The optimal Hardy--Littlewood constants for $2$-homogeneous polynomials on $\ell_{p}(\mathbb{R}^{2})$ for $2<p\leq4$ are $2^{2/p}$

TL;DR

This paper determines the exact optimal constants for Hardy--Littlewood inequalities concerning 2-homogeneous polynomials on two-dimensional real ℓₚ spaces for p between 2 and 4, showing they are 2^{2/p}.

Contribution

It provides the precise values of the Hardy--Littlewood constants for 2-homogeneous polynomials on ℓₚ(ℝ²) in the range 2<p<4, filling a gap in the existing literature.

Findings

Optimal constants are 2^{2/p} for 2<p<4.

The result applies specifically to 2-homogeneous polynomials.

The constants are exact and proven for the specified range.

Abstract

We show that the optimal constants for the Hardy--Littlewood inequalities for $2$-homogeneous polynomials on $\ell_{p}(\mathbb{R}^{2})$ are precisely $2^{2/p}$ for all $2<p<4.$