Toeplitz operators with uniformly continuous symbols
Toeplitz operators \(T_f\) on Bergman spaces of bounded symmetric domains (BSD) \(\Omega\) in complex space \(\mathbb{C}^n\) or on the Segal-Bargmann space of Gaussian square-integrable complex functions on \(\mathbb{C}^n\) are studied. The symbols are uniformly continuous (UC) functions with respect to the Bergman (hyperbolic) distance function \(\beta(\cdot , \cdot)\) on \(\Omega\) or with respect to the usual Euclidean distance \(d(\cdot , \cdot)\) on \(\mathbb{C}^n\). For \(f\) in UC, the authors prove in Theorem 1.1 that the operator \(T_f\) is bounded iff \(f\) is bounded; \(T_f\) is compact iff \(f\) vanishes at the boundary \(\partial\Omega\) (or at \(\infty\) for \(\mathbb{C}^n\)). The Bergman metric induced by the kernel \(K\) up to a factor coincides with the Euclidean metric \(d(z,w) := |z-w|\) on \(\mathbb{C}^n\). The authors write \(\mathrm{Lip}(\mathbb{C}^n)\) and \(\mathrm{UC}(\mathbb{C}^n)\) for the space of all Lipschitz continuous and uniformly continuous functions with respect to \(d\). Then \(\mathrm{Lip}(\mathbb{C}^n)\subset\mathrm{UC}(\mathbb{C}^n)\). Let \(f \in\mathrm{UC}(\mathbb{C}^n)\). They write \(\mathcal{T}{\mathbb{C}^n}\) for the ``symbol space'' of all functions \(f : \mathbb{C}^n \rightarrow \mathbb{C}\) satisfying \(f(w- \cdot) \in L^2(\mathbb{C}^n, d\mu)\) for all \(w \in \mathbb{C}^n\). For a given function \(f \in \mathcal{T}{\mathbb{C}^n}\) and \(t \in (0, 1/4]\), they define the heat transform of \(f\) at time \(t\) as \[ \tilde{f}^{(t)}(w) := \frac{1}{(4\pi t)^n}\int_{\mathbb{C}^n} f(w-z)e^{-\frac{|z|^2}{4t}} dv(z). \] Then the authors prove in Theorem 2.1 that \(\lim_{t \downarrow 0} \tilde{f}^{(t)} = f\) uniformly on \(\mathbb{C}^n\). The mean oscillation of \(f\) is denoted by \(\mathrm{MO}(f,w)\). The authors write \(\mathrm{BMO}^2(\mathbb{C}^n)\) for the space of all functions of bounded mean oscillation \(f \in \mathcal{T}{\mathbb{C}^n}\) satisfying \(\| f\|_{\mathrm{BMO}} := \sup_{z \in \mathbb{C}^n} \mathrm{MO}(f,z)^{1/2} < \infty\). Let \(f \in\mathrm{BMO}^2(\mathbb{C}^n)\). Then the authors prove in Theorem 2.2 that (i) if \(T_f\) is bounded, then the heat transform \(\tilde{f}^{(t)}\) is bounded for all \(t > 0\), and (ii) if \(T_f\) is compact, then \(\tilde{f}^{(t)} \in C_0(\mathbb{C}^n)\) for all \(t > 0\). Let \(f \in \mathrm{UC}(\mathbb{C}^n)\). Then the authors prove in Theorem 2.3 that (i) \(T_f\) is bounded if and only if \(f\) is bounded on \(\mathbb{C}^n\), (ii) \(T_f\) is compact if and only if \(f \in C_0 (\mathbb{C}^n)\). Let \(g \in L^1(\Omega, dv)\) and \(\lambda\geq p\), then it is shown that the integral \[ B_\lambda (g)(w) := c_\lambda \int_\Omega g\circ \phi_w(z)h(z, z)^{\lambda -p}dv(z) \] exists and defines a real analytic function on \(\Omega\), which is called a Berezin-Harish-Chandra flow. Let \(\Omega \subset \mathbb{C}^n\) be a BSD of genus \(p\) and let \(f \in\mathrm{UC}(\Omega)\). Then the authors prove in Theorem 3.3 that the integral transforms \(\{B_\lambda (f)\}_{\lambda\geq p}\) are contained in \(\mathrm{Lip}(\Omega)\) with \(\lim_{\lambda \rightarrow \infty} B_\lambda(f) = f\), where the convergence is uniform on \(\Omega\). Let \(\lambda > p-1\) and fix \(g \in L^2(\Omega, d\nu_\lambda)\). Then the authors prove in Theorem 3.6 that there is a constant \(C(r, a, b) > 0\) depending on the type \((r, a, b)\) of \(\Omega\) such that for all \(\lambda_0 > C(r, a, b)\) the estimate \(\| B_{\lambda_0}(g)\|_\infty \leq C(\lambda, \lambda_0)\| T_g\|\) holds, where \(C(\lambda, \lambda_0)\) is independent of \(g\). In particular, if \(T_g\) is bounded, then \(B_{\lambda_0}(g)\) is a bounded function for sufficiently large weight \(\lambda_0 > 0\). Let \(f \in\mathrm{UC}(\Omega)\). Then on the weighted Bergman space \(A^2_\lambda(\Omega)\) with \(\lambda > p-1\) the authors prove in Theorem 3.8 that (i) \(T_f\) is bounded if and only if \(f\) is bounded on \(\Omega\), (ii) \(T_f\) is compact if and only if \(f \in C_0(\Omega)\).
scientific article; zbMATH DE number 6497570