{
  "id": "hypercontractivity-on-a-torus-and-juntas",
  "title": "Hypercontractivity on a torus and juntas for Lipschitz functions",
  "description": "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.",
  "units": [
    {
      "id": "trigonometric-polynomials-and-the-heat-operators",
      "title": "Trigonometric polynomials and the heat operators on a torus",
      "reader": "trigonometric-polynomials-and-the-heat-operators.html",
      "source": "src/trigonometric-polynomials-and-the-heat-operators.md",
      "source_sha256": "37520ee7283f635df6759a9c2eabd72ef4215df97305989ca023b376da86e2a6",
      "status": "author_self_checked"
    },
    {
      "id": "a-logarithmic-sobolev-inequality-on-the-circle",
      "title": "A logarithmic Sobolev inequality on the circle",
      "reader": "a-logarithmic-sobolev-inequality-on-the-circle.html",
      "source": "src/a-logarithmic-sobolev-inequality-on-the-circle.md",
      "source_sha256": "9ec219e3570d4ac244df271f8de4cfdbe0c91e180cd0b9993585e6f08efc74a4",
      "status": "author_self_checked"
    },
    {
      "id": "hypercontractivity-of-the-heat-operators",
      "title": "Hypercontractivity of the heat operators",
      "reader": "hypercontractivity-of-the-heat-operators.html",
      "source": "src/hypercontractivity-of-the-heat-operators.md",
      "source_sha256": "609ac44003b8a9a4a179afebbf0579301838ead3de9e8d8e1b1c2e2e59139285",
      "status": "author_self_checked"
    },
    {
      "id": "lipschitz-functions-are-close-to-juntas",
      "title": "Lipschitz functions are close to juntas",
      "reader": "lipschitz-functions-are-close-to-juntas.html",
      "source": "src/lipschitz-functions-are-close-to-juntas.md",
      "source_sha256": "174ab5295b42a00d5140116398defdcc6a4de34633fb49a0f9b27d14b9bce4a0",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Measure and Integration (D10): the Lebesgue integral, Fubini's theorem, dominated convergence, the inequalities of Jensen, Hoelder and Cauchy-Schwarz",
    "calculus of one and several variables (differentiation under the integral, integration by parts, the Gaussian integral)",
    "the lessons prove every other fact they use"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}