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Introduction to Coercive Inequalties |
Prof. B Zegarlinski |
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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 : |
Bibliography Lecture Notes Guionnet A, Zegarlinski B, Lectures on logarithmic Sobolev inequalities, Lecture Notes Math, 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.
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