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Electronic Research Announcements

ISSN 1079-6762



On Bojarski's index formula for nonsmooth interfaces

Author: Marius Mitrea
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 40-46
MSC (1991): Primary 58G10, 42B20; Secondary 34L40, 30D55
Published electronically: April 6, 1999
MathSciNet review: 1679452
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Abstract: Let $D$ be a Dirac type operator on a compact manifold ${\mathcal{M}}$ and let $\Sigma $ be a Lipschitz submanifold of codimension one partitioning ${\mathcal{M}}$ into two Lipschitz domains $\Omega _{\pm }$. Also, let ${\mathcal{H}}^{p}_{\pm }(\Sigma ,D)$ be the traces on $\Sigma $ of the ($L^{p}$-style) Hardy spaces associated with $D$ in $\Omega _{\pm }$. Then $({\mathcal{H}}^{p}_{-}(\Sigma ,D),{\mathcal{H}}^{p}_{+}(\Sigma ,D))$ is a Fredholm pair of subspaces for $L^{p}(\Sigma )$ (in Kato's sense) whose index is the same as the index of the Dirac operator $D$ considered on the whole manifold ${\mathcal{M}}$.

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  • [Ar] N. Aronszajn, A unique continuation theorem for solutions of elliptic differential equations or inequalities of second order, Journ. de Math. 36 (1957), 235-249. MR 19:1056c
  • [Bo] B. Bojarski, The abstract linear conjugation problem and Fredholm pairs of subspaces, In Memoriam I. N. Vekua, Tbilisi University, Tbilisi, 1979, 45-60. MR 83d:47021
  • [BBW1] B. Booß-Bavnbek and K. P. Wojciechowski, Desuspension of splitting elliptic symbols II, Ann. Global. Anal. Geom. 4 (1986), 349-400. MR 89f:58126
  • [BBW2] B. Booß-Bavnbek and K. P. Wojciechowski, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston-Basel-Berlin, 1993. MR 94h:58168
  • [CMcM] R. Coifman, A. McIntosh and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $L^{2}$ pour les courbes Lipschitziennes, Ann. of Math. 116 (1982), 361-387. MR 84m:42027
  • [Co] H. O. Cordes, Über die eindeutige Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. 11 (1956), 239-258. MR 19:148a
  • [Da] G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. E.N.S. 17 (1984), 157-189. MR 85k:42026
  • [GM] J. Gilbert and M. A. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, Vol.26, 1991. MR 93e:42027
  • [Ka] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1976. MR 53:11389
  • [Ke] C. E. Kenig, Weighted $H^{p}$ spaces on Lipschitz domains, Amer. J. Math. 102 (1980), 129-163. MR 81d:30060
  • [LMcS] C. Li, A. McIntosh and S. Semmes, Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc. 5 (1992), 455-481. MR 93b:42029
  • [MeCo] Y. Meyer and R. R. Coifman, Ondelettes et opérateurs, Vol.III, Hermann, editeurs des sciences et des arts, Paris, 1990. MR 93i:42004
  • [Mi] M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Mathematics No.1575, Springer-Verlag, Berlin, New York, 1994. MR 96e:31005
  • [Mi2] M. Mitrea, Generalized Dirac operators on nonsmooth manifolds and Maxwell's equations, in prepartion (1998).
  • [Mu] N. I. Musheli\v{s}vili, Singular Integral Equations, English translation of the 1946 edition: Noordhoff, Groningen, 1953. MR 15:434e
  • [Ni] L. Nicolaescu, The Maslov index, the spectral flow, and decompositions of manifolds, Duke Math. J. 80 (1995), 485-533. MR 96k:58208
  • [Ta] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991. MR 92j:35193

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Additional Information

Marius Mitrea
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211

Received by editor(s): December 2, 1998
Published electronically: April 6, 1999
Additional Notes: Partially supported by NSF
Communicated by: Stuart Antman
Article copyright: © Copyright 1999 American Mathematical Society

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