Schatten class Hankel operators on the harmonic Bergman space of the unit ball (Q2464662)

Let \(B_n\) denote the open unit ball in \({\mathbb R}^n\) and let \(V\) denote the normalized Lebesgue measure on \(B_n\). The harmonic Bergman space \(L_h^2(B_n)\) is the set of all harmonic functions on \(B_n\) that are also in \(L^2(B_n)\). The present paper deals with the characterization of Hankel operators \(H_f\) in the Schatten \(p\)-class \(S_p\) on the space \(L_h^2(B_n)\). For \(r\in(0,1)\) and \(x\in B_n\), let \(B_r(x)=\{y\in B_n:| y-x| <r(1-| x| )\}\) and \[ MO_r(f,x)=\frac{1}{V(B_r(x))}\int_{B_r(x)}\left| f(y)-\frac{1}{V(B_r(x))}f(z)dV(z)\right| ^2dV(y). \] The following two results are obtained. Theorem 1. Suppose that \(f\in L^2(B_n)\), \(2\leq p<\infty\), and \(0<r<1\). Then \(H_f\in S_p\) if and only if \(\int_{B_n}{MO_r(f,x)^{p/2}\over (1-| x| )^n}\,dV(x)<\infty\). Theorem 2. Suppose that \(f\in L_h^2(B_n)\) and \(n\geq 3\). Then (1) for \(0<p\leq n-1\), \(H_f\in S_p\) if and only if \(f\) is constant; (2) for \(n-1<p<\infty\), \(H_f\in S_p\) if and only if \(\int_{B_n}{| \nabla f(x)| ^p(1-| x| )^p\over (1-| x| )^n}\,dV(x)<\infty\).