New upper bounds for the constants in the Bohnenblust–Hille inequality

A classical inequality due to Bohnenblust and Hille states that for every positive integer m   there is a constant Cm>0Cm>0 so that(∑i1,…,im=1N|U(ei1,…,eim)|2mm+1)m+12m⩽Cm‖U‖ for every positive integer N and every m  -linear mapping U:ℓ∞N×⋯×ℓ∞N→C, where Cm=mm+12m2m−12. The value of CmCm was improved to Cm=2m−12 by S. Kaijser and more recently H. Quéffelec and A. Defant and P. Sevilla-Peris remarked that Cm=(2π)m−1 also works. The Bohnenblust–Hille inequality also holds for real Banach spaces with the constants Cm=2m−12. In this note we show that a recent new proof of the Bohnenblust–Hille inequality (due to Defant, Popa and Schwarting) provides, in fact, quite better estimates for CmCm for all values of m∈Nm∈N. In particular, we will also show that, for real scalars, if m   is even with 2⩽m⩽242⩽m⩽24, thenCR,m=212CR,m/2. We will mainly work on a paper by Defant, Popa and Schwarting, giving some remarks about their work and explaining how to, numerically, improve the previously mentioned constants.