×

zbMATH — the first resource for mathematics

Blei’s inequality and coordinatewise multiple summing operators. (English) Zbl 1286.26017
Summary: Two inequalities resembling the multilinear Hölder inequality for mixed-norm Lebesgue spaces are proved. When specialized to single-function inequalities they include a pair of inequalities due to Blei and a recent extension of Blei’s inequality. The first of these inequalities is applied to give explicit indices in a known result for coordinatewise multiple summing operators. The second is used to prove a complementary result to the known one, again with explicit indices. As an application of the complementary result, a sufficient condition is given for a composition of operators to be multiple summing.

MSC:
26D15 Inequalities for sums, series and integrals
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47H60 Multilinear and polynomial operators
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] R. Blei, “Analysis in integer and fractional dimensions” , Cambridge Studies in Advanced Mathematics 71 , Cambridge University Press, Cambridge, 2001. \small<span class=”texttt”>D</span>OI: 10.1017/CBO9780511543012. · Zbl 1006.46001
[2] R. C. Blei, Fractional Cartesian products of sets, Ann. Inst. Fourier (Grenoble) 29(2) (1979), 79\Ndash105. · Zbl 0381.43003 · doi:10.5802/aif.744 · numdam:AIF_1979__29_2_79_0 · eudml:74413
[3] R. C. Blei and J. J. Fournier, Mixed-norm conditions and Lorentz norms, in: “Commutative harmonic analysis” (Canton, NY, 1987), Contemp. Math. 91 , Amer. Math. Soc., Providence, RI, 1989, pp. 57\Ndash78. \small<span class=”texttt”>D</span>OI: 10.1090/conm/091/1002588.
[4] H. F. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32(3) (1931), 600\Ndash622. \small<span class=”texttt”>D</span>OI: 10.2307/1968255. · Zbl 0001.26901 · doi:10.2307/1968255
[5] F. Bombal, D. Pérez-García, and I. Villanueva, Multilinear extensions of Grothendieck’s theorem, Q. J. Math. 55(4) (2004), 441\Ndash450. \small<span class=”texttt”>D</span>OI: 10.1093/qjmath/55.4.441. · Zbl 1078.46030 · doi:10.1093/qmath/hah017
[6] A. Defant and K. Floret, “Tensor norms and operator ideals” , North-Holland Mathematics Studies 176 , North-Holland Publishing Co., Amsterdam, 1993. · Zbl 0774.46018
[7] A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounaïes, and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174(1) (2011), 485\Ndash497. \small<span class=”texttt”>D</span>OI: 10.4007/annals.2011.174.1.13. · Zbl 1235.32001 · doi:10.4007/annals.2011.174.1.13
[8] A. Defant, M. Maestre, and U. Schwarting, Bohr radii of vector valued holomorphic functions, Adv. Math. 231(5) (2012), 2837\Ndash2857. \small<span class=”texttt”>D</span>OI: 10.1016/j.aim.2012.07.016. · Zbl 1252.32001 · doi:10.1016/j.aim.2012.07.016
[9] A. Defant, D. Popa, and U. Schwarting, Coordinatewise multiple summing operators in Banach spaces, J. Funct. Anal. 259(1) (2010), 220\Ndash242. \small<span class=”texttt”>D</span>OI: 10.1016/j.jfa.2010.01.008. · Zbl 1205.46026 · doi:10.1016/j.jfa.2010.01.008
[10] J. Diestel, H. Jarchow, and A. Tonge, “Absolutely summing operators” , Cambridge Studies in Advanced Mathematics 43 ,Cambridge University Press, Cambridge, 1995. \small<span class=”texttt”>D</span>OI: 10.1017/ \small<span class=”texttt”>C</span>BO9780511526138. · Zbl 0855.47016
[11] J. J. F. Fournier, Mixed norms and rearrangements: Sobolev’s inequality and Littlewood’s inequality, Ann. Mat. Pura Appl. (4) 148 (1987), 51\Ndash76. \small<span class=”texttt”>D</span>OI: 10.1007/BF01774283. · Zbl 0639.46034 · doi:10.1007/BF01774283
[12] J. E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quart. J. Math. Oxford Ser. 1(1) (1930), 164\Ndash174. \small<span class=”texttt”>D</span>OI: 10.1093/qmath/os-1.1.164. · JFM 56.0335.01
[13] M. C. Matos, Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math. 54(2) (2003), 111\Ndash136. · Zbl 1078.46031 · eudml:43048
[14] M. Milman, Notes on interpolation of mixed norm spaces and applications, Quart. J. Math. Oxford Ser. (2) 42(167) (1991), 325\Ndash334. \small<span class=”texttt”>D</span>OI: 10.1093/qmath/42.1.325. · Zbl 0760.46059 · doi:10.1093/qmath/42.1.325
[15] D. Pérez-García and I. Villanueva, Multiple summing operators on \(C(K)\) spaces, Ark. Mat. 42(1) (2004), 153\Ndash171. · Zbl 1063.46032 · doi:10.1007/BF02432914
[16] A. Pietsch, “Operator ideals” , Mathematische Monographien [Mathematical Monographs] 16 , VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.
[17] U. Schwarting, Vector valued Bohnenblust-Hille inequalities, Ph.D. Thesis, Carl von Ossietzky University (2013).
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.