Geodesic
Some Sharp $L^2$ Inequalities for Dirac Type Operators
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We use the spectra of Dirac type operators on the sphere $S^n$ to produce sharp $L^2$ inequalities on the sphere. These operators include the Dirac operator on $S^n$, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers of the Dirac operator and their inverses in $\mathbb R^n$.
Keywords: Dirac operator; Clifford algebra; conformal Laplacian; Paenitz operator. 
@article{SIGMA_2007_3_a113,
     author = {Alexander Balinsky and John Ryan},
     title = {Some {Sharp} $L^2$ {Inequalities} for {Dirac} {Type} {Operators}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2007},
     volume = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a113/}
}
TY  - JOUR
AU  - Alexander Balinsky
AU  - John Ryan
TI  - Some Sharp $L^2$ Inequalities for Dirac Type Operators
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2007
VL  - 3
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a113/
LA  - en
ID  - SIGMA_2007_3_a113
ER  - 
%0 Journal Article
%A Alexander Balinsky
%A John Ryan
%T Some Sharp $L^2$ Inequalities for Dirac Type Operators
%J Symmetry, integrability and geometry: methods and applications
%D 2007
%V 3
%U http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a113/
%G en
%F SIGMA_2007_3_a113
Alexander Balinsky; John Ryan. Some Sharp $L^2$ Inequalities for Dirac Type Operators. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a113/

[1] Ahlfors L. V., “Old and new in Möbius groups”, Ann. Acad. Sci. Fenn. Math., 9 (1984), 93–105 | MR | Zbl

[2] Beckner W., “Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality”, Ann. of Math., 138 (1993), 213–242 | DOI | MR | Zbl

[3] Bojarski B., Conformally covariant differential operators, Proceedings of XXth Iranian Math. Congress, Tehran, 1989

[4] Brackx F., Delanghe R., Sommen F., Clifford analysis, Pitman, London, 1982 | MR | Zbl

[5] Branson T., Ørsted B., “Spontaneous generators of eigenvalues”, J. Geom. Phys., 56 (2006), 2261–2278 ; math.DG/0506047 | DOI | MR | Zbl

[6] Calderbank D., Dirac operators and Clifford analysis on manifolds with boundary, Preprint no. 53, Institute for Mathematics, Syddansk University, 1997; available at http://bib.mathematics.dk/imada/

[7] Cnops J., Malonek H., An introduction to Clifford analysis, Textos de Matematica, Serie B, Universidade de Coimbra, Departmento de Matematica, Coimbra, 1995 | MR

[8] Davies E. B., Hinz A., “Explicit construction for Rellich inequalities”, Math. Z., 227 (1998), 511–523 | DOI | MR | Zbl

[9] Erdos L., Solovej J. P., “The kernel of Dirac operators on $S^3$ and $\mathbb R^3$”, Rev. Math. Phys., 10 (2001), 1247–1280 ; math-ph/0001036 | DOI | MR

[10] Lieb E., Loss M., Analysis, Graduate Texts in Mathematics, 14, American Mathematical Society, Providence, 2001 | MR | Zbl

[11] Liu H., Ryan J., “Clifford analysis techniques for spherical PDE”, J. Fourier Anal. Appl., 8 (2002), 535–564 | DOI | MR

[12] Ryan J., “Iterated Dirac operators in $\mathbb C^n$”, Z. Angew. Math. Phys., 9 (1990), 385–401 | MR | Zbl

[13] Ryan J., “Dirac operators on spheres and hyperbolae”, Bol. Soc. Mat. Mexicana, 3 (1996), 255–269 | MR

[14] Sommen F., “Spherical monogenic functions and analytic functionals on the unit sphere”, Tokyo J. Math., 4 (1981), 427–456 | DOI | MR | Zbl

[15] Sudbery A., “Quaternionic analysis”, Math. Proc. Cambridge Philos. Soc., 85:2 (1979), 199–225 | DOI | MR | Zbl

[16] Van Lanker P., “Clifford analysis on the sphere”, Clifford Algebras and their Applications in Mathematical Physics, eds. V. Dietrich et al., Kluwer, Dordrecht, 1998, 201–215 | MR | Zbl