Sharp interpolation inequalities for discrete operators and applications

TL;DR

This paper develops a general method for deriving sharp interpolation inequalities for discrete operators, providing precise constants and extremal elements, with applications to spectral theory and Carlson's inequalities.

Contribution

It introduces a novel method for finding sharp constants and extremal elements in interpolation inequalities for discrete operators.

Findings

Derived sharp constants for interpolation inequalities

Identified extremal sequences and correction terms

Applied results to spectral theory of discrete operators

Abstract

We consider interpolation inequalities for imbeddings of the $l^2$-sequence spaces over $d$-dimensional lattices into the $l^\infty_0$ spaces written as interpolation inequality between the $l^2$-norm of a sequence and its difference. A general method is developed for finding sharp constants, extremal elements and correction terms in this type of inequalities. Applications to Carlson's inequalities and spectral theory of discrete operators are given.