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Analysis on Regular Corner Spaces

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Abstract

We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel’s calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11–51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective \(2\times 2\) operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11–51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind.

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Acknowledgements

The paper is based on lectures which were given by the authors during the 2019 International Workshop on Geometric Analysis and Harmonic Analysis that was held from May 27 to May 30 at National Center for Theoretical Sciences, Taipei, Taiwan. We would like to express their profound gratitude to Director Professor Junk-Kai Chen for his invitation and for the warm hospitality extended to them during their stay in Taiwan. The first author is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Georgetown University, Washington, DC, 20057, USA

    Der-Chen Chang

  2. Graduate Institute of Business Administration, College of Management, Fu Jen Catholic University, New Taipei City, 242, Taiwan, ROC

    Der-Chen Chang

  3. Institute of Mathematics, Jadara University, Irbid, Jordan

    Sara Khalil

  4. Institute of Mathematics, University of Potsdam, 14476, Potsdam, Germany

    Bert-Wolfgang Schulze

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  1. Der-Chen Chang
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Correspondence to Der-Chen Chang.

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Dedicated to Professor Guido Weiss on his 90th birthday.

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Chang, DC., Khalil, S. & Schulze, BW. Analysis on Regular Corner Spaces. J Geom Anal 31, 9199–9240 (2021). https://doi.org/10.1007/s12220-021-00614-3

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