Remarks on an inequality of hardy and littlewood

Abstract

An inequality of Hardy and Littlewood for m-homogeneous polynomials on ℓp 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.

Mathematics Subject Classication (2010): 47B10, 26D15, 46B25.

Key words: Hardy and Littlewood inequalities, multiple summing operators.