Schatten classes of generalized Hilbert operators

Abstract

Let ${\mathcal{D}}_v$ denote the Dirichlet type space in the unit disc induced by a radial weight $v$ for which $\hat{v}(r)={\int }_r^{1}v(s)ds$ satisfies the doubling property ${\int }_r^{1}v(s)ds\le C{\int }_{\frac{1+r}{2}}^{1}v(s)ds.$ In this paper, we characterize the Schatten classes $S_p({\mathcal{D}}_v)$ of the generalized Hilbert operators \begin{equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt \end{equation*} acting on ${\mathcal{D}}_v$, where $v$ satisfies the Muckenhoupt-type conditions $$\underset{0<r<1}{\sup }{\left({\int }_r^1\frac{\hat{v}(s)}{(1-s{)}^2}ds\right)}^{1/2}{\left({\int }_0^r\frac{1}{\hat{v}(s)}ds\right)}^{1/2}<\infty$$ and $$\underset{0<r<1}{\sup }{\left({\int }_0^r\frac{\hat{v}(s)}{(1-s{)}^4}ds\right)}^{\frac{1}{2}}{\left({\int }_r^1\frac{(1-s{)}^2}{\hat{v}(s)}ds\right)}^{\frac{1}{2}}<\infty .$$ For $p\ge 1$, it is proved that ${\mathcal{H}}_g\in S_p({\mathcal{D}}_v)$ if and only if \begin{equation*} \int_0^1 \left((1-r)\int_{-π}^π|g'(re^{iθ})|^2\,dθ\right)^{\frac{p}{2}}\frac{dr}{1-r} <\infty. \end{equation*}