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Motivated by the study of layer potentials on manifolds with straight conicalor cylindrical ends, we introduce and study two classes (or calculi) ofpseudodifferential operators defined on manifolds with cylindrical ends: theclass of pseudodifferential operators that are ``translation invariant atinfinity'' and the class of ``essentially translation invariant operators.''These are ``minimal'' classes of pseudodifferential operators containing thelayer potential operators of interest. Both classes are close to the$b$-calculus considered by Melrose and Schulze and to the $c$-calculusconsidered by Melrose and Mazzeo-Melrose. Our calculi, however, are differentand, while some of their properties follow from those of the $b$- or$c$-calculi, many of their properties do not. In particular, we prove that the``essentially translation invariant calculus'' is spectrally invariant, aproperty not enjoyed by the ``translation invariant at infinity'' calculus orthe $b$-calculus. For our calculi, we provide easy, intuitive proofs of theusual properties: stability for products and adjoints, mapping and boundednessproperties for operators acting between Sobolev spaces, regularity properties,existence of a quantization map, topological properties of our algebras, andthe Fredholm property. Since our applications will be to the Stokes operator,we systematically work in the setting of \ADN-elliptic operators. We also showthat our calculi behave well with respect to restrictions to (suitable)submanifolds, which is crucial for our applications to layer potentialoperators.

Layer potentials and essentially translation invariant pseudodifferential operators on manifolds with cylindrical ends