Hypercontractivity on a torus and juntas for Lipschitz functions

Trigonometric polynomials and Fejer means on a torus; the heat operators and three inequalities for them. A logarithmic Sobolev inequality on the circle, with Rothaus's lemma. Hypercontractivity of the heat operators on the circle and on a torus of any dimension. Austin's theorem: a function on a cube that is Lipschitz for the sup distance is close to a function of a bounded number of coordinates, with an explicit bound that does not depend on the dimension. All proofs are complete; no theorem of the literature is assumed.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Trigonometric polynomials and the heat operators on a torus
2. A logarithmic Sobolev inequality on the circle
3. Hypercontractivity of the heat operators
4. Lipschitz functions are close to juntas

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• The Lebesgue integral with Fubini's theorem and the inequalities of Jensen and Hoelder (core course Measure and Integration); calculus.