Remarks on an inequality of Hardy and Littlewood

Abstract

An inequality of Hardy and Littlewood for m-homogeneous polynomials on ℓ_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 and obtain the optimal constant, when restricted to the case of 2-homogeneous polynomials on ℓ₂(ℝ2). In an Appendix we justify why, curiously, the optimal exponents of some Hardy-Littlewood type inequalities do not behave smoothly.