Multidimensional lower density versions of Pl\"unnecke's inequality

We investigate the lower asymptotic density of sumsets in ${\mathbb{N}}^2$ by proving certain Pl"unnecke type inequalities for various notions of lower density in ${\mathbb{N}}^2$. More specifically, we introduce a notion of lower tableaux density in ${\mathbb{N}}^2$ which involves averaging over convex tableaux-shaped regions in ${\mathbb{N}}^2$ which contain the origin. This generalizes the well known Pl"unnecke type inequality for the lower asymptotic density of sumsets in $\mathbb{N}$. We also provide a conjectural Pl"unnecke inequality for the more basic notion of lower rectangular asymtpotic density in ${\mathbb{N}}^2$ and prove certain partial results.