWLINENEWLINE(3) The harmonic polynomials are dense in \(L^2_h(\omega)\).NEWLINENEWLINENEWLINEA unit vector \(f\) in \(L^2_a(\omega)\) is called \(L^2_a(\omega)\)-inner if NEWLINE\[NEWLINEh(z)=\int_D h(z)\bigl|f(z)\bigr |^2\omega (z)dA(z)NEWLINE\]NEWLINE for all bounded harmonic functions \(h\) in \(D\). If \(Z\) is a zero set for \(L^2_a(\omega)\), and if \(Z\) does not contain 0, then the extremal problem NEWLINE\[NEWLINE\sup\bigl \{\text{Re} f(0): f\in I_Z,|f|\leq 1\}NEWLINE\]NEWLINE has a unique solution \(G_Z\), where \(I_Z\) consists of all functions in \(L^2_a (\omega)\) that vanish on \(Z\) (counting multiplicities). The paper under review proves several properties about the zero-based extremal functions \(G_Z\) when \(Z=\{\lambda\}\) is a singleton; the notation \(G_\lambda\) is used in this case. Among the main results we mention the following two. First, if \(|G_\lambda(z) |\geq 1\) for every \(z\) on the unit circle, and if \(\lambda\) is the only zero of \(G_\lambda\) in \(D\), then \(G_\lambda\) is an expansive multiplier on \(L^2_a(\omega)\), that is, NEWLINE\[NEWLINE\|G_\lambda f\|\geq\|f\|NEWLINE\]NEWLINE for every polynomial \(f\). Second, if NEWLINE\[NEWLINE\liminf_{|\lambda|\to 1^-}\bigl(1-|\lambda^2)^{-2}\bigl (\|G_\lambda f\|- \|f\|\bigr)\geq 0NEWLINE\]NEWLINE for every polynomial \(f\), then every \(L^2_a(\omega)\)-inner function \(G\) that is continuous on the closed unit disk satisfies \(|G(z)|\geq 1\) for all \(z\) on the unit circle.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].