×

zbMATH — the first resource for mathematics

On the mixed \((\ell_1,\ell_2)\)-Littlewood inequalities and interpolation. (English) Zbl 1429.47001
Summary: It is well-known that the optimal constant of the bilinear Bohnenblust-Hille inequality (i.e., Littlewood’s \(4/3\) inequality) is obtained by interpolating the bilinear mixed \((\ell_1,\ell_2)\)-Littlewood inequalities. We remark that this cannot be extended to the 3 -linear case and, in the opposite direction, we show that the asymptotic growth of the constants of the \(m\)-linear Bohnenblust-Hille inequality is the same of the constants of the mixed \(\left(\ell_{\frac{2m+2}{m+2}},\ell_2\right)\)-Littlewood inequality. This means that, contrary to what the previous works seem to suggest, interpolation does not play a crucial role in the search of the exact asymptotic growth of the constants of the Bohnenblust-Hille inequality. In the final section we use mixed Littlewood type inequalities to obtain the optimal cotype constants of certain sequence spaces.

MSC:
47H60 Multilinear and polynomial operators
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] F. ALBIAC ANDN. KALTON, Topics in Banach Space Theory, Graduate Texts in Mathematics 233, Springer-Verlag 2005.
[2] N. ALBUQUERQUE, G. ARAUJO´, D. NUNEZ˜-ALARCON´, D. PELLEGRINO ANDP. RUEDA, Bohnenblust-Hille and Hardy-Littlewood inequalities by blocks,arXiv:1409.6769 [math.FA].
[3] G. ARAUJO, D. PELLEGRINO ANDD. DINIZP.DASILVA ESILVA, On the upper bounds for the constants of the Hardy-Littlewood inequality, J. Funct. Anal. 267, 6 (2014), 1878–1888.
[4] F. BAYART, D. PELLEGRINO,ANDJ. B. SEOANE-SEPULVEDA´, The Bohr radius of the n - dimensional polydisk is equivalent to(logn)/n , Adv. Math. 264 (2014), 726–746. · Zbl 1331.46037
[5] O. BLASCO, G. BOTELHO, D. PELLEGRINO ANDP. RUEDA, Summability of multilinear mappings: Littlewood, Orlicz and beyond, Monatsh. Math. 163, 2 (2011), 131–147. · Zbl 1246.46045
[6] H. F. BOHNENBLUST ANDE. HILLE, On the absolute convergence of Dirichlet series, Ann. of Math. 32 (1931), 600–622. · JFM 57.0266.05
[7] N. CARO, D. NUNEZ˜-ALARCON AND´D.M. SERRANO-RODR´IGUEZ, On the generalized Bohnenblust–Hille inequality for real scalars, Positivity 21 (2017), 1439–1455. · Zbl 06816310
[8] J. DIESTEL, H. JARCHOW ANDA. TONGE, Absolutely summing operators, Cambridge University Press, 1995. · Zbl 0855.47016
[9] D. J. H. GARLING, Inequalities: A Journey into Linear Analysis, Cambridge University Press, Cambridge, 2007. · Zbl 1135.26014
[10] J. E. LITTLEWOOD, On bounded bilinear forms in an infinite number of variables, Quart. J. Math. 1 (1930), 164–174. · JFM 56.0335.01
[11] D. PELLEGRINO, The optimal constants of the mixed(1,2)-Littlewood inequality, J. Number Theory 160 (2016), 11–18. · Zbl 1431.46024
[12] D. POPA ANDG. SINNAMON, Blei’s inequality and coordinatewise multiple summing operators, Publ. Mat. 57, 2 (2013), 455–475. · Zbl 1286.26017
[13] P. RUEDA ANDE.A. S ´ANCHEZ-P ´EREZ, Factorization of p -dominated polynomials through Lp- spaces, Michigan Math. J. 63, 2 (2014), 345–353. · Zbl 1305.46042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.