Weyl’s Laws and Connes’ Integration Formulas for Matrix-Valued LlogL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\!\log \!L$$\end{document}-Orlicz Potentials

Thanks to the Birman-Schwinger principle, Weyl’s laws for Birman-Schwinger operators yields semiclassical Weyl’s laws for the corresponding Schrödinger operators. In a recent preprint Rozenblum established quite general Weyl’s laws for Birman-Schwinger operators associated with pseudodifferential operators of critical order and potentials that are product of LlogL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\!\log \!L$$\end{document}-Orlicz functions and Alfhors-regular measures supported on a submanifold. In this paper, we show that, for matrix-valued LlogL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\!\log \!L$$\end{document}-Orlicz potentials supported on the whole manifold, Rozenblum’s results are direct consequences of the Cwikel-type estimates on tori recently established by Sukochev–Zanin. As applications we obtain CLR-type inequalities and semiclassical Weyl’s laws for critical Schrödinger operators associated with matrix-valued LlogL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\!\log \!L$$\end{document}-Orlicz potentials. Finally, we explain how the Weyl’s laws of this paper imply a strong version of Connes’ integration formula for matrix-valued LlogL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\!\log \!L$$\end{document}-Orlicz potentials.