Szegö's theorem on Hardy spaces induced by rotation-invariant Borel measures (Q6610401)

Let \(\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}\) denote the open unit disk in the complex plane, and let \(A(\mathbb{D})\) denote the disk algebra consisting of continuous functions on the closed unit disk \(\overline{\mathbb{D}}\) whose restrictions to the open unit disk \(\mathbb{D}\) are analytic.\N\NA celebrated theorem of G. Szegő states that if \(K\) is a nonnegative function in \(L^1(\frac{d\theta}{2\pi})\) with \(\log K\in L^1(\frac{d\theta}{2\pi})\), then\N\[\N\inf_{p\in A_0(\mathbb{D})}\int_0^{2\pi}|1-p|^2K \frac{d\theta}{2\pi}=\exp\left(\int_0^{2\pi}\log K \frac{d\theta}{2\pi}\right),\N\]\Nwhere \(A_0(\mathbb{D})=\{f\in A(\mathbb{D}):f(0)=0\}\). This theorem has profound impact in many areas of pure and applied mathematics such as theoretical physics, stochastic processes, and numerical analysis.\N\NThe goal of this paper is to find an analog of this theorem for Hardy spaces induced by rotation-invariant measures on the closed unit disk. Further, some connections between cyclic vectors in these spaces (\(f\) is cyclic if the polynomial multiples of \(f\) are dense in the space) and function theoretic aspects of these spaces are revealed. \N\NIt is well known that (see [\textit{K. Hoffman}, Banach spaces of analytic functions. Englewood Cliffs, N.J.: Prentice-Hall (1962; Zbl 0117.34001)]) a necessary and sufficient condition for a nonnegative function \(K\in L^1(\frac{d\theta}{2\pi})\) to satisfy \(\log K\in L^1(\frac{d\theta}{2\pi})\) is that \(K=|f|^2\) for some \(f\in H^2(\frac{d\theta}{2\pi})\). Therefore, Szegő's theorem can be rewritten as\N\[\N\inf_{p\in A_0(\mathbb{D})}\int_0^{2\pi}|1-p|^2|f|^2\frac{d\theta}{2\pi}=\exp\left(\int_0^{2\pi}\log |f|^2 \frac{d\theta}{2\pi}\right),\N\]\Nfor every nonzero function \(f\in H^2(\frac{d\theta}{2\pi})\). To state the main result of this paper, we need to recall that a Borel measure \(d\mu\) on the closed unit disk is said to be rotation-invariant provided that for every Borel subset \(E\subseteq \overline{\mathbb{D}}\) and every complex number \(\xi\) with \(|\xi|=1\) we have \(\mu(E)=\mu(\xi E)\). It is further assumed that \(\mu\) satisfies the condition \(\sup\{|z|:z\in \mathrm{support}(\mu)\}=1\). \par The authors then define \(H^2(d\mu)\) as the norm-closure of \(A(\mathbb{D})\) in \(L^2(d\mu)\). Note that when \(d\mu=\frac{d\theta}{2\pi}\), then \(H^2(d\mu)\) becomes the classical Hardy space \(H^2(\mathbb{D})\) consisting of all analytic functions in the unit disk for which\N\[\N\Vert f\Vert^2_{H^2(\mathbb{D})}=\sup_{0<r<1}\frac{1}{2\pi}\int_0^{2\pi} |f(re^{i\theta})|^2d\theta<\infty.\N\]\NNeedless to say that functions in the Hardy space \(H^2(\mathbb{D})\) are identified with their boundary functions in \(L^2(d\mu, \mathbb{T})\) where \(\mathbb{T}\) stands for the unit circle. Similarly, when \(d\mu(re^{i\theta})=\frac{\alpha+1}{\pi}(1-r^2)^\alpha rdrd\theta\), for \(\alpha>-1\), we get the standard weighted Bergman space \(A^2_\alpha (\mathbb{D})\) on the unit disk which consists of all analytic functions \(f\) in the open unit disk for which \[\Vert f\Vert_{A^2_\alpha(\mathbb{D})}^2=\frac{\alpha+1}{\pi}\int_\mathbb{D} |f(z)|^2(1-|z|^2)^\alpha dxdy<\infty.\] Now, let \(\varphi_w\) be the Möbius map \[\varphi_w(z)=\frac{w-z}{1-\overline w z},\quad z\in\overline{\mathbb{D}},\] and let \(\mu_w(E)=\mu(\varphi_w^{-1}(E))\) be the probability measure induced by \(\varphi_w\) on the closed unit disk. The main result of the paper under review reads as follows:\N\NTheorem. Let \(d\mu\) satisfy the following conditions:\N\begin{itemize}\N\item for every \(w\in \mathbb{D}\), the measure \(\mu_w\) is absolutely continuous with respect to \(\mu\);\N\item for every \(f\in L^1 (d\mu)\), \(f\circ \varphi_w\in L^1 (d\mu)\), and \(\hat{f}(w)=\int_{\overline{\mathbb{D}}}f\circ \varphi_w d\mu\) is continuous on \(\mathbb{D}\).\N\end{itemize}\NAssume further that \(K\) is a nonnegative function in \(L^1(d\mu)\) with \(\log K\in L^1(d\mu)\). Then\N\[\N\inf_{p\in A_0(\mathbb{D})}\int_{\overline{\mathbb{D}}}|1-p|^2K d\mu=\exp \left(\int_{\overline{\mathbb{D}}}\log K d\mu \right)\N\]\Nif and only if \(K=|h|^2\) for some cyclic vector \(h\in H^2(d\mu)\).\N\NFor the entire collection see [Zbl 1531.47001].