Let p ≥ 2. We improve the bound f p f 2 ≤ (p − 1) s/2 for a polynomial f of degree s on the boolean cube {0, 1} n , which comes from hypercontractivity, replacing the right hand side of this inequality by an explicit bivariate function of p and s, which is smaller than (p − 1) s/2 for any p > 2 and s > 0. We show the new bound to be tight, within a smaller order factor, for the Krawchouk polynomial of degree s.This implies several nearly-extremal properties of Krawchouk polynomials and Hamming spheres (equivalently, Hamming balls). In particular, Krawchouk polynomials have (almost) the heaviest tails among all polynomials of the same degree and ℓ 2 norm 1 . The Hamming spheres have the following approximate edge-isoperimetric property: For all 1 ≤ s ≤ n 2 , and for all even distances 0 ≤ i ≤ 2s(n−s) n , the Hamming sphere of radius s contains, up to a multiplicative factor of O(i), as many pairs of points at distance i as possible, among sets of the same size 2 . This also implies that Hamming spheres are (almost) stablest with respect to noise among sets of the same size. In coding theory terms this means that a Hamming sphere (equivalently a Hamming ball) has the maximal probability of undetected error, among all binary codes of the same rate.We also describe a family of hypercontractive inequalities for functions on {0, 1} n , which improve on the 'usual' "q → 2" inequality by taking into account the concentration of a function (expressed as the ratio between its ℓ r norms), and which are nearly tight for characteristic functions of Hamming spheres. p 2 . However, if we allow p to grow with n, the two bounds can be significantly different, even for small s. This will be important in estimates which take into account higher moments of polynomials, as is the cases we discuss below.Let us also observe that both bounds hold in somewhat higher generality -for all polynomials of degree at most s on {0, 1} n (see Corollary 1.4 below).We proceed with an informal description of several applications of (1). The formal statements and a more extensive discussion of these results will be given below, in Section 1.2. First, it will be shown that (1) is "nearly tight" (in the sense that will be clarified below) if f is the Krawchouk polynomial K s defined byRecalling that K s is proportional to the Fourier transform of the characteristic function of the Hamming sphere of radius s around zero, this says, alternatively, that Fourier transforms of Hamming spheres are nearly extremal with respect to (1). This will be shown to imply that Krawchouk polynomials and Hamming spheres have certain nearly extremal properties, compared to other objects with similar characteristics. Specifically, we will show that, up to at most polynomial in n error, the following facts hold for functions on {0, 1} n :• Krawchouk polynomials have (almost) the heaviest tails among all polynomials of the same degree and ℓ 2 norm. That is, for a polynomial f of degree s with f 2 = K s 2 , and for a threshold T > 0 holdswhere T ′ is not much larger than T . ...
