Introduction 3 - Hyper-contractivity

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Defines the noise operator on the Boolean cube (resample each bit independently with probability 1−ε), proves Tε acts on Fourier characters by ε^{|S|} and that ||Tε f||_q ≤ ||f||_2 for ε ≤ 1/√(q−1). Deduces ||f||_q ≤ (q−1)^{d/2}||f||_2 for homogeneous degree‑d polynomials, a level‑d inequality bounding low‑degree Fourier mass of an α‑dense indicator (roughly ≤ α^2 × polylog factors), and small‑set expansion / edge isoperimetry: small sets leave under a 1/2‑noise step, giving Influence ≥ C α log(1/α).

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" chapters (12)

1. 0:00Introduction to hypercontractivity
2. 3:24The hypercontractivity theorem
3. 8:02Proof of the theorem
4. 12:34Properties of the noise operator
5. 16:40Moment bounds and applications
6. 20:55Fourier spectrum and influence
7. 25:01Correlation and distributions
8. 30:40Level-D inequality
9. 37:11Upper bounding mass
10. 42:03Small set expansion theorem
11. 52:00KKL and isoperimetry
12. 55:42Advanced topics and conclusion