Lower bounds for the complex polynomial Hardy–Littlewood inequality

Abstract

The Hardy–Littlewood inequality for complex homogeneous polynomials asserts that given positive integers m≥2 and n≥1, if P is a complex homogeneous polynomial of degree m on ℓpn with m<p≤∞ given by P(x1,…,xn)=∑|α|=maαxα, then there exists a constant CC,m,ppol≥1 (which does not depend on n) such that(∑|α|=m|aα|ρ)1ρ≤CC,m,ppol⋅supz∈Bℓpn⁡|P(z)|, with ρ=pp−m if m<p<2m and ρ=2mpmp+p−2m if 2m≤p≤∞. In this short note we provide nontrivial lower bounds for the constants CC,m,ppol. For instance we prove that, for m≥2 and m<p<∞,CC,m,ppol≥2mp for m even, andCC,m,ppol≥2m−1p for m odd. Estimates for the case p=∞ (this is the particular case of the complex polynomial Bohnenblust–Hille inequality) were recently obtained by D. Nuñez-Alarcón in 2013.