On singular values of Hankel operators on Bergman spaces (Q2139174)

Let \(\omega:=e^{-\varphi}\) be a weight on the unit disc \(\mathbb{D}\) such that \(\varphi\) is a regular subharmonic function and let \(A_\omega^2\) the weighted Bergman space of holomorphic functions \(f\) in \(\mathbb{D}\) such that \[ \|f\|_\omega= \left( \int_{\mathbb{D}}|f(z)|^2\, dA_\omega(z)\right)^\frac12<\infty, \] where \(dA_\omega:=\omega\, dA\). If \(P_\omega\) is the orthogonal projection from \(L^2(dA_\omega)\) onto \(A_\omega^2\), the Hankel operator with anti-analytic symbol \(\overline{\Phi}\) is the operator \(H_{\overline{\Phi}}: L^2(dA_\omega)\to L^2(dA_\omega)\) defined by \[ H_{\overline{\Phi}}= \overline{\Phi}f- P_\omega (\overline{\Phi}f). \] If \(K\) denotes the reproducing kernel of the space \(A_\omega^2\), let \[ \tau_\omega(z)=\frac{1}{\omega^\frac12 (z)\|K_z\|_\omega}. \] The class of weights \(\omega\) satisfying that \(\tau_\omega(z)=O(1-|z|)\), \(z\in\mathbb{D},\) and that there exists \(\eta>0\) such that for \(|z-\zeta|\leq\eta\tau_\omega(z)\), \[ \tau_\omega(z) \simeq \tau_\omega (\zeta)\quad\text{and } \|K_z\|_\omega\|K_\zeta\|_\omega \lesssim |K(\zeta, z)|, \] is denoted by \(\mathcal{W}\). If \(\omega =e^{-\varphi}\), the subclass of \(\mathcal{W}\) of weights for which \(\tau_\omega^2(z) \nabla\varphi(z)\gtrsim 1\), \(z\in \mathbb{D},\) is denoted by \(\mathcal{W}^*\). The goal of this paper is to study the asymptotic behavior of the singular values \((s_n(H_{\overline{\Phi}}))_n\) of the Hankel operator \(H_{\overline{\Phi}}\), extending analogous results for the classical and weighted Bergman spaces. Theorem. Let \(\omega\in \mathcal{W}^*\) such that \(\tau_\omega^2(z)\simeq (1-|z|)^{2+\beta}\), \(\beta\geq 0\). Let \(p=\frac{2(1+\beta)}{2+\beta}\) and let \(\Phi\in \mathcal{B}_0^\omega\). Then \[ s_n(H_{\overline{\Phi}})=(n^{-1/p})\iff \Phi'\in H^p. \] In addition, if \(\omega\) is a radial weight and \(\lim_n \frac{s_n(H_{\overline{z}})}{\gamma n^{-1/p}}=1\), for some \(\gamma>0\), then \[ \lim_n \frac{s_n(H_{\overline{\Phi}})}{\|\Phi'\|_{H^p}\gamma n^{-1/p}}=1,\quad \Phi'\in H^p. \] Here, \(\mathcal{B}_0^\omega\) is the class of analytic functions such that \(\lim_{|z|\to 1}\tau_\omega(z) |\Phi'(z)|=0\), and \(H^p\) is the Hardy space on the unit disc.