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In this paper, we generalize the log-Sobolev inequalities to R\'enyi--Sobolevinequalities by replacing the entropy with the two-parameter entropy, which isa generalized version of entropy closely related to R\'enyi divergences. Wederive the sharp nonlinear dimension-free version of this kind of inequalities.Interestingly, the resultant inequalities show a phase transition depending onthe parameters. We then connect R\'enyi--Sobolev inequalities to the spectralgraph theory. Our proofs in this paper are based on the information-theoreticcharacterization of the R\'enyi--Sobolev inequalities, as well as the method oftypes.

R\'enyi--Sobolev Inequalities and Connections to Spectral Graph Theory