Functional inequalities for uniformly integrable semigroups and application to essential spectrums

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Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry (2004, background): “Their motivation was a criterion for the generator to have a non-empty essential spectrum (see [30,58] for connections with super-Poincaré inequalities).”
Functional Inequalities and Spectrum Estimates: The Infinite Measure Case (2002, background): “1) is a generalization to the known Poincaré–Sobolev-type inequalities (see [6, 13, 14]) as well as the Nash-type ones (see e.”
Spectral gap of positive operators and applications (2000, background): “Proof. By Lemma 4.1 and Proposition 4.2 in [32] or see Theorem 1.2 in [ 9 ], condition (i) is equivalent to the uniform integrability of Pt , t> 0, in L2(µ) (i.e., Pt tail = 0 for all”