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

Abstract

A classical inequality due to Bohnenblust and Hille states that for every positive integer m there is a constant C(m) > 0 so that (Sigma(N)(i1...., im=1) vertical bar U(e(i1), ..., e(im))vertical bar(2m/m+1))(m+1/2m) <= C(m)parallel to U parallel to for every positive integer N and every m-linear mapping U : l(infinity)(N) x ... x l(infinity)(N) -> C, where C(m) = m(m+1/2m)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. Queffelec and A. Defant and P. Sevilla-Peris remarked that C(m) = (2/root pi)(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 is an element of 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.