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

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.

### MSC:

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

### Citations:

Zbl 1158.32001; Zbl 1124.47018
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