The Becker-Bonami operator $T_p$ on a function $f(x) : \{0,1\}^d \to \mathbb{R}$ is defined as follows. Let $\nu(x)$ be a perturbation of $x$ so that every bit remains the same with probability $p$ or is chosen uniformly at random from $\{0,1 \}$ with probability $1-p$. Then $T_p(f(x)) = E_{y =\nu(x)} f(y)$ . It is well known, by a famous inductive proof:

\begin{equation}||T_{\sqrt{p-1}} f ||_2 \leq ||f||_p \end{equation}

I am curious first, as to what other proofs are known besides the traditional inductive one I see in most lecture notes on boolean functions?

(Note: It would be especially nice if there was a proof that leverages the fact that $T_p$ attenuates the coefficients of the parity basis functions. i.e, if $\chi_S: \{0,1\}^d \to \{-1,1\}$ is the function such that $\chi_S(x) = -1^{|x \cap S|}$, it is known that $T_p(\chi_S(x)) = p^{|S|} \chi_S(x)$.)



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