  We consider the mapping properties of generalized Laplace-type operators
${\mathcal L} = \nabla^* \nabla + {\mathcal R}$ on the class of
quasi-asymptotically conical (QAC) spaces, which provide a Riemannian
generalization of the QALE manifolds considered by Joyce. Our main result gives
conditions under which such operators are Fredholm when between certain
weighted Sobolev or weighted H\"older spaces. These are generalizations of
well-known theorems in the asymptotically conical (or asymptotically Euclidean)
setting, and also sharpen and extend corresponding theorems by Joyce. The
methods here are based on heat kernel estimates originating from old ideas of
Moser and Nash, as developed further by Grigor'yan and Saloff-Coste. As
demonstrated by Joyce's work, the QAC spaces here contain many examples of
gravitational instantons, and this work is motivated by various applications to
manifolds with special holonomy.
