Integral operators in the Schatten class on Dirichlet spaces

Abstract

We characterize three integral operators in Schatten p-classes on Dirichlet spaces $D_{\alpha }$ in the unit disk $D$ for $\alpha >0$ and $0<p<\infty$. The main results are threefold:

• (1)
  If $0<\alpha <1<p<\infty$ and g is a holomorphic function in $D$, then the Volterra operator $T_g$, defined by$T_{g}f\left(z\right)=\underset{0}{\overset{z}{\int }}f\left(\zeta \right)g^{\prime }\left(\zeta \right)d\zeta ,$ is in the Schatten p-class on $D_{\alpha }$ if and only if$\underset{D}{\int }{\left({(1-|w{|}^2)}^{\alpha }\underset{D}{\int }\frac{\left|g^{\prime }\right(z\left){|}^{2}d{A}_{\alpha }\right(z)}{|1-\overline{w}z{|}^{2+2\alpha }}\right)}^{\frac{p}{2}}{(1-|w{|}^2)}^{p-2}dA\left(w\right)<\infty .$
• (2)
  If $\alpha >0$ and $0<p<\infty$, μ is a finite Borel measure on $D$, then the Toeplitz operator $Q_{\mu }^{\alpha }$ acting on $D_{\alpha }$ is in the Schatten p-class if and only if$\underset{D}{\int }{\left({(1-|w{|}^2)}^{\alpha +2t}\underset{D}{\int }\frac{d\mu \left(z\right)}{|1-\overline{w}z{|}^{2\alpha +2t}}\right)}^p{(1-|w{|}^2)}^{-2}dA\left(w\right)<\infty ,$ where t is any (or some) nonnegative number such that $\alpha +2t>\max \{1,\frac{1}{p}\}$.
• (3)
  If g is a holomorphic function in $D$, $\alpha >0$ and $0<p\le 1$, then the small Hankel operator $h_g^{\alpha }$ acting from $D_{\alpha }$ to the Sobolev space $L_{\alpha }^2$ is in the Schatten p-class if and only if g belongs to the Besov space $B_p$.

These results answer the corresponding open problems left by Pau-Peláez [12] (J. Anal. Math. 120 (2013), 255-289).