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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.