For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev inequality
$||u||_{L^2(\mathbb{R}^2)} \leq C ||\nabla u||_{L^1(\mathbb{R}^2)}$
or the Poincare inequality
$||u - \tilde{u}||_{L^2(\Omega)} \leq C ||\nabla u||_{L^1(\Omega)}$
where $\tilde{u} = \int_{\Omega} u \ dx$.
It is not so hard to prove the Poincare inequality via contradiction. For particular domains $\Omega$ (e.g. disc, polygon) one can also exhibit direct proofs that give either the optimal $C$ and something not too far from optimal.
My question is this: has there been any work on "discrete Poincare inequalities" of the type
$||u - Pu||_2 \leq C ||Du||_1 $
where $D: R^n \to R^m$ is some linear transformation and $P$ is the orthogonal projection onto its kernel.
For what it's worth, for general $D$ one can prove $||u-Pu||_2 \leq \frac{\sqrt{m}}{\min \sigma_i} ||Du||_1$ via the singular value decomposition of $D$ (here the minimum is over the nonzero singular values of $D$). But this is not taking into account any special properties of $D$ (e.g. block toeplitz, block circulant, etc) and is not as strong as I would like (specifically, the $\sqrt{m}$ factor).
The situation I have in mind is when $D$ is some finite difference matrix (although perhaps nonstandard). For standard finite differences on $\mathbb{Z}^2$ it is not too hard to show directly
$||u||_{L^2(\mathbb{Z}^2)} \leq C ||Du||_{L^1(\mathbb{Z}^2)}$
in a manner analogous to the classical proof. The discrete Poincare inequality would be more work (and the constant there would depend on the boundary conditions of the difference operator). But really, I would also like this to work for e.g. centered finite differences, or finite difference kernels with higher order of approximation. Is there any literature on this sort of thing?