Bohnenblust--Hille inequality for cyclic groups

Abstract

For any $K>2$ and the multiplicative cyclic group $\hat{I}{\text{c◯}}_K$ of order $K$, consider any function $f:\hat{I}{\text{c◯}}_K^n\to \mathbf{C}$ and its Fourier expansion $f(z)={\sum }_{\hat{I}\pm \in \{0,1,\dots ,K-1{\}}^n}{a}_{\hat{I}}\pm z^{\hat{I}}\pm$, 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 $C(d,K)\Vert f{\Vert }_{\infty }$ 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 $\hat{I}{\text{c◯}}_K^n$. 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.

35 pages. Final version based on the referee's comments. To appear in Adv. Math