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A new multilinear insight on Littlewood’s 4/3-inequality. (English) Zbl 1171.46034
For a positive integer \(m\) and \(1\leq p\leq q\), the authors let \(\rho={2m\over m+2({1\over p}-\max\{{{1\over q},{1\over 2}\}})}\) if \(p\leq 2\) and \(\rho = p\) if \(p\geq 2\). They show that there is \(C>0\) such that for every \(m\)-linear mapping \(A: c_0\times\dots \times c_0\to\ell_p\),
\[ \left(\sum_{i_1,\dots, i_m}^\infty\|A(e_{i_1},\dots, e_{i_m})\|_q^\rho\right)^{1/ \rho}\leq C\|A\|, \] where \(\|A\|=\sup_{\|x_j\|\leq 1}\|A(x_1,\dots, x_m)\|\). Moreover, this value of \(\rho\) is the best possible.
This result unifies Littlewood’s 4/3-inequality [J. E. Littlewood, Q. J. Math., Oxf. Ser. 1, 164–174 (1930; JFM 56.0335.01)], its multilinear extension due to H. F. Bohnenblust and E. Hille [Ann. Math. (2) 32, 600–622 (1931; Zbl 0001.26901 and JFM 57.0266.05)], a vector-valued inequality for operators from \(c_0\) to \(\ell_p\) discovered independently by G. Bennett [J. Funct. Anal. 13, 20–27 (1973; Zbl 0255.47033)] and B. Carl [Math. Nachr. 63, 353–360 (1974; Zbl 0292.47019)], and a recent result of F. Bombal, D. Pérez–García and I. Villanueva [Q. J. Math. 55, No. 4, 441–450 (2004; Zbl 1078.46030)].
As an application, it is shown that, if \(\sum_{\alpha\in {\mathbb N}^{(\mathbb N)}}c_\alpha(P) z^\alpha\) is the monomial expansion of an \(m\) homogeneous polynomial \(P: c_0\to\ell_p\), then \[ \left(\sum_{\alpha\in {\mathbb N}^{({\mathbb N})}}\|c_\alpha (P)\|_q^\rho\right)^{1/\rho}\leq C\|P\| \] with \(\rho\) the best possible constant.
The authors say that a linear operator between Banach spaces, \(v: X\to Y\), is \((r,1)\)-summing of order \(m\) is there is \(C>0\) such that for every \(m\)-linear mapping \(A: c_0\times\dots \times c_0\to X\) \[ \left(\sum_{i_1,\dots, i_m}^\infty\|v\circ A(e_{i_1},\dots, e_{i_m})\|^r\right) ^{1/r}\leq C\|A\|. \] They examine the indices \(\inf\{r:v\text{ is }(r,a)\text{-summing of order m}\}\) for certain identity and inclusion mappings and relate \((r,1)\)-summing operators of a given order to summing operators.

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
46B07 Local theory of Banach spaces
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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[1] Aron, R.M.; Globevnik, J., Analytic functions on \(c_0\), Congress on functional analysis, Madrid, 1988, Rev. mat. complut., 2, Suppl., 27-33, (1989) · Zbl 0748.46021
[2] Bennett, G., Inclusion mappings between \(l^p\) spaces, J. funct. anal., 13, 20-27, (1973) · Zbl 0255.47033
[3] Bergh, J.; Löfström, J., Interpolation spaces. an introduction, Grundlehren math. wiss., vol. 223, (1976), Springer-Verlag Berlin · Zbl 0344.46071
[4] Bohnenblust, H.F.; Hille, E., On the absolute convergence of Dirichlet series, Ann. of math. (2), 32, 3, 600-622, (1931) · Zbl 0001.26901
[5] Bohr, H., Über die bedeutung der potenzreihen unendlich vieler variabeln in der theorie der dirichletschen reihen \(\sum \frac{a_n}{n^2}\), Nachr. ges. wiss. Göttingen, math. phys. kl., 441-488, (1913) · JFM 44.0306.01
[6] Bombal, F.; Pérez-García, D.; Villanueva, I., Multilinear extensions of Grothendieck’s theorem, Q. J. math., 55, 4, 441-450, (2004) · Zbl 1078.46030
[7] Bonet, J.; Peris, A., On the injective tensor product of quasinormable spaces, Results math., 20, 1-2, 431-443, (1991) · Zbl 0754.46044
[8] G. Botelho, H.-A. Braunss, H. Junek, D. Pellegrino, Inclusions and coincidences for multiple multilinear mappings, preprint · Zbl 1175.46037
[9] Carl, B., Absolut-\((p, 1)\)-summierende identische operatoren von \(l_u\) in \(l_v\), Math. nachr., 63, 353-360, (1974) · Zbl 0292.47019
[10] Choi, Y.S.; Kim, S.G.; Meléndez, Y.; Tonge, A., Estimates for absolutely summing norms of polynomials and multilinear maps, Q. J. math., 52, 1, 1-12, (2001) · Zbl 0991.46024
[11] Davie, A.M., Quotient algebras of uniform algebras, J. London math. soc. (2), 7, 31-40, (1973) · Zbl 0264.46055
[12] Defant, A.; Floret, K., Tensor norms and operator ideals, North-holland math. stud., vol. 176, (1993), North-Holland Amsterdam · Zbl 0774.46018
[13] Defant, A.; Maestre, M., Property (BB) and holomorphic functions on fréchet – montel spaces, Math. proc. Cambridge philos. soc., 115, 2, 305-313, (1994) · Zbl 0847.46021
[14] A. Defant, C. Prengel, Volume estimates in spaces of homogeneous polynomials, Math. Z., doi:10.1007/s00209-008-0358-x · Zbl 1171.46033
[15] A. Defant, P. Sevilla-Peris, Convergence of Dirichlet polynomials in Banach spaces, preprint
[16] Defant, A.; Díaz, J.C.; García, D.; Maestre, M., Unconditional basis and gordon – lewis constants for spaces of polynomials, J. funct. anal., 181, 1, 119-145, (2001) · Zbl 0986.46031
[17] Defant, A.; Maestre, M.; Sevilla-Peris, P., Cotype 2 estimates for spaces of polynomials on sequence spaces, Israel J. math., 129, 291-315, (2002) · Zbl 1009.46032
[18] Defant, A.; García, D.; Maestre, M., Bohr’s power series theorem and local Banach space theory, J. reine angew. math., 557, 173-197, (2003) · Zbl 1031.46014
[19] A. Defant, D. García, M. Maestre, D. Pérez-García, Bohr’s strip for vector valued Dirichlet series, Math. Ann., doi:10.1007/s00208-008-0246-z
[20] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely summing operators, Cambridge stud. adv. math., vol. 43, (1995), Cambridge Univ. Press Cambridge · Zbl 0855.47016
[21] Dineen, S., Complex analysis on infinite-dimensional spaces, Springer monogr. math., (1999), Springer London · Zbl 1034.46504
[22] Floret, K., Natural norms on symmetric tensor products of normed spaces, Note mat., 17, 153-188, (1997) · Zbl 0961.46013
[23] Floret, K., The extension theorem for norms on symmetric tensor products of normed spaces, (), 225-237 · Zbl 1010.46019
[24] Harris, L.A., Bounds on the derivatives of holomorphic functions of vectors, (), 145-163
[25] Kaijser, S., Some results in the metric theory of tensor products, Studia math., 63, 2, 157-170, (1978) · Zbl 0392.46047
[26] König, H., Eigenvalue distribution of compact operators, Oper. theory adv. appl., vol. 16, (1986), Birkhäuser Basel · Zbl 0618.47013
[27] Kwapień, S., Some remarks on \((p, q)\)-absolutely summing operators in \(l_p\)-spaces, Studia math., 29, 327-337, (1968) · Zbl 0182.17001
[28] Lindenstrauss, J.; Tzafriri, L., Classical Banach spaces. I, sequence spaces, Ergeb. math. grenzgeb., vol. 92, (1977), Springer Berlin · Zbl 0362.46013
[29] Lindenstrauss, J.; Tzafriri, L., Classical Banach spaces. II, function spaces, Ergeb. math. grenzgeb., vol. 97, (1979), Springer Berlin · Zbl 0403.46022
[30] Littlewood, J.E., On bounded bilinear forms in an infinite number of variables, Quart. J. (Oxford ser.), 1, 164-174, (1930) · JFM 56.0335.01
[31] Maurey, B.; Pisier, G., Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia math., 58, 1, 45-90, (1976) · Zbl 0344.47014
[32] Mellon, P., The polarisation constant for \(\operatorname{JB}^*\)-triples, Extracta math., 9, 3, 160-163, (1994) · Zbl 0903.46067
[33] Orlicz, W., Über unbedingte konvergenz in funktionenräumen, I, Studia math., 4, 33-37, (1933) · Zbl 0008.31501
[34] Pérez-García, D.; Villanueva, I., Multiple summing operators on Banach spaces, J. math. anal. appl., 285, 1, 86-96, (2003) · Zbl 1044.46037
[35] Pietsch, A., Eigenvalues and s-numbers, Cambridge stud. adv. math., vol. 13, (1987), Cambridge Univ. Press Cambridge · Zbl 0615.47019
[36] Praciano-Pereira, T., On bounded multilinear forms on a class of \(l^p\) spaces, J. math. anal. appl., 81, 2, 561-568, (1981) · Zbl 0497.46007
[37] Queffélec, H., H. Bohr’s vision of ordinary Dirichlet series; old and new results, J. anal., 3, 43-60, (1995) · Zbl 0881.11068
[38] Sawa, J., The best constant in the Khintchine inequality for complex Steinhaus variables, the case \(p = 1\), Studia math., 81, 1, 107-126, (1985) · Zbl 0612.60017
[39] Talagrand, M., Cotype and \((q, 1)\)-summing norm in a Banach space, Invent. math., 110, 3, 545-556, (1992) · Zbl 0814.46010
[40] Tomczak-Jaegermann, N., Banach – mazur distances and finite-dimensional operator ideals, Pitman monographs surveys pure appl. math., vol. 38, (1989), Longman Harlow · Zbl 0721.46004
[41] Tonge, A., Polarization and the two-dimensional Grothendieck inequality, Math. proc. Cambridge philos. soc., 95, 2, 313-318, (1984) · Zbl 0563.46041
[42] Zalduendo, I., An estimate for multilinear forms on \(\ell^p\) spaces, Proc. roy. irish acad. sect. A, 93, 1, 137-142, (1993) · Zbl 0790.46016
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