Extended Sobolev Scale on $\mathbb{Z}^n$

Abstract

In analogy with the definition of ``extended Sobolev scale" on $\mathbb{R}^n$ by Mikhailets and Murach, working in the setting of the lattice $\mathbb{Z}^n$, we define the ``extended Sobolev scale" $H^{\varphi}(\mathbb{Z}^n)$, where $\varphi$ is a function which is $RO$-varying at infinity. Using the scale $H^{\varphi}(\mathbb{Z}^n)$, we describe all Hilbert function-spaces that serve as interpolation spaces with respect to a pair of discrete Sobolev spaces $[H^{(s_0)}(\mathbb{Z}^n), H^{(s_1)}(\mathbb{Z}^n)]$, with $s_0<s_1$. We use this interpolation result to obtain the mapping property and the Fredholmness property of (discrete) pseudo-differential operators (PDOs) in the context of the scale $H^{\varphi}(\mathbb{Z}^n)$. Furthermore, starting from a first-order positive-definite (discrete) PDO $A$ of elliptic type, we define the ``extended discrete $A$-scale" $H^{\varphi}_{A}(\mathbb{Z}^n)$ and show that it coincides, up to norm equivalence, with the scale $H^{\varphi}(\mathbb{Z}^n)$. Additionally, we establish the $\mathbb{Z}^n$-analogues of several other properties of the scale $H^{\varphi}(\mathbb{R}^n)$.