Polynomial growth of Bohnenblust--Hille constants on the Hamming cube

Abstract

We prove that the Bohnenblust--Hille constants for Walsh polynomials on the Hamming cube grow at most polynomially in the degree. More precisely, there is an absolute constant $K$ such that every $f:\{-1,1{\}}^n\to \mathbb{C}$ of degree at most $m$ satisfies $${\left(\sum _{|S|\le m}|\hat{f}(S){|}^{2m/(m+1)}\right)}^{(m+1)/(2m)}\le K{m}^{27}\Vert f{\Vert }_{\infty }.$$ The estimate is uniform in the dimension. The exponent $27$ is not optimized.

arXiv comment11 pages