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Optimal Hardy-Littlewood inequalities uniformly bounded by a universal constant. (English) Zbl 1410.46027
The original Hardy-Littlewood inequality, published in [G. H. Hardy and J. E. Littlewood, Q. J. Math., Oxf. Ser. 5, 241–254 (1934; JFM 60.0335.01)], relates some \(\ell_{r}\)-norm of the coefficients of a bilinear form on \(\mathbb{C}^{n} \times \mathbb{C}^{n}\) with the supremum on \(B_{\ell_{p}^{n}} \times B_{\ell_{q}^{n}}\). This was later extended to \(m\)-linear forms. This article contributes in this direction. Given \(1 < p_{1}, \ldots , p_{m} \leq \infty\), denote \(\frac{1}{\mathbf{p}} := \frac{1}{p_{1}} + \cdots + \frac{1}{p_{m}}\) (here we use the convention \(\frac{1}{\infty}=0\)). If \(\frac{1}{2} \leq \frac{1}{\mathbf{p}} <1\), then, for every \(n \in \mathbb{N}\) and every \(m\)-linear \(T : \mathbb{C}^{n} \times \cdots \times \mathbb{C}^{n} \to \mathbb{C}\), we have \[ \bigg( \sum_{i_{1}, \dots , i_{m}=1}^{n} | T(e_{i_{1}}, \dots , e_{i_{m}}) |^{\frac{1}{1-\frac{1}{\mathbf{p}}}} \bigg)^{1-\frac{1}{\mathbf{p}}} \leq 2^{(m-1) \big( 1 - \frac{1}{\mathbf{p} }\big)} \sup_{\substack{ {\| x_{j} \|_{p_{j}} < 1}\\ {j=1, \dots ,m}}} | T(x_{1} , \ldots , x_{m}) | . \] Some other similar results, involving mixed sums, are also given.

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
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