zbMATH — the first resource for mathematics

Anisotropic regularity principle in sequence spaces and applications. (English) Zbl 1411.46037
The paper under review provides a very interesting version of the inclusion theorem for absolutely summing operators in the multilinear framework. Recall that absolutely summing operators are linear operators between Banach spaces that improve the convergence of series.
The main illustration of this phenomenon is given by Grothendieck’s theorem: every continuous linear operator \(T:\ell_1\rightarrow\ell_2\) is absolutely \((1;1)\)-summing. This means that every continuous linear operator \(T:\ell _1\rightarrow\ell_2\) is such that \[ \sum_{j=1}^{\infty}\left\| T(x_{j})\right\| <\infty \] whenever \((x_{j})_{j=1}^{\infty}\) is weakly summable in \(\ell_1\). This is specially dramatic because this result does not happen when \((\ell_{p},\ell_{q})\neq(\ell_1,\ell_2)\). The theory of absolutely summing operators has a profound impact in functional analysis and applications, and in the last decades it was extended to the multilinear setting, with interesting connections with quantum information theory.
The notion of absolutely \((p;q)\)-summing operators is a natural extension of the concept of absolutely \((1;1)\)-summing operators and one of the cornerstones of the theory is Pietsch’s domination theorem. A corollary of Pietsch’s domination theorem is that every absolutely \((r;r)\)-summing operator is also absolutely \((s;s)\)-summing whenever \(1\leq r\leq s\). The inclusion theorem refines this information when we deal with \((p;q)\)-summing operators. It asserts that every absolutely \((p_1;q_1)\)-summing operator is absolutely \((p_2 ;q_2)\)-summing whenever \(1\leq p_1\leq p_2\), \(1\leq q_1\leq q_2\) and \[ \frac{1}{q_1}-\frac{1}{p_1}\leq\frac{1}{q_2}-\frac{1}{p_2}. \] The existence of this kind of result in the multilinear framework was investigated by several authors. For instance, D. Pérez-García [Stud. Math. 165, No. 3, 275–290 (2004; Zbl 1064.47057)] showed that the inclusion theorem has an intriguing behaviour in the context of multiple summing operators. Essentially, Pérez-García proved that every multiple \((r;r)\)-summing operator is also multiple \((s;s)\)-summing whenever \(1\leq r\leq s<2\) and, in general, it cannot be extended for other parameters \(s\geq2\). D. Pellegrino et al. [Bull. Sci. Math. 141, No. 8, 802–837 (2017; Zbl 1404.46041)] and F. Bayart [J. Funct. Anal. 274, No. 4, 1129–1154 (2018; Zbl 1391.46057)] investigated cases that were not covered by Pérez-García’s inclusion theorem, providing results encompassing multiple \((p;q)\)-summing operators with \(p\neq q\) and also for values bigger that \(2\) (that were not covered before).
In the paper under review, using a tricky and original approach, the authors push this subject further and improve the previous theorems by providing a new anisotropic inclusion for multilinear operators. It is proved that, if \(m\) is a positive integer, \(r\geq1\), \(\mathbf{s}:=(s_1,\dots,s_{m})\), and \(\mathbf{p}:=(p_1,\dots ,p_{m})\), \(\mathbf{q}:=(q_1,\dots,q_{m})\in[1,\infty)^{m}\) are such that \(q_{k}\geq p_{k}\), \(k=1,\dots,m\), and \[ \frac{1}{r}-\sum_{i=1}^{m}\frac{1}{p_{i}}+\sum_{i=1}^{m}\frac{1}{q_{i}}>0, \] then every multiple \((r;\mathbf{p})\)-summing operator is multiple \((\mathbf{s} ;\mathbf{q})\)-summing whenever \[ \frac{1}{s_{k}}-\sum_{i=k}^{m}\frac{1}{q_{i}}=\frac{1}{r}-\sum_{i=k}^{m} \frac{1}{p_{i}} \] for each \(k\in\{1,\dots,m\}\).
Albeit technical, the above result has immediate and relevant consequences. For instance, applications to the theory of Hardy-Littlewood inequalities for \(m\)-linear forms are provided. In particular, it is proved that, for a fixed a positive integer \(m\) and \(p\in(m,2m]\), we have \[ \left(\sum_{j_1=1}^{n}\left(\ldots\left(\sum_{j_{m}=1}^{n}\left| T(e_{j_1},\dots,e_{j_{m}})\right| ^{s_{m}}\right)^{\frac{s_{m-1}}{s_{m}}}\dots\right)^{\frac{s_1}{s_2}}\right)^{\frac{1}{s_1}}\leq(\sqrt{2})^{m-1}\| T\| \] with \[ s_1=\frac{p}{p-m},\;\dots,\;s_{m}=\frac{2mp}{mp+p-2m}, \] for all positive integers \(n\) and all \(m\)-linear forms \(T:\ell_{p}^{n} \times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}\). This result is a beautiful generalization of the multilinear version of the Hardy-Littlewood inequality presented in [V. Dimant and P. Sevilla-Peris, Publ. Mat., Barc. 60, No. 2, 289–310 (2016; Zbl 1378.46032)].

46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
47J22 Variational and other types of inclusions
Full Text: DOI arXiv
[1] Albuquerque, N.; Bayart, F.; Pellegrino, D.; Seoane-Sepúlveda, J. B., Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators, Israel J. Math., 211, 1, 197-220, (2016) · Zbl 1342.26040
[2] Araújo, G.; Pellegrino, D.; da S. e. Silva, D. D. P., On the upper bounds for the constants of the Hardy-Littlewood inequality, J. Funct. Anal., 267, 6, 1878-1888, (2014) · Zbl 1298.26066
[3] Aron, R.; Pellegrino, D.; Núñez-Alarcón, D.; Serrano-Rodríguez, D., Optimal exponents for Hardy-Littlewood inequalities for \(m\)-linear operators, Linear Algebra its Appl., 531, 399-422, (2017) · Zbl 06770622
[4] F. Bayart, Multiple summing maps: Coordinatewise summability, inclusion theorems and \(p\)-Sidon sets, to appear in J. Funct. Anal. (2017); doi:10.1016/j.jfa.2017.08.013.
[5] Benedek, A.; Panzone, R., The space \(L_{\mathbf{p}}\) with mixed norm, Duke Math. J., 28, 301-324, (1961) · Zbl 0107.08902
[6] W. V. Cavalcante, Some applications of the regularity principle in sequence spaces, to appear in Positivity (2017); doi:10.1007/s11117-017-0506-9.
[7] Cavalcante, W. V.; Núñez-Alarcón, D., Remarks on an inequality of Hardy and Littlewood, Quaest. Math., 39, 8, 1101-1113, (2016)
[8] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely Summing Operators, 43, 197-210, (1995), Cambridge University Press, Cambridge · Zbl 0855.47016
[9] Dimant, V.; Sevilla-Peris, P., Summation of coefficients of polynomials on \(\ell_p\) spaces, Publ. Mat., 60, 2, 289-310, (2016) · Zbl 1378.46032
[10] Garling, D. J. H., Inequalities: A Journey into Linear Analysis, (2007), Cambridge University Press, Cambridge · Zbl 1135.26014
[11] Hardy, G.; Littlewood, J. E., Bilinear forms bounded in space \([p, q]\), Quart. J. Math., 5, 241-254, (1934) · Zbl 0010.36101
[12] Littlewood, J. E., On bounded bilinear forms in an infinite number of variables, Quart. J. Math. Oxford, 1, 164-174, (1930) · JFM 56.0335.01
[13] Maia, M.; Nogueira, T.; Pellegrino, D., The Bohnenblust-Hille inequality for polynomials whose monomials have a uniformly bounded number of variables, Integral Equations Operator Theory, 88, 143-149, (2017) · Zbl 1378.32001
[14] Matos, M. C., Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math., 54, 111-136, (2003) · Zbl 1078.46031
[15] Nunes, A., A new estimate for the constants of an inequality due to Hardy and Littlewood, Linear Algebra Appl., 526, 27-34, (2017) · Zbl 1376.46033
[16] Pellegrino, D.; Santos, J.; Seoane-Sepulveda, J. B., Some techniques on nonlinear analysis and applications, Adv. Math., 229, 2, 1235-1265, (2012) · Zbl 1248.47024
[17] D. Pellegrino, J. Santos, D. Serrano-Rodrigueź and E. Teixeira, A regularity principle in sequence spaces and applications, Bulletin des Sciences Mathématiques 141(8) (2017) 802-837. · Zbl 1404.46041
[18] D. Pellegrino and E. Teixeira, Towards sharp Bohnenblust-Hille constants, to appear in Comm. Contemp. Math.; doi:10.1142/S0219199717500298.
[19] D. Pérez-García, Operadores multilineales absolutamente sumantes, Ph.D. thesis, Universidad Complutense de Madrid (2004).
[20] Praciano-Pereira, T., On bounded multilinear forms on a class of \(\ell_p\) spaces, J. Math. Anal. Appl., 81, 561-568, (1981) · Zbl 0497.46007
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.