Modified log-Sobolev inequalities and isoperimetry

  • Alexander V. Kolesnikov

    Moscow State University, Russian Federation


We find sufficient conditions for a probability measure μ\mu to satisfy an inequality of the type

Rdf2F(f2Rdf2dμ)dμCRdf2c(ff)dμ+BRdf2dμ,\int_{\R^d} f^2 F\Bigl( \frac{f^2}{\int_{\R^d} f^2 \,d \mu} \Bigr) \,d \mu \le C \int_{\R^d} f^2 c^{*}\Bigl( \frac{|\nabla f|}{|f|} \Bigr) \,d \mu + B \int_{\R^d} f^2 \,d \mu,

where FF is concave and cc (a~cost function) is convex. We show that under broad assumptions on cc and FF the above inequality holds if for some δ>0\delta>0 and ε>0\varepsilon>0 one has

0εΦ(δc[tF(1t)Iμ(t)])dt<,\int_{0}^{\varepsilon} \Phi\Bigl(\delta c\Bigl[\frac{t F(\frac{1}{t})}{{\mathcal I}_{\mu}(t)} \Bigr] \Bigr) \,dt < \infty,

where Iμ{\mathcal I}_{\mu} is the isoperimetric function of μ\mu and Φ=(yF(y)y)\Phi = (y F(y) -y)^{*}. In a partial case

Iμ(t)ktφ11α(1/t),{\mathcal I}_{\mu}(t) \ge k t \varphi ^{1-\frac{1}{\alpha}} (1/t),

where φ\varphi is a concave function growing not faster than log\log, k>0k>0, 1<α21 < \alpha \le 2 and t1/2t \le 1/2, we establish a family of tight inequalities interpolating between the FF-Sobolev and modified inequalities of log-Sobolev type. A basic example is given by convex measures satisfying certain integrability assumptions.

Cite this article

Alexander V. Kolesnikov, Modified log-Sobolev inequalities and isoperimetry. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 2, pp. 179–208

DOI 10.4171/RLM/489