Bohnenblust–Hille inequality for cyclic groups.

For any K > 2 and the multiplicative cyclic group Ω K of order K , consider any function f : Ω K n → C and its Fourier expansion f (z) = ∑ α ∈ { 0 , 1 , ... , K − 1 } n a α z α , with d : = deg ⁡ (f) denoting its degree as a multivariate polynomial. We prove a Bohnenblust–Hille (BH) inequality in this setting: the ℓ 2 d / (d + 1) norm of the Fourier coefficients of f is bounded by C (d , K) ‖ f ‖ ∞ 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 = ∞) 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 Ω K n. By known reductions, the cyclic group BH inequality also entails a noncommutative BH inequality for tensor products of the K × 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. [ABSTRACT FROM AUTHOR]