Positive Schatten(-Herz) class Toeplitz operators on pluriharmonic Bergman spaces. (English) Zbl 1189.47029

Na, Kyunguk

Integral Equations Oper. Theory 64, No. 3, 409-428 (2009).

Let \(B\) be the open unit ball in \({\mathbb C}^n\) and \(b_\alpha^2\) be the weighted pluriharmonic Bergman space of all pluriharmonic functions on \(B\) such that \(\|f\|_2=(\int_B|f|^2 \,dV_\alpha)^{1/2}<\infty\), where \(\alpha>-1\), \(dV_\alpha(z)=(1-|z|^2)^\alpha\, dV(z)\), and \(V\) denotes the Lebesgue volume measure on \(B\). Further, let \(L^p(\lambda)\) and \({\mathcal K}_q^p(\lambda)\) be the Lebesgue and Herz spaces with respect to the measure \(d\lambda(z)=R_z^0(z) \,dV(z)\), where \(R_z^\alpha(w)=R^\alpha(z,w)\) is the reproducing kernel for \(b_\alpha^2\). For a positive Borel measure \(\mu\) on \(B\), define the Berezin transform \(\widetilde{\mu}\) by \(\widetilde{\mu}(z)=\int_B |R_z^\alpha(w)|/\|R_z^\alpha\|_2\, d\mu(w)\). Let \(E_r(z)\) be the pseudohyperbolic ball centered at \(z\in B\) and of radius \(r\in(0,1)\). Consider the averaging function \(\widehat{\mu}_r\) defined by \(\widehat{\mu}_r(z)=\mu[E_r(z)]/V_\alpha[E_r(z)]\). By \(T_\mu\) denote the Toeplitz operator generated by a positive Borel measure \(\mu\) on \(B\). Let \(S_p\) be the Schatten class, by \(S_{p,q}\) denote the Schatten-Herz class of Toeplitz operators.
The main results of the paper are the following. Theorem 1. Let \(0<p<1\), \(\alpha>-1\), and \(r\in(0,1)\). The following conditions are equivalent: (a) \(T_\mu\in S_p\); (b) \(\widehat{\mu}_r\in L^p(\lambda)\). Moreover, if \(n/(n+\alpha+1)<p\), then the above statements are also equivalent to (c) \(\widetilde{\mu}\in L^p(\lambda)\). Theorem 2. Let \(0<p\leq\infty\), \(0\leq q\leq\infty\), \(\alpha>-1\), and \(r\in(0,1)\). The following conditions are equivalent: (a) \(T_\mu\in S_{p,q}\); (b) \(\widehat{\mu}_r\in {\mathcal K}_q^p(\lambda)\). Moreover, if \(n/(n+\alpha+1)<p\leq\infty\), then the above statements are also equivalent to (c) \(\widetilde{\mu}\in {\mathcal K}_q^p(\lambda)\).
For the case \(1\leq p<\infty\) and \(\alpha=0\), these results were obtained by E. S. Choi [“Positive Toeplitz operators on pluriharmonic Bergman spaces”, J. Math. Kyoto Univ. 47, No. 2, 247–267 (2007; Zbl 1158.32001)] and E. S. Choi and K. Na [“Schatten-Herz type positive Toeplitz operators on pluriharmonic Bergman spaces”, J. Math. Anal. Appl. 327, No. 1, 679–694 (2007; Zbl 1124.47018)], respectively.

Reviewer: Alexei Yu. Karlovich (Lisboa)

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions

Keywords:

Toeplitz operator; Schatten class; Schatten-Herz class; Herz space; pluriharmonic Bergman space; Berezin transfom; reproducing kernel

Citations:

Zbl 1158.32001; Zbl 1124.47018