Extended Sobolev Scale on Non-Compact Manifolds

Abstract

Adapting the definition of ``extended Sobolev scale" on compact manifolds by Mikhailets and Murach to the setting of a (generally non-compact) manifold of bounded geometry $X$, we define the ``extended Sobolev scale" $H^{\varphi}(X)$, where $\varphi$ is a function which is $RO$-varying at infinity. With the help of the scale $H^{\varphi}(X)$, we obtain a description of all Hilbert function-spaces that serve as interpolation spaces with respect to a pair of Sobolev spaces $[H^{(s_0)}(X), H^{(s_1)}(X)]$, with $s_0<s_1$. We use this interpolation property to establish a mapping property of proper uniform pseudo-differential operators (PUPDOs) in the context of the scale $H^{\varphi}(X)$. Additionally, using a first-order positive-definite PUPDO $A$ of elliptic type we define the ``extended $A$-scale" $H^{\varphi}_{A}(X)$ and show that it coincides, up to norm equivalence, with the scale $H^{\varphi}(X)$. Besides the mentioned results, we show that further properties of the $H^{\varphi}$-scale, originally established by Mikhailets and Murach on $\mathbb{R}^n$ and on compact manifolds, carry over to manifolds of bounded geometry.