Schatten classes of generalized Hilbert operators

Abstract

Let Dv denote the Dirichlet type space in the unit disc induced by a radial weight v for which vˆ(r)=∫1rv(s)ds satisfies the doubling property ∫1rv(s)ds≤C∫11+r2v(s)ds. In this paper, we characterize the Schatten classes Sp(Dv) of the generalized Hilbert operators

Hg(f)(z)=∫10f(t)g′(tz)dt acting on Dv, where v satisfies certain Muckenhoupt type conditions. For p≥1, it is proved that Hg∈Sp(Dv) if and only if

∫10((1−r)∫π−π|g′(reiθ)|2dθ)p2dr1−r<∞.