Schatten classes of generalized Hilbert operators

Let Dv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}_v$$\end{document} denote the Dirichlet type space in the unit disc induced by a radial weight v for which v^(r)=∫r1v(s)ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{v}(r)=\int _r^1 v(s)\,\text {d}s$$\end{document} satisfies the doubling property ∫r1v(s)ds≤C∫1+r21v(s)ds.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _r^1 v(s)\,\text {d}s\le C \int _{\frac{1+r}{2}}^1 v(s)\,\text {d}s.$$\end{document} In this paper, we characterize the Schatten classes Sp(Dv)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_p(\mathcal {D}_v)$$\end{document} of the generalized Hilbert operators Hg(f)(z)=∫01f(t)g′(tz)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {H}_g(f)(z)=\int _0^1f(t)g'(tz)\,\text {d}t \end{aligned}$$\end{document}acting on Dv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}_v$$\end{document}, where v satisfies certain Muckenhoupt type conditions. For p≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 1$$\end{document}, it is proved that Hg∈Sp(Dv)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}_{g}\in S_p(\mathcal {D}_v)$$\end{document} if and only if ∫01(1-r)∫-ππ|g′(reiθ)|2dθp2dr1-r<∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^1 \left( (1-r)\int _{-\pi }^\pi |g'(r\text {e}^{i\theta })|^2\,\text {d}\theta \right) ^{\frac{p}{2}}\frac{{\text {d}}r}{1-r} <\infty . \end{aligned}$$\end{document}.