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

翻译：本文通过将熵替换为双参数熵（一种与Rényi散度密切相关的广义熵），将log-Sobolev不等式推广至Rényi–Sobolev不等式。我们推导了此类不等式的严格非线性无维版本。有趣的是，所得不等式根据参数呈现出相变现象。随后，我们建立了Rényi–Sobolev不等式与谱图理论之间的联系。本文的证明基于Rényi–Sobolev不等式的信息论刻画以及类型方法。