Schatten class Toeplitz operators acting on large weighted Bergman spaces (Q2787135)

For a certain class of rapidly decreasing weights \(\omega\) on the unit disk \(\mathbb{D}\) in \(\mathbb{C}\), the authors give a complete description of the membership in the Schatten ideal \(S_p(A^2_\omega)\) of Toeplitz operators with positive measure symbols acting on the weighted Bergman space \(A^2_\omega\). Let \(\omega(z)=e^{-2\varphi(z)}\) be a radial decreasing weight, where \(\varphi\in C^2(\mathbb{D})\) is a radial function such that \(\bigl(\Delta\varphi(z)\bigr)^{-1/2}\approx \tau(|z|)\), for some positive function \(\tau\) satisfying: {\parindent= 0.5 cm \begin{itemize}\item[1)] \(\tau(r)\) decreases to 0 as \(r\to 1^{-}\), \item[2)] \(\lim_{r\to 1^{-}}\tau'(r)=0\), \item[3)] either there exists \(c>0\) such that \(\tau(r)(1-r)^{-c}\) increases as \(r\to 1^{-}\), or \(\lim_{r\to 1^{-}}\tau'(r)\,\log \tau(r)=0\). NEWLINENEWLINE\end{itemize}} Examples of weights satisfying these conditions are NEWLINE\[NEWLINE \omega(z)=\exp(-\gamma \exp(\beta(1-|z|^2)^{-\alpha})),\,\,\alpha,\beta,\gamma>0. NEWLINE\]NEWLINENEWLINENEWLINELet \(\mu\) be a finite positive Borel measure on \(\mathbb{D}\). The Toeplitz operator with symbol \(\mu\) is defined by NEWLINE\[NEWLINE T^\omega_\mu(f)(z)=\int_{\mathbb{D}} f(w)K_\omega(z,w)\omega (w)d\mu(w), NEWLINE\]NEWLINE where \(K_\omega(z,w)\) is the Bergman kernel for the Hilbert space \(A^2_\omega\).NEWLINENEWLINEThe main result of this article is the following:NEWLINENEWLINEFor \(0<p<\infty\), \(T^\omega_\mu\in S_p(A^2(\omega))\) if and only if for any \(\delta>0\) small enough, the averaging function NEWLINE\[NEWLINE \hat\mu_\delta(z)=\frac{\mu\{w\in\mathbb{D}:|w-z|\leq\delta \tau(|z|)\}}{\tau(|z|)^2} NEWLINE\]NEWLINE is in \(L^p\bigl(\mathbb{D}, \tau(|z|)^{-2}dA(z)\bigr)\).NEWLINENEWLINEThis result solves a problem posed in [\textit{P. Lin} and \textit{R. Rochberg}, Pacific J. Math., No. 173, 127--146 (1996; Zbl 0853.47015)], where the authors studied the properties of \(T^\omega_\mu\) acting on Bergman spaces with exponential-type weights.NEWLINENEWLINEThe authors also prove that \(T^\omega_\mu\) is bounded (resp. compact) on \(A^2_\omega\) if and only if for each \(\delta>0\) small enough, \(\hat\mu_\delta\) is a bounded function (resp. \( \lim_{r\to 1^{-}}\sup_{|z|>r}\hat\mu_\delta(z)=0\)).