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

W. Cavalcante; D. Nunez-Alarcon; D. Pellegrino

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

Functional Analysis 2015-08-27 v3

Authors: W. Cavalcante , D. Nunez-Alarcon , D. Pellegrino

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.

Keywords

Hardy--Littlewood inequality m-homogeneous polynomials m-linear forms optimal constant ℓ_p spaces optimal exponents

Cite

@article{arxiv.1507.06099,
  title  = {Remarks on the Hardy--Littlewood inequality for $m$-homogeneous polynomials and $m$-linear forms},
  author = {W. Cavalcante and D. Nunez-Alarcon and D. Pellegrino},
  journal= {arXiv preprint arXiv:1507.06099},
  year   = {2015}
}

Comments

One minor change in the text

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