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Introduction to Coercive Inequalties



Prof. B Zegarlinski



Imperial HXLY 6M42

Friday's 10 am-12noon (October 13 – December 8)



The Course Content:



    I. Rapid Review of Measure & Integration Theory, L_p and Orlicz Spaces, Convexity of L_p norms, Differentiablity of the norms, Holder and Minkowski inequality,

    II. Weak Differentiation and Dirichlet Forms, Sobolev Inequality (CSI), Gagliardo-Niremberg Approach, Nash inequality (NI), Ultracontractivity of heat semigroup, Equivalence of (CSI) & (NI).

    III. Logarithmic Sobolev Inequality (LSI): Product property, Poincare Inequality, Exponential Bounds, Perturbation Lemma, Hypercontractivity,

    IV. How to prove Coercive inequalities for Probability Measures: Bakry-Emery criterion and beyond.



Course Material :

Lec1 ; Lec2 ; Lec3 ; Lec4 ; Lec5 ; Lec6 ; Lec7 ; Lec8 ; Lec9 ; Lec10

PS.1; PS.2; PS.3; PS.4; PS.5; PS.6 ; PS.7; PS.8; PS.9 ; PS.10;

ExampleExam

Bibliography

Lecture Notes

Guionnet A, Zegarlinski B, Lectures on logarithmic Sobolev inequalities, Lecture Notes Mmath, 2003, Vol:1801, Pages:1-134,

Sur Les Inegalites de Sobolev Logarithmiques - S. Blanchere, D. Chafai, P. Fougeres, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer, Societe Mathematique de France, 2000.

Books

AH Lieb & M Loss, Analysis, Graduate Studies in Math Vol 14, AMS 1997

WP Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, Springer-Verlag

EB Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics vol 92

D Bakry, I Gentil & M Ledoux, Analysis and Geometry of Markov Diffusion Operators , Grundlehren der Mathematischen Wissenschaften vol 348, Springer 2014

MM Res & ZD Rao, Theory of Orlicz Spaces, Pure and Applied Mathematics, Marcel Dekker, Inc.

Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford University Press, Clarendon Press, 1985.