Asymptotic Contractivity of Bohnenblust–Hille Constants with Bounded Monomial Support

Abstract We study the optimal Bohnenblust – Hille constants $$K_{ m, M}$$ 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 $$ K_{ m, M} \\le A_{ M}^{ M/ m } \\cdot m^{ ( M^2 - 1 ) / ( 2\\,m ) } \\,, \\quad \\text {for} \\ m \\ge M\\,, $$ K m , M ≤ A M M / m · m ( M 2 - 1 ) / ( 2 m ) , for m ≥ M , where $$A_{ M} \\ge 1$$ A M ≥ 1 depends only on $$M$$ M . In particular, $$ K_{ m, M} \\rightarrow 1 \\quad \\text {as} \\ m \\rightarrow \\infty \\,, $$ 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.