Bloch functions and asymptotic tail variance (Q529251)

Given a function \(f\in L^1(\mathbb{D})\), its Bergman projection is defined as \[ \mathbf{P} f(z) = \int_{\mathbb{D}} \frac{f(w)}{(1-z\overline{w})^2}\, dA(w), \quad z\in \mathbb{D}, \] where \(dA\) denotes the normalized area measure on the unit disk \(\mathbb{D}\). For \(\mu\in L^\infty(\mathbb{D})\), \(\|\mu\|_{L^\infty(\mathbb{D})} \leq 1\), and \(0 < a<1\), the author proves that \[ \int_{\mathbb{T}} \exp\left(a\frac{r^4 |\mathbf{P}\mu(r\zeta)|^2}{\log\frac{1}{1-r^2}} \right)\,ds(\zeta) \leq C(a), \quad 0 < r < 1, \] where \(C(a) =10(1-a)^{-3/2}\) and \(ds\) is length measure on the unit circle \(\mathbb{T}\). If \(1< a < \infty\), then there exists \(\mu_0\in L^\infty(\mathbb{D})\), \(\|\mu_0\|_{L^\infty(\mathbb{D})} = 1\), such that \[ \lim_{r\to 1-} \int_{\mathbb{T}} \exp\left(a\frac{r^4 |\mathbf{P}\mu_0(r\zeta)|^2}{\log\frac{1}{1-r^2}} \right)\,ds(\zeta) = +\infty. \] The main aspects of this result may be formulated as follows: The uniform asymptotic tail variance of the unit ball in \(\mathbf{P} L^\infty(\mathbb{D})\) equals one. The above result has various important applications. So, given a holomorphic function \(g: \mathbb{D}\to \mathbb{C}\), define the exponential-type spectrum \(\beta_g: \mathbb{C} \to [0, +\infty]\) of the zero-free function \(e^g\) as \[ \beta_g(t) = \limsup_{r\to 1-} \frac{\log\int_{\mathbb{T}} |e^{tg(r\zeta)}|\, ds(\zeta)}{\log\frac{1}{1-r^2}}. \] For \(g=\mathbf{P}\mu\), \(\|\mu\|_{L^\infty(\mathbb{D})}\leq 1\), the author proves that \(\beta_g(t) \leq |t|^2 /4\) for \(|t|\leq 2\), and \(\beta_g(t) \leq |t|-1\) for \(|t|\geq 2\). For \(\mathbb{D}_e\), the exterior unit disk, and a holomorphic function \(g:\mathbb{D}_e \to \mathbb{C}\), the exponential-type spectrum is defined analogously: \[ \beta_g(t) = \limsup_{R\to 1+} \frac{\log\int_{\mathbb{T}} |e^{tg(R\zeta)}|\, ds(\zeta)}{\log\frac{R^2}{R^2-1}}. \] Let \(\Sigma\) denote the class of conformal mappings \(\psi: \mathbb{D}_e \to \mathbb{C}\) with asymptotics \(\psi(z) = z + \mathcal{O}(1)\) as \(z\to\infty\). For \(0 < k <1\), let \(\Sigma^k\) denote the collection of those \(\psi\in\Sigma\) which have a \(k\)-quasiconformal extension \(\widetilde{\psi}:\mathbb{C}_\infty \to \mathbb{C}_\infty\), that is, \(\widetilde{\psi}\) is a homeomorphism of Sobolev class \(W^{1,2}\) with dilation estimate \[ |\overline{\partial}_z \widetilde{\psi}(z)| \leq k |{\partial}_z \widetilde{\psi}(z)|, \quad z\in\mathbb{C}. \] The universal spectra \(B(k,t)\), \(0 < k\leq 1\), \(t\in\mathbb{C}\), are defined as \[ B(1, t) =\sup_{\psi\in\Sigma} \beta_{\log \psi^\prime} (t), \quad B(k, t) =\sup_{\psi\in\Sigma^k} \beta_{\log \psi^\prime} (t). \] The author proves that \(B(k,t) \leq \frac{1}{4} k^2 |t|^2 (1+ 7k^2)\), \(0 < k <1\), \(t\in\mathbb{C}\). Also, he gives an application to the Minkowski dimension of quasicircles.