  Landau-Kolmogorov inequalities have been extensively studied on both
continuous and discrete domains for an entire century. However, the research is
limited to the study of functions and sequences on $\Bbb R$ and $\Bbb Z$, with
no equivalent inequalities in higher-dimensional spaces. The aim of this paper
is to obtain a new class of discrete Landau-Kolmogorov type inequalities of
arbitrary dimension: $$ \|\varphi\|_{\ell^\infty(\Bbb Z^d)} \leq
\mu_{p,d}\|\nabla_D\varphi\|^{p/2^d}_{\ell^2(\Bbb Z^d)}\,
\|\varphi\|^{1-p/2^d}_{\ell^2(\Bbb Z^d)}, % $$ where the constant $\mu_{p,d}$
is explicitly specified. In fact, this also generalises the discrete Agmon
inequality to higher dimension, which in the corresponding continuous case is
not possible.
