Around a Sobolev-Orlicz inequality for operators of given spectral density

TL;DR

This paper establishes general Sobolev-Orlicz, Nash, and Faber-Krahn inequalities for positive operators with specific spectral decay, linking spectral density to geometric and cohomological properties of coverings.

Contribution

It introduces a novel connection between spectral decay rates and Sobolev inequalities, extending the understanding of spectral density functions in geometric analysis.

Findings

Spectral decay characterizes ultracontractive norms of spectral projectors.

Polynomial decay relates Novikov-Shubin invariants to Sobolev inequalities.

Vanishing torsion in cohomology is linked to spectral properties of operators.

Abstract

We prove some general Sobolev-Orlicz, Nash and Faber-Krahn inequalities for positive operators of given ultracontractive norms of the spectral projectors on ]0, lambda]. For invariant operators on coverings of finite simplicial complexes this "ultracontractive spectral decay" is equivalent to von-Neumann's spectral density function. This allows in the polynomial decay case to relate the Novikov-Shubin numbers of such coverings to Sobolev inequalities on exact $\ell^2$-cochains, and to the vanishing of the torsion of the $\ell^{p,2}$-cohomology for some $p \geq 2$.