Asymptotic Contractivity of Bohnenblust-Hille Constants with Bounded Monomial Support

Abstract

We study the optimal Bohnenblust-Hille constants $K_{m,M}$ for complex $m$-homogeneous polynomials in any number of variables whose monomials with nonzero coefficient involve at most $M$ variables. For every fixed $M$, we prove that these constants satisfy \[ K_{ m , M } \leq A_{ M }^{ M / m } \cdot m^{ ( M^2 - 1 ) / ( 2m ) } \, , \quad \text{for} \ m \geq M \, , \] where $A_M\ge 1$ depends only on $M$. In particular, \[ K_{ m , M } \to 1 \quad \text{as} \ m \to \infty \, , \] 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.

Expanded and revised. To appear in Bulletin of the Brazilian Mathematical Society