Bohnenblust--Hille inequality for cyclic groups

For any $K>2$ and the multiplicative cyclic group ${\mathrm{\Omega }}_K$ of order $K$, consider any function $f:{\mathrm{\Omega }}_K^n\to \mathbf{C}$ and its Fourier expansion $f(z)={\sum }_{\alpha \in \{0,1,\dots ,K-1{\}}^n}{a}_{\alpha }{z}^{\alpha }$, with $d:=\text{deg}(f)$ denoting its degree as a multivariate polynomial. We prove a Bohnenblust–Hille (BH) inequality in this setting: the ${\mathrm{\ell }}_{2d\mathrm{/}(d+1)}$ norm of the Fourier coefficients of $f$ is bounded by $C(d,K)\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ 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 ($K=\mathrm{\infty }$) 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 ${\mathrm{\Omega }}_K^n$. 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.