Hypercontractive inequality for pseudo-Boolean functions of bounded Fourier width

A function f: ^-1,1n→R is called pseudo-Boolean. It is well-known that each pseudo-Boolean function f can be written as f(x)=∑ _I∈Ff̂(I) ^χI(x), where F⊆I:I⊆[n], [n]=1,2,...,n, ^χI(x)= ∏ _i∈I ^xi and f̂(I) are non-zero reals. The degree of f is max|I|:I∈F and the width of f is the minimum integer ρ such that every i∈[n] appears in at most ρ sets in F. For i∈[n], let x _i be a random variable taking values 1 or -1 uniformly and independently from all other variables x _j, j≠i. Let x=(x ₁,...,x _n). The p-norm of f is || _f||p=( ^E[|f(x)|p])1p for any p<1. It is well-known that || _f||q<|| _f||p whenever q>p<1. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on f: if f is of degree d and q>p>1 then || _f||q≤( ^q-1p-1)d2|| _f||p. This inequality is called the Hypercontractive Inequality. We show that one can replace d by ρ in the Hypercontractive Inequality for each q>p<2 as follows: || _f||q≤( ^(2r)!ρr-1)1(2r)|| _f||p, where r=q2⌉. For the case q=4 and p=2, which is important in many applications, we prove a stronger inequality: || _f||4≤( ^2ρ+1)14|| _f||2.