Remarks on the Hardy--Littlewood inequality for $m$-homogeneous polynomials and $m$-linear forms

Abstract

The Hardy--Littlewood inequality for $m$-homogeneous polynomials on ${\ell }_p$ spaces is valid for $p>m.$ In this note, among other results, we present an optimal version of this inequality for the case $p=m.$ We also show that the optimal constant, when restricted to the case of $2$-homogeneous polynomials on ${\ell }_2({\mathbb{R}}^2)$ is precisely $2$. In an Appendix we justify why, curiously, the optimal exponents of the Hardy--Littlewood inequality do not behave smoothly.

One minor change in the text

References in corpus (2)

• Estimates on the norm of polynomials and applications
• The optimal Hardy--Littlewood constants for $2$-homogeneous polynomials on ${\ell }_p({\mathbb{R}}^2)$ for $2<p\le 4$ are $2^{2/p}$

Cited by in corpus (1)

• Optimal exponents for Hardy--Littlewood inequalities for $m$-linear operators