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Polynomial and multilinear Hardy-Littlewood inequalities: analytical and numerical approaches. (English) Zbl 1405.46028
Estimates for the constants in the polynomial and the multilinear Hardy-Littlewood inequalities are given. The multilinear inequality states that, given \( 2 \leq m \) and \(1 \leq p \leq \infty \) with \(2m \leq p\), there is a (best possible) constant \(C_{m,p}\geq 1\) such that, for every \(n\) and every \(m\)-linear form \(T\) on \(\ell_p^n\), \[ \Big(\sum_{i_1, \dots,i_m=1 }^\infty |T(e_{i_1}, \dots, e_{i_m})|^{ \frac{2mp}{mp+p-2m}}\Big)^{ \frac{mp+p-2m}{2mp}} \leq C_{m,p} \|T\|\,, \] where \(\|T\|\) stands for the sup norm of \(T\) on the unit ball of \(\ell^n_p\). Moreover, the exponent in the left sum is optimal. Changing the constant \(C_{m,p}\) by some other constant \(D_{m,p}\), a similar inequality holds for \(m\)-homogeneous polynomials \(P\) on \(\ell_p^n\) (instead of multilinear forms \(T\)). In the special case \(p=\infty\), these are the so-called polynomial and multilinear Bohnenblust-Hille inequalities which have recently shown significant applications in many different topics of modern analysis. For many of these applications, good upper and lower estimates for the constants \(C_{m,p}\) and \(D_{m,p}\) are essential, and in this context for the Bohnenblust-Hille inequalities, the breakthroughs came with [A. Defant et al., Ann. Math. (2) 174, No. 1, 485–497 (2011; Zbl 1235.32001)] and [F. Bayart et al., Adv. Math. 264, 726–746 (2014; Zbl 1235.32001)]. In the present article, the authors continue their intensive study of upper and lower estimates for the constants of the polynomial and the multilinear Hardy-Littlewood inequalities, in particular, present some computer-aided estimates.

46G25 (Spaces of) multilinear mappings, polynomials
47L22 Ideals of polynomials and of multilinear mappings in operator theory
47H60 Multilinear and polynomial operators
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