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Operators on manifolds with conical singularities

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Abstract

We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ 0, σ 1) where σ 0 is the standard (degenerate) homogeneous principal symbol, and σ 1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ 1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dim X, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ 0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ 1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges.

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Authors and Affiliations

  1. Beijing, People’s Republic of China

    L. Ma

  2. Potsdam, Germany

    B.-W. Schulze

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  1. L. Ma
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  2. B.-W. Schulze
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Correspondence to B.-W. Schulze.

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Ma, L., Schulze, BW. Operators on manifolds with conical singularities. J. Pseudo-Differ. Oper. Appl. 1, 55–74 (2010). https://doi.org/10.1007/s11868-010-0002-5

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  • DOI: https://doi.org/10.1007/s11868-010-0002-5

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