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 , we define the “extended Sobolev scale” , where is a function which is ‐varying at infinity. With the help of the scale , we obtain a description of all Hilbert function‐spaces that serve as interpolation spaces with respect to a pair of Sobolev spaces , with . We use this interpolation property to establish a mapping property of proper uniform pseudo‐differential operators (PUPDOs) in the context of the scale . Additionally, using a first‐order positive‐definite PUPDO of elliptic type we define the “extended ‐scale” and show that it coincides, up to norm equivalence, with the scale . Besides the mentioned results, we show that further properties of the ‐scale, originally established by Mikhailets and Murach on and on compact manifolds, carry over to manifolds of bounded geometry.