Introduction to Analysis on the Discrete Cube IV
summary
Shows the discrete-cube log‑Sobolev inequality holds with constant 2 and derives hypercontractivity bounds that recover Bonami/Kahane moment comparisons for chaoses of degree D (moment growth like ((p‑1)/(q‑1))^{D/2}). Raises an explicit open problem for degree‑2 chaoses: the optimal comparison constant is at least 4, is proved for n ≤ 6, and the constructive search for a counterexample at n ≥ 7 is proposed. Also treats the decoupled chaos case (elementary e^2–type bounds).
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chapters (12)
- 0:00Start
- 0:22Optimal constant K2,1-proof
- 4:16Open problems
- 18:45Elementary moment comparison
- 22:51Moment comparison - proof
- 27:00Comparison example computation
- 37:00Logarithmic Sobolev inequalities - continued
- 40:38Logarithmic Sobolev inequality - equivalent versions
- 43:57Logarithmic Sobolev inequalities - proof of the equivalence
- 46:40Logarithmic Sobolev inequality - semigroup application
- 51:41Hypercontractivity-explanation
- 52:40Hypercontractivity - explanation
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