Absolutely summing multilinear operators via interpolation

We use an interpolative technique from [1] to introduce the notion of multiple N  -separately summing operators. Our approach extends and unifies some recent results; for instance we recover the best known estimates of the multilinear Bohnenblust–Hille constants due to F. Bayart, D. Pellegrino and J. Seoane-Sepúlveda. More precisely, as a consequence of our main result, for 1≤t<21≤t<2 and m>1m>1 we prove that(∑i1,…,im=1∞|U(ei1,…,eim)|2tm2+(m−1)t)2+(m−1)t2tm≤[∏j=2mΓ(2−2−tjt−2t+2)t(j−2)+22t−2jt]‖U‖ for all complex m  -linear forms U:c0×⋯×c0→CU:c0×⋯×c0→C.