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" $H^{\ddot{I}}†(X)$, where $\ddot{I}†$ is a function which is $RO$-varying at infinity. With the help of the scale $H^{\ddot{I}}†(X)$, 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 $H^{\ddot{I}}†(X)$. Additionally, using a first-order positive-definite PUPDO $A$ of elliptic type we define the ``extended $A$-scale" $H^{\ddot{I}}{†}_A(X)$ and show that it coincides, up to norm equivalence, with the scale $H^{\ddot{I}}†(X)$. Besides the mentioned results, we show that further properties of the $H^{\ddot{I}}†$-scale, originally established by Mikhailets and Murach on ${\mathbb{R}}^n$ and on compact manifolds, carry over to manifolds of bounded geometry.

We clarified the exposition in several places and we added a few references. arXiv admin note: text overlap with arXiv:2310.10894