Cwikel-Solomyak estimates on tori and Euclidean spaces

Abstract

We revise Cwikel-type estimate for the singular values of the operator $(1-{\Delta }_{{\mathbb{T}}^d}{)}^{-\frac{d}{4}}{M}_f(1-{\Delta }_{{\mathbb{T}}^d}{)}^{-\frac{d}{4}}$ on the torus ${\mathbb{T}}^d$ for the ideal ${\mathcal{L}}_{1,\infty }$, established by M.Z. Solomyak in even dimensions in \cite{Solomyak1995}, and extend it to odd dimensions. We obtain a new result for (symmetrized) Cwikel type estimates for Laplacians on ${\mathbb{R}}^d$ for arbitrary positive integer $d.$

References in corpus (2)

• Traces of compact operators and the noncommutative residue
• Measures from Dixmier Traces and Zeta Functions

Cited by in corpus (4)

• Weyl's laws and Connes' integration formulas for matrix-valued $L\log L$-Orlicz potentials
• Connes Integration Formula without singular traces
• Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori
• Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori