Agmon-Kolmogorov inequalities on ${\ell }^2({\mathbb{Z}}^d)$

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 $\mathbb{R}$ and $\mathbb{Z}$, 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: $$\Vert \phi {\Vert }_{{\ell }^{\infty }({\mathbb{Z}}^d)}\le {\mu }_{p,d}\Vert {\nabla }_D\phi {\Vert }_{{\ell }^2({\mathbb{Z}}^d)}^{p/2^d}\Vert \phi {\Vert }_{{\ell }^2({\mathbb{Z}}^d)}^{1-p/2^d},$$ 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.