Asymptotic Contractivity of Bohnenblust–Hille Constants with Bounded Monomial Support

We study the optimal Bohnenblust – Hille constants Km, M K m , M for complex m -homogeneous polynomials in any number of variables whose monomials with nonzero coefficient involve at most M M variables. For every fixed M M , we prove that these constants satisfy Km, M ≤ AMM/ m · m( M² - 1 ) / ( 2\,m ) \,, for \ m ≥ M\,, K m , M ≤ A M M / m · m ( M 2 - 1 ) / ( 2 m ) , for m ≥ M , where AM ≥ 1 A M ≥ 1 depends only on M M . In particular, Km, M → 1 as \ m → ∞ \,, K m , M → 1 as m → ∞ , and so the corresponding Bohnenblust – Hille constants are asymptotically contractive. The proof exploits the homogeneous structure through a decomposition according to exact monomial-support levels, partitions of the set of variables, multilinear Bohnenblust – Hille estimates, and interpolation with Parseval ’s identity.