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 C m > 0 so that ( ∑ i 1 , … , i m = 1 N | U ( e i 1 , … , e i m ) | 2 m m + 1 ) m + 1 2 m ⩽ C m ‖ U ‖ for every positive integer N and every m-linear mapping U : ℓ ∞ N × ⋯ × ℓ ∞ N → C , where C m = m m + 1 2 m 2 m − 1 2 . The value of C m was improved to C m = 2 m − 1 2 by S. Kaijser and more recently H. Quéffelec and A. Defant and P. Sevilla-Peris remarked that C m = ( 2 π ) m − 1 also works. The Bohnenblust–Hille inequality also holds for real Banach spaces with the constants C m = 2 m − 1 2 . 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 C m for all values of m ∈ N . In particular, we will also show that, for real scalars, if m is even with 2 ⩽ m ⩽ 24 , then C R , m = 2 1 2 C R , 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.