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We prove some general Sobolev-Orlicz, Nash and Faber-Krahn inequalities forpositive operators of given ultracontractive norms of the spectral projectorson ]0, lambda]. For invariant operators on coverings of finite simplicialcomplexes this "ultracontractive spectral decay" is equivalent to von-Neumann'sspectral density function. This allows in the polynomial decay case to relatethe 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 $p \geq 2$.

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