Abstract
We construct Mellin quantisations or operator conventions, applied to corner–degenerate pseudo-differential symbols, referring to geometric corners of singularity order \( k \in \mathbb{N} \), and we obtain holomorphic Mellin symbols in \(z \in {\mathbb{C}}^{k}\), also with a corresponding degenerate behaviour.
Mathematics Subject Classification (2010). Primary 35S35; Secondary 35J70, 47G30, 58J40.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Calvo, C.-I. Martin, and B.-W. Schulze, Symbolic structures on corner manifolds, RIMS Conf. dedicated to L. Boutet de Monvel on “Microlocal Analysis and Asymptotic Analysis”, Kyoto, August 2004, Keio University, Tokyo, 2005, pp. 22–35.
D. Calvo and B.-W. Schulze, Edge symbolic structures of second generation, Math. Nachr. 282 (2009), 348–367.
M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341, Springer-Verlag, Berlin, Heidelberg, 1988.
N. Dines, Elliptic Operators on Corner Manifolds, Ph.D. thesis, University of Potsdam, 2006.
G.I. Eskin, Boundary Value Problems for Elliptic Pseudo-Differential Equations, Transl. of Nauka, Moskva, 1973, Math. Monographs, Amer. Math. Soc. 52, Providence, Rhode Island 1980.
R.C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, American Mathematical Soc., Rhode Island, 1965.
N. Habal and B.-W. Schulze, Holomorphic corner symbols, J. Pseudo-D iffer. Oper. Appl. 2, 4 (2011), 419–465.
G. Harutyunyan and B.-W. Schulze, The relative index for corner singularities, Integr. Equ. Oper. Theory 54, 3 (2006), 385–426.
G. Harutyunyan and B.-W. Schulze, Elliptic Mixed, Transmission and Singular Crack Problems, European Mathematical Soc., Zürich, 2008.
D. Kapanadze and B.-W. Schulze, Crack Theory and Edge Singularities, Kluwer Academic Publ., Dordrecht, 2003.
A.I. Komech, Elliptic boundary problems for pseudo-differential operators on manifolds with conical points, Mat. Sb. 86, 2 (1971), 268–298.
V.A. Kondratyev, Boundary value problems for elliptic equations in domains with conical points, Trudy Mosk. Mat. Obshch. 16 (1967), 209–292.
T. Krainer, Parabolic Pseudo-Differential Operators and Long-time Asymptotics of Solutions, Ph.D. thesis, University of Potsdam, 2000.
L. Maniccia and B.-W. Schulze, An algebra of meromorphic corner symbols, Bull. des Sciences Math. 127, 1 (2003), 55–99.
Pham The Lai, Problème de Dirichlet dans un cône avec paramètre spectral pour une classe d’espaces de Sobolev à poids, Comm. Part. Diff. Equ. 4 (1979), 389–445.
B.A. Plamenevskij, On the boundedness of singular integrals in spaces with weight, Mat. Sb. 25, 4 (1968), 573–592.
B.A. Plamenevskij, Algebras of Pseudodifferential Operators, Nauka: Moscow, 1986.
B.A. Plamenevskij and V.N. Senichkin, On the representation of algebras of pseudodifferential operators on manifolds with discontinuous symbols (Russian), Izv. Akad. Nauk SSSR 51, 4 (1987), 833–859.
V.S. Rabinovich, Pseudo-differential operators in non-bounded domains with conical structure at infinity, Mat. Sb. 80 (1969), 77–97.
V.S. Rabinovich, Mellin pseudo-differential operators with operator symbols and their applications, Operator Theory: Adv. and Appl., Birkhäuser Verlag, Basel, 78 (1995), 271–279.
S. Rempel and B.-W. Schulze, Parametrices and boundary symbolic calculus for elliptic boundary problems without transmission property, Math. Nachr. 105 (1982), 45–149.
S. Rempel and B.-W. Schulze, Complete Mellin and Green symbolic calculus in spaces with conormal asymptotics, Ann. Glob. Anal. Geom. 4, 2 (1986), 137–224.
B.-W. Schulze, Pseudo-Differential Operators on Manifolds with Singularities, North- Holland, Amsterdam, 1991.
B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, J. Wiley, Chichester, 1998.
B.-W. Schulze, Operators with symbol hierarchies and iterated asymptotics, Publications of RIMS, Kyoto University 38, 4 (2002), 735–802.
B.-W. Schulze, The iterative structure of the corner calculus, Oper. Theory: Adv. Appl. 213, Pseudo-Differential Operators: Analysis, Application and Computations (L. Rodino et al. eds.), Birkhäuser Verlag, Basel, 2011, 79–103.
B.-W. Schulze and J. Seiler, The edge algebra structure of boundary value problems, Ann. Glob. Anal. Geom. 22 (2002), 197–265.
B.-W. Schulze and M.W. Wong, Mellin and Green operators of the corner calculus, J. Pseudo-Differ. Oper. Appl. 2, 4 (2011), 467–507.
J. Seiler, Pseudodifferential Calculus on Manifolds with Non-Compact Edges, Ph.D . thesis, University of Potsdam, 1997.
I. Witt, Explicit algebras with the Leibniz-Mellin translation product, Math. Nachr. 280, 3 (2007), 326–337.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to the 70th anniversary of V. Rabinovich
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Habal, N., Schulze, BW. (2013). Mellin Quantisation in Corner Operators. In: Karlovich, Y., Rodino, L., Silbermann, B., Spitkovsky, I. (eds) Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory: Advances and Applications, vol 228. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0537-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0537-7_8
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0536-0
Online ISBN: 978-3-0348-0537-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)