Publication: New upper bounds for the constants in the Bohnenblust–Hille inequality
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Publication Date
2012
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Elsevier
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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 (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.
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O. Blasco, G. Botelho, D. Pellegrino, P. Rueda, Summability of multilinear mappings: Littlewood, Orlicz and beyond, Monatsh. Math. 163 (2011) 131–147.
H.F. Bohnenblust, E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. 32 (1931) 600–622.
F. Bombal, D. Pérez-García, I. Villanueva, Multilinear extensions of Grothendieck’s theorem, Q. J. Math. 55 (2004) 441–450.
G. Botelho, H.A. Braunss, H. Junek, D. Pellegrino,Inclusions and coincidences for multiple summing multilinear mappings, Proc. Amer. Math. Soc. 137 (2009) 991–1000.
G. Botelho, C. Michels, D. Pellegrino, Complex interpolation and summability properties of multilinear operators, Rev. Mat. Complut. 23 (2010) 139–161.
A. Defant, K. Floret, Tensor Norms and Operator Ideals,North-Holland Math. Stud., vol. 176, 1993.
A. Defant, P. Sevilla-Peris, A new multilinear insight on Littlewood’s 4/3-inequality, J. Funct. Anal. 256 (2009) 1642–1664.
A. Defant, D. Popa, U. Schwarting, Coordinatewise multiple summing operators in Banach spaces, J. Funct. Anal. 259 (2010) 220–242.
A. Defant, L. Frerick, J. Ortega-Cerdá, M. Ounaïes, K.Seip, The Bohnenblust–Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174 (2011) 485–497, doi:10.4007/annals.2011.174.1.13.
J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing perators, Cambridge University Press, Cambridge, 1995.
U. Haagerup, The best constants in Khinchine inequality, Studia Math. 70 (1982) 231–283.
S. Kaijser, Some results in the metric theory of tensor products, Studia Math. 63 (1978) 157–170.
J.E. Littlewood, On bounded bilinear forms in an infinite number of variables, Q. J. Math. 1 (1930) 164–174.
M.C. Matos, Fully absolutely summing and Hilbert–Schmidt multilinear mappings, Collect. Math. 54 (2003) 111–136.
H. Queffélec, H. Bohr’s vision of ordinary Dirichlet series: old and new results, J. Anal. 3 (1995) 43–60