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A classical inequality due to Bohnenblust and Hille states that for every positive integer $m$ there is a constant $C_{m}>0$ so that $$(\sum\limits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}},...,e_{i_{m}})| ^{\frac{2m}{m+1}}) ^{\frac{m+1}{2m}}\leq C_{m}| U|$$ for every positive integer $N$ and every $m$-linear mapping $U:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow\mathbb{C}$, where $C_{m}=m^{\frac{m+1}{2m}}2^{\frac{m-1}{2}}.$ The value of $C_{m}$ was improved to $C_{m}=2^{\frac{m-1}{2}}$ by S.
H. F. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. 32 (1931), 600-622
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G. Botelho, H.-A. Braunss, H. Junek, D. Pellegrino, Inclusions and coincidences for multiple summing multilinear mappings, Proc. Amer. Math. Soc. 137
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G. Botelho, C. Michels and D. Pellegrino, Complex interpolation and summability properties of multilinear operators, Rev. Mat. Complut. 23
2010
Closest in time.
A. Defant. D. Popa, U. Schwarting, Coordinatewise multiple summing operators in Banach spaces, J. Funct. Anal. 259 (2010), 220-242
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O. Blasco, G. Botelho, D. Pellegrino and P. Rueda, Summability of multilinear mappings: Littlewood, Orlicz and beyond, Monatshefte fur Mathematik, to appear
Cited in the paper.