A simple proof that the power $\frac{2m}{m+1}$ in the Bohnenblust--Hille inequalities is sharp

TL;DR

This paper provides a straightforward proof confirming that the exponent m/(m+1) in the Bohnenblust--Hille inequalities is the best possible, simplifying the understanding of this fundamental result.

Contribution

It offers a simple, accessible proof that the exponent in the Bohnenblust--Hille inequalities is sharp, improving on the complexity of previous proofs.

Findings

The exponent m/(m+1) is proven to be optimal.

The proof simplifies the understanding of the inequality's sharpness.

The result confirms the known optimality with a more straightforward approach.

Abstract

The power $\frac{2m}{m+1}$ in the polynomial (and multilinear) Bohnenblust--Hille inequality is optimal. This result is well-known but its proof highly nontrivial. In this note we present a quite simple proof of this fact.