Anisotropic regularity principle in sequence spaces and applications.

*(English)*Zbl 1411.46037The 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)].

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)].

Reviewer: Daniel Pellegrino (João Pessoa)

##### MSC:

46G25 | (Spaces of) multilinear mappings, polynomials |

47H60 | Multilinear and polynomial operators |

47J22 | Variational and other types of inclusions |

##### Keywords:

inclusion theorem; multilinear operators; regularity principle; Hardy-Littlewood inequalities; absolutely \((p;q)\)-summing operator##### References:

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