6533b7d4fe1ef96bd1261c33

RESEARCH PRODUCT

FREDHOLM THEORY FOR DEGENERATE PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBERED BOUNDARIES

Sergiu MoroianuRobert Lauter

subject

Pure mathematicsExact sequenceApplied MathematicsMathematical analysisFibrationFredholm integral equationOperator theoryFredholm theoryManifoldSobolev spacesymbols.namesakeMathematics::K-Theory and HomologyBounded functionsymbolsAnalysisMathematics

description

We consider the calculus Ψ*,* de(X, deΩ½) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects. We construct a scale of weighted double-edge Sobolev spaces on which double-edge pseudodifferential operators act as bounded operators, characterize the Fredholm elements in Ψ*,* de(X) by means of the invertibility of an appropriate symbol map, and describe a K-theoretical formula for the Fredholm index extending the Atiyah–Singer formula for closed manifolds. The algebra of operators of order (0, 0) is shown to be a Ψ*-algebra, hence its K-theory coincides with that of its C *-closure, and we give a description of the corresponding cyclic 6-term exact sequence. We define a Wodzicki-type residue trace on an ideal in Ψ*,* de(X, deΩ½), and we show that it coincides with Dixmier's trace for operators of order –dim X in ...

yearjournalcountryeditionlanguage
2001-01-31Communications in Partial Differential Equations
https://doi.org/10.1081/pde-100001754