Complex polynomial Bohnenblust--Hille inequality with polynomial bounds

The Bohnenblust-Hille inequality and its variants have found applications in several areas of Mathematics and related fields. The control of the constants for the variant for complex $m$-homogeneous polynomials is of special interest for applications in Harmonic Analysis and Number Theory. Up to now, the best known estimates for its constants are dominated by $\kappa\left(1+\varepsilon\right) ^{m}$, where $\varepsilon>0$ is arbitrary and $\kappa>0$ depends on the choice of $\varepsilon$. For the special cases in which the number of variables in each monomial is bounded by some fixed number $M$, it has been shown that the optimal constant is dominated by a constant depending solely on $M$. In this note, based on a deep result of Bayart, we prove an inequality for any subset of the indices, showing how summability of arbitrary restrictions on monomials can be related to the combinatorial dimension associated with them.