Results on heat operators on a torus, logarithmic Sobolev inequalities, hypercontractivity and juntas

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Poincare inequality for means over coordinates — Lemma 1.1. For a trigonometric polynomial f on T^N and a set S of coordinates, |f - E_S f|_2^2 <= (1/(4 pi^2)) times the sum over n outside S of |d_n f|_2^2.
• Fejer means on a torus — Lemma 2.2. The Fejer means of a bounded Borel function on T^N are trigonometric polynomials; they keep bounds a <= f <= b and the Lipschitz constant for the sup distance, commute with the means over coordinates, and converge uniformly for continuous f.
• The heat operators on characters — Lemma 3.2. P_t f(x) = E f(x + sqrt(2t) Z) satisfies P_t e_k = exp(-4 pi^2 |k|^2 t) e_k; on trigonometric polynomials P_t is a semigroup, commutes with derivatives and solves the heat equation.
• The heat operator moves a function slowly in L^1 — Lemma 4.1. For a real trigonometric polynomial f on T^N: |f - P_t f|_1 <= sqrt(2t) times the sum of |d_n f|_1.
• Derivatives of P_t f in L^2 — Lemma 4.2. For a trigonometric polynomial f on T^N and t > 0: the sum of |d_n P_t f|_2^2 is at most |f|_2^2 / (2 e t).
• Gibbs' inequality and the sign of the entropy — Lemma 1.1. For bounded h >= 0 and a density p on a probability space, the integral of h log p is at most Ent(h); Ent(h) >= 0 and Ent(c h) = c Ent(h).
• Rothaus's lemma — Lemma 4.1. For a bounded real function g on a probability space and a real number a: Ent((a+g)^2) <= Ent(g^2) + 2 times the integral of g^2.
• A logarithmic Sobolev inequality on the circle — Theorem 5.1. For every real continuously differentiable function f on the circle R/Z: Ent(f^2) <= (7/4) |f'|_2^2.
• Hypercontractivity of the heat operators on the circle — Theorem 1.1. If 1 < p <= q and (q-1)/(p-1) <= exp(4t/c_0) with c_0 = 7/4, then |P_t f|_q <= |f|_p for every real continuous function on the circle; in particular |P_t f|_2 <= |f|_p for p = 1 + exp(-4t/c_0).
• Hypercontractivity of the heat operators on a torus — Theorem 2.2. For every N, t >= 0 and p = 1 + exp(-4t/c_0): |P_t f|_2 <= |f|_p for every real continuous function on T^N. The exponent does not depend on N.
• The heat operator makes a function with small L^1 norm small in L^2 — Corollary 3.1. With b = tanh(2t/c_0) and a = 1 - b: |P_t f|_2 <= |f|_1^b |f|_2^a for real continuous f on T^N.
• Juntas for trigonometric polynomials with a bounded sum of derivatives — Theorem 1.1. For 0 < eps <= 1 there is an explicit q(eps) such that every real trigonometric polynomial f on T^N with sum |d_n f|_1 <= 1 and |f - mean|_2 <= 1 satisfies |f - E_S f|_1 < eps for the set S of the at most q(eps) coordinates with |d_n f|_1 >= eta(eps).
• Lipschitz functions on a torus are close to juntas — Theorem 3.2. Every H: T^N -> [-1,1] that is L-Lipschitz for the sup distance is within tau in L^2 of E_S H for a set S of at most J_T(L,tau) coordinates; J_T does not depend on N.
• Austin's junta theorem on the cube — Theorem 4.1. For L >= 0 and tau > 0 there is J(L,tau) such that every f: [0,1]^N -> [-1,1] with |f(x) - f(y)| <= L max |x_k - y_k| has a set S of at most J(L,tau) coordinates with |f - E_S f|_2 < tau, for every N. One can take J(L,tau) = J_T(2L,tau).

Reading order

Four lessons with complete proofs and no hypothesis from the literature. The first gives the tools. The second proves a logarithmic Sobolev inequality on the circle. The third derives hypercontractivity of the heat operators, first on the circle, then on a torus of any dimension. The fourth proves that a function on a cube that is Lipschitz for the sup distance is close to a function of a bounded number of coordinates. The course on large independent sets in three-colorable graphs uses this theorem.