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3. Bohnenblust-Hille inequalities

See [MR3961323] and the references therein for more information.

    1. Best constants in Bohnenblust-Hille inequalities for hypercubes


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      Problem 3.1.

      Fix $d\ge 1$. Show that there exists $C>0$ such that for all $n\ge 1$ and all polynomials $f=\sum_{|S|\le d}\widehat{f}(S)\chi_S$ on $\{-1,1\}^n$ of degree at most $d$, we have \begin{equation*} \left(\sum_{|S|\le d}|\widehat{f}(S)|^{\frac{2d}{d+1}}\right)^{\frac{d+1}{2d}}\le d^C \|f\|_{\infty}. \end{equation*}
          It is known that the inequality holds with constant $C^{\sqrt{d\log d}}$ for some universal $C>0$.

        • Homogeneous case

              The homogeneous case is simpler.

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          Problem 3.2.

          Fix $d\ge 1$. Show that there exists $C>0$ such that for all $n\ge 1$ and all $d$-homogeneous polynomials $f=\sum_{|S|= d}\widehat{f}(S)\chi_S$ on $\{-1,1\}^n$, we have \begin{equation*} \left(\sum_{|S|= d}|\widehat{f}(S)|^{\frac{2d}{d+1}}\right)^{\frac{d+1}{2d}}\le d^C \|f\|_{\infty}. \end{equation*}
              Similarly, the inequality holds with constant $C^{\sqrt{d\log d}}$ for some universal $C>0$.

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