Polynomial Bohnenblust--Hille bounds for product of cyclic groups

Abstract

Fix an integer $K\ge 2$, and let $C_K^n=\{(e^{\frac{2\pi ij}{K}}{)}_{j=0}^{K-1}{\}}^n$ be the product of cyclic groups of order $K$. For a Fourier character ${\chi }_{\alpha }$, let $s(\alpha )$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust--Hille constants governed by interaction order grow polynomially: if $p_d=2d/(d+1)$ and \[ γ_2=\frac12, \qquad γ_K=\frac{K\log(K-1)}{4(K-2)}\quad(K\ge3), \] then the ${\ell }^{p_d}$ norm of Fourier coefficients $\{\hat{f}(\alpha )\}$ is bounded by $L^{\infty }$ norm of $f$ multiplied by $C(K)d^{4{\gamma }_K+5}$. The constant $C(K)$ is actually at most of the order $K^{5/2}$.

20 pages