Bohnenblust--Hille inequality for cyclic groups

Abstract

For any $K>2$ and the multiplicative cyclic group $\Omega_K$ of order $K$, consider any function $f:\Omega_K^n\to\mathbf{C}$ and its Fourier expansion $f(z)=\sum_{\alpha\in\{0,1,\ldots,K-1\}^n}a_\alpha z^\alpha$, with $d:=\text{deg}(f)$ denoting its degree as a multivariate polynomial. We prove a Bohnenblust--Hille (BH) inequality in this setting: the $\ell_{2d/(d+1)}$ norm of the Fourier coefficients of $f$ is bounded by $C(d,K)\|f\|_\infty$ with $C(d,K)$ independent of $n$. This is the interpolating case between the now well-understood BH inequalities for functions on the poly-torus ($K =\infty$) and the hypercube ($K=2$) but those extreme cases of $K$ have special properties whose absence for intermediate $K$ prevent a proof by the standard BH framework. New techniques are developed exploiting the group structure of $\Omega_K^n$. By known reductions, the cyclic group BH inequality also entails a noncommutative BH inequality for tensor products of the $K \times K$ complex matrix algebra (or in the language of quantum mechanics, systems of $K$-level qudits). These new BH inequalities generalize several applications in harmonic analysis and statistical learning theory to broader classes of functions and operators.