We discuss how Hardy inequalities can be obtained and optimality can be shown. We illustrate this by various examples which also show a disparity to the continuum setting. Finally, we discuss how Hardy inequalities can be used to derive Rellich inequalities. Such inequalities are essential to estimate the decay of eigenfunctions in the spirit of Agmon estimates.
(This includes joint work with F. Fischer, M. Lemm, M. Nietschmann, F. Pogorzelski and Y. Pinchover)