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5. Moment comparison inequalities

The moment comparison inequalities are related to many topics such as hypercontractivity and some problems in previous sections. See [MR3951277] and [MR4146538] for more information.

    1. Moment comparison


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      Problem 5.1.

      Consider the following problems:
      1. Find the best $C>0$ such that for all $f:\{-1,1\}^n\to \mathbb{R}$, \begin{equation*} \|f\|_2\le C^{\deg(f)}\|f\|_1. \end{equation*}
      2. For $1< p < q \le 2$, find the best $C_{p,q}>0$ such that \begin{equation*} \|f\|_q\le C_{p,q}^{\deg(f)}\|f\|_p. \end{equation*}
      3. How about the $d$-homogeneous analogs of the above two problems? Of particular interest, find the best $C>0$ such that for all $2$-homogeneous polynomial $f(x)=\sum_{i < j}a_{ij}x_i x_j$ over $\{-1,1\}^n$ for all $n\ge 1$, we have \begin{equation*} \|f\|_2\le C\|f\|_1. \end{equation*}
        1. Remark. [Paata Ivanisvili] In problem 5.2. part 3 it is conjectured that C=2.

              Cite this as: AimPL: Analysis on the hypercube with applications to quantum computing, available at http://aimpl.org/hypercubequantum.