Lower bounds for the constants of the Hardy-Littlewood inequalities

Given an integer $m\geq2$, the Hardy--Littlewood inequality (for real scalars) says that for all $2m\leq p\leq\infty$, there exists a constant $C_{m,p}% ^{\mathbb{R}}\geq1$ such that, for all continuous $m$--linear forms $A:\ell_{p}^{N}\times\cdots\times\ell_{p}^{N}\rightarrow\mathbb{R}$ and all positive integers $N$, \[ \left( \sum_{j_{1},...,j_{m}=1}^{N}\left\vert A(e_{j_{1}},...,e_{j_{m}% })\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}\leq C_{m,p}^{\mathbb{R}}\left\Vert A\right\Vert . \] The limiting case $p=\infty$ is the well-known Bohnenblust--Hille inequality; the behavior of the constants $C_{m,p}^{\mathbb{R}}$ is an open problem. In this note we provide nontrivial lower bounds for these constants.