Hypercontractivity and its applications

Hypercontractive inequalities are a useful tool in dealing with extremal questions in the geometry of high-dimensional discrete and continuous spaces. In this survey we trace a few connections between different ways of stating hypercontractivity, and also present some relatively recent applications of these techniques in computer science. 1 Preliminaries and notation Fourier analysis on the hypercube. The uniform probability measure on the Boolean cube {−1, 1}n gives rise to a natural inner product 〈f, g〉 = Ex f(x)g(x) on functions f, g : {−1, 1}n → R. The multilinear polynomials χS(x) = ∏ i∈S xi (where S ranges over subsets of [n]) form an orthogonal basis under this inner product; they are called the Fourier basis. Thus, for any function f : {−1, 1}n → R, we have f = ∑ S⊆[n] f̂(S)χS(x), where the Fourier coefficients f̂(S) = 〈f, χS〉 obey Plancherel’s relation ∑ f̂(S)2 = 1. It is easy to verify that Ex f(x) = f̂(0) and Varx f(x) = ∑ S 6=∅ f̂(S) 2. Norms. For 1 ≤ p < ∞, define the `p norm ‖f‖p = (Ex |f(x)|p)1/p. These norms are monotone in p: for every function f , p ≥ q implies ‖f‖p ≥ ‖f‖q. For a linear operator M carrying functions f : {−1, 1}n → R to functions Mf = g : {−1, 1}n → R, we define the p-to-q operator norm ‖M‖p→q = supf ‖Mf‖q/‖f‖q. M is said to be a contraction from `p to `q when ‖M‖p→q ≤ 1. Because of the monotonicity of norms, a contraction from `p to `p is automatically a contraction from `p to `q for any q < p. When q > p and ‖M‖p→q ≤ 1, then M is said to be hypercontractive. Convolution operators. The convolution (f∗g)(x) = Ey f(x)g(xy) of two functions f, g : {−1, 1}n → R defines a linear operator f 7→ f ∗ g, where xy is the coordinate-wise product. Convolution is commutative and associative, and the Fourier coefficients of a convolution satisfy the useful property f̂ ∗ g = f̂ ĝ. We shall be particularly interested in the convolution properties of the following functions • The Dirac delta δ : {−1, 1}n → R, given by δ(1, . . . , 1) = 1 and δ(x) = 0 otherwise. It is the identity for convolution and has δ̂(S) = 1 for all S ⊆ [n].