A moment ratio bound for polynomials and some extremal properties of Krawchouk polynomials and Hamming spheres

Understand

Let $p \ge 2$.

• We improve the bound $\frac{\|f\|_p}{\|f\|_2} \le (p-1)^{s/2}$ 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).