Agmon-Kolmogorov inequalities on $\ell^2(\Bbb Z^d)$

Abstract

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.