Improving the closed formula for subpolynomial constants in the multilinear Bohnenblust–Hille inequalities

Abstract

For K=R or C, the Bohnenblust–Hille inequality asserts that there exists a sequence of scalars CK,mm=1∞ such that∑i1,…,im=1N∣U(ei1,…,eim)∣2mm+1m+12m⩽CK,msupz1,…,zm∈DNtU(z1,…,zm)∣ for all m-linear forms U:KN×⋯×KN→K and every positive integer N, where eii=1N denotes the canonical basis of KN and DN represents the open unit polydisc in KN. Very recently (2012) it was shown that there exist constants CK,mm=1∞ with subpolynomial growth satisfying this inequality. However, these constants were obtained via a complicated recursive formula. We improve the best known closed (non-recursive) approximation for these constants.