An Information Theoretic View of Hypercontractivity

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" summary

Provides an entropy-based proof of Beckner’s hypercontractive inequality (primal) and revisits a 2001 entropy proof of the dual that yields a 28^(1/4) base instead of the optimal sqrt(3). Defines T_epsilon via noisy correlated pairs and eigenvalues epsilon^d, partitions the ground set into seven regions to count quadruples, uses a fractional Shira's lemma with t=2 to compare fourth and second moments, and constructs a discrete random variable z (integers r,s with ratios (1±epsilon)/2) so that bitwise chain-rule entropy plus a Lagrange-multiplier calculus step gives the inequality.

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" chapters (10)

1. 0:00Intro
2. 1:23Overview
3. 1:58Applications
4. 3:02The Fourier-Walsh Basis
5. 4:33The tensor definition of the operator
6. 6:03The spectral definition of the operator
7. 9:12The inequality and its dual
8. 18:47Plan of proof
9. 28:00Theorems
10. 32:21The Boolean case

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