A simple proof that the power $\frac{2m}{m+1}$ in the Bohnenblust--Hille   inequalities is sharp
  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. 
Daniel Nu\~nez-Alarc\'on and Daniel Pellegrino
math.FA