Optimal upper bound for the Boolean Bohnenblust–Hille constant

Determine the optimal upper bound, as a function of the degree d, for the best constant BH_{≤d}^{ {−1,1} } in the Boolean Bohnenblust–Hille inequality: for every n ≥ 1 and every real-valued function f: {−1,1}^n → ℝ of degree at most d, establish the smallest C(d) such that ∥f∥_{2d/(d+1)} ≤ C(d) ∥f∥_{∞} holds uniformly over n. Equivalently, ascertain the sharp growth rate in d of BH_{≤d}^{ {−1,1} }.

Background

The Boolean Bohnenblust–Hille inequality states that for functions f: {−1,1}^n of degree at most d, one has ∥f∥{2d/(d+1)} ≤ C(d) ∥f∥{∞} with a constant C(d) independent of n. Denoting the best such constant by BH_{≤d}^{ {−1,1} }, current results give a sub‑exponential upper bound BH_{≤d}^{ {−1,1} } ≤ C√(d log d).

While recent work shows optimal constants when restricting to Boolean-valued functions f: {−1,1}^n → {−1,1}, the optimal bound for general bounded real-valued functions remains unknown. Resolving this would refine sample complexity bounds in low-degree learning tasks and sharpen our understanding of hypercontractive phenomena on the discrete cube.