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 \(H^{\varphi }(\mathbb {R}^n)\).