= Beckner's inequality
{c}
{title2=$\|T_\rho f\|_q\leq\|f\|_p$}
For $1<p\leq q<\infty$ and $0\leq\rho\leq\sqrt{(p-1)/(q-1)}$, the <noise operator on the Boolean hypercube> satisfies $\|T_\rho f\|_q\leq\|f\|_p$ for the uniform <probability measure>. In particular,
$$
\sum_S\rho^{2|S|}|\widehat f(S)|^2\leq\|f\|_{1+\rho^2}^2
\qquad(0\leq\rho\leq1).
$$
Here $\widehat f(S)$ are <Fourier-Walsh transform> coefficients. The endpoint $\rho=0$ follows by averaging. This <hypercontractive inequality on the Boolean hypercube> controls <Fourier-Walsh transform> coefficients of <discrete derivatives of a Boolean function> and is a principal ingredient in the <Kahn-Kalai-Linial theorem>.
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