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.