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 $\widehat{v}(r)=\int_r^1 v(s)\,ds$ 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 $$ \sup_{0<r<1}\left(\int_r^1 \frac{\widehat{v}(s)}{(1-s)^2} \,ds\right)^{1/2} \left(\int_0^r \frac{1}{\widehat{v}(s)} \,ds\right)^{1/2}<\infty $$ and $$\sup_{0< r<1}\left(\int_{0}^r \frac{\widehat{v}(s)}{(1-s)^4}\,ds\right)^{\frac{1}{2}} \left(\int_{r}^1\frac{(1-s)^2}{\widehat{v}(s)}\,ds\right)^\frac{1}{2}<\infty. $$ 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_{-\pi}^\pi |g'(re^{i\theta})|^2\,d\theta\right)^{\frac{p}{2}}\frac{dr}{1-r} <\infty. \end{equation*}