Abstract
We characterize the groupoids for which an operator is Fredholm if and only if its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called Fredholm . Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that are asymptotically Euclidean or asymptotically hyperbolic, on products of such manifolds, and on many other non-compact manifolds. Moreover, we show that the desingularization of groupoids preserves the class of Fredholm groupoids.
Highlights
We obtain necessary and sufficient conditions for operators modeled by groupoids to be Fredholm
One of the main example is that of operators obtained by desingularization of singular spaces by successively blowing up the lowest dimensional singular strata
We begin with a general study of Fredholm conditions for pseudodifferential operators in the framework of “Fredholm groupoids.”
Summary
We obtain necessary and sufficient conditions for operators modeled by groupoids to be Fredholm. We begin with a general study of Fredholm conditions for pseudodifferential operators in the framework of “Fredholm groupoids.”. We obtain a general characterization of Fredholm groupoids. Using some results of Renault [22, 23] and Ionescu and Williams [9], we show that an almost amenable, second-countable, Hausdorff groupoid is Fredholm. The “desingularization” [[ G : L ]] of G along L [18] is the a groupoid modeling the analysis on the space obtained by blowing-up L. We use the explicit structure of the desingularized groupoid [[ G : L ]] (see [18]) to show that it is Fredholm if G is.
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