On the Schatten class membership of Hankel operators on the unit ball
Let \(L^2_a(B_n, dV)\) denote the Bergman space which is the subspace in \(L^2(B_n, dV)\) consisting of analytic functions on the unit ball \(B_n\subset \mathbb{C}^n\) and let \(H_f =(I-P)f P\) be the Hankel operator with symbol \(f\) where \(P:L^2(B_n, dV) \to L^2_a(B_n, dV)\) is the projection. For \(f\in L^2_a(B_n, dV)\), denote by \(\widetilde f\) its Berezin transform \(\widetilde f (z)=\int f | k_z| ^2 \,dV\) where the kernel function \(k_z\) has the form \(k_z (\zeta)=(1-| z| ^2)^{(n+1)/2}/(1-\langle z,\zeta\rangle)^{n+1}\), \(\langle z,\zeta\rangle= z_1 \overline \zeta_1+\cdots+ z_n \overline \zeta_n \). Finally, by \({\mathcal C}_p\), \(1\leq p <\infty\), denote the Schatten class, by \(MO(f)(z)\) the mean oscillation \([MO(f)(z)]^2=\int | f-\widetilde f (z)| ^2| k_z| ^2 \,dV\) and let \(d\lambda\) denote the Möbius-invariant measure on \(B_n\). The main result of the present paper contains the following Theorem. Let \(2n/(n+1)<p<2\) and \(f\in L^2(B_n, \,dV)\). Then \(H_f\in {\mathcal C}_p\) and \(H_{\overline f}\in {\mathcal C}_p\) if and only if \(MO(f)\in L^p(B_n, d\lambda)\). This theorem is an extension of the same assertion which was earlier proved by K. Zhu for \(2\leq p<\infty\), and as it is noted in the paper, it cannot be extended to the case \(p\leq 2n/(n+1).\)
scientific article; zbMATH DE number 1865540