Lower bounds for the constants of the Hardy–Littlewood inequalities

Given an integer m≥2m≥2, the Hardy–Littlewood inequality (for real scalars) says that for all 2m≤p≤∞2m≤p≤∞, there exists a constant Cm,pR≥1 such that, for all continuous m  -linear forms A:ℓpN×⋯×ℓpN→R and all positive integers N,(∑j1,...,jm=1N|A(ej1,...,ejm)|2mpmp+p−2m)mp+p−2m2mp≤Cm,pR‖A‖. The limiting case p=∞p=∞ is the well-known Bohnenblust–Hille inequality; the behavior of the constants Cm,pR is an open problem. In this note we provide nontrivial lower bounds for these constants.