## Bernoulli

• Bernoulli
• Volume 24, Number 1 (2018), 333-353.

### Functional inequalities for Gaussian convolutions of compactly supported measures: Explicit bounds and dimension dependence

#### Abstract

The aim of this paper is to establish various functional inequalities for the convolution of a compactly supported measure and a standard Gaussian distribution on $\mathbb{R}^{d}$. We especially focus on getting good dependence of the constants on the dimension. We prove that the Poincaré inequality holds with a dimension-free bound. For the logarithmic Sobolev inequality, we improve the best known results (Zimmermann, JFA 2013) by getting a bound that grows linearly with the dimension. We also establish transport-entropy inequalities for various transport costs.

#### Article information

Source
Bernoulli, Volume 24, Number 1 (2018), 333-353.

Dates
Revised: June 2016
First available in Project Euclid: 27 July 2017

https://projecteuclid.org/euclid.bj/1501142446

Digital Object Identifier
doi:10.3150/16-BEJ879

Mathematical Reviews number (MathSciNet)
MR3706760

Zentralblatt MATH identifier
06778331

#### Citation

Bardet, Jean-Baptiste; Gozlan, Nathaël; Malrieu, Florent; Zitt, Pierre-André. Functional inequalities for Gaussian convolutions of compactly supported measures: Explicit bounds and dimension dependence. Bernoulli 24 (2018), no. 1, 333--353. doi:10.3150/16-BEJ879. https://projecteuclid.org/euclid.bj/1501142446

#### References

• [1] Aida, S. (1998). Uniform positivity improving property, Sobolev inequalities, and spectral gaps. J. Funct. Anal. 158 152–185.
• [2] Aida, S. and Shigekawa, I. (1994). Logarithmic Sobolev inequalities and spectral gaps: Perturbation theory. J. Funct. Anal. 126 448–475.
• [3] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les Inégalités de Sobolev Logarithmiques. Panoramas et Synthèses [Panoramas and Syntheses] 10. Paris: Société Mathématique de France. With a preface by Dominique Bakry and Michel Ledoux.
• [4] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Cham: Springer.
• [5] Bobkov, S.G., Gentil, I. and Ledoux, M. (2001). Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 80 669–696.
• [6] Bobkov, S.G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1–28.
• [7] Cattiaux, P., Guillin, A. and Wu, L.-M. (2010). A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality. Probab. Theory Related Fields 148 285–304.
• [8] Chafaï, D. and Malrieu, F. (2010). On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 46 72–96.
• [9] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.
• [10] Gozlan, N. (2010). Poincaré inequalities and dimension free concentration of measure. Ann. Inst. Henri Poincaré Probab. Stat. 46 708–739.
• [11] Gozlan, N., Roberto, C., Samson, P.-M. and Tetali, P. (2014). Kantorovich duality for general transport costs and applications. Preprint. Available at arXiv:1412.7480.
• [12] Houdré, C. (2001). Mixed and isoperimetric estimates on the log-Sobolev constants of graphs and Markov chains. Combinatorica 21 489–513.
• [13] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Providence, RI: Amer. Math. Soc.
• [14] Ledoux, M. (2001). Logarithmic Sobolev inequalities for unbounded spin systems revisited. In Séminaire de Probabilités XXXV. Lecture Notes in Math. 1755 167–194. Berlin: Springer.
• [15] Maurey, B. (1991). Some deviation inequalities. Geom. Funct. Anal. 1 188–197.
• [16] Menz, G. and Schlichting, A. (2014). Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape. Ann. Probab. 42 1809–1884.
• [17] Milman, E. (2010). Isoperimetric and concentration inequalities: Equivalence under curvature lower bound. Duke Math. J. 154 207–239.
• [18] Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361–400.
• [19] Royer, G. (2007). An Initiation to Logarithmic Sobolev Inequalities. SMF/AMS Texts and Monographs 14. Providence, RI: Amer. Math. Soc.; Paris: Société Mathématique de France. Translated from the 1999 French original by Donald Babbitt.
• [20] Schlichting, A. (2012). The Eyring–Kramers formula for Poincaré and logarithmic Sobolev inequalities Ph.D. thesis, Leipzig University.
• [21] Wang, F.-Y. and Wang, J. (2016). Functional inequalities for convolution probability measures. Ann. Inst. Henri Poincaré Probab. Stat. 52 898–914.
• [22] Zimmermann, D. (2013). Logarithmic Sobolev inequalities for mollified compactly supported measures. J. Funct. Anal. 265 1064–1083.
• [23] Zimmermann, D. (2014). Bounds for logarithmic Sobolev constants for Gaussian convolutions of compactly supported measures. Preprint. Available at arxiv:1405.2581.
• [24] Zimmermann, D. (2016). Elementary proof of logarithmic Sobolev inequalities for Gaussian convolutions on $\mathbb{R}$. Ann. Math. Blaise Pascal 23 129–140.