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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.

##### MSC:
 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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