Geodesic
Singular Integrals on Ultraspherical Series
Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 60-71

Voir la notice de l'article provenant de la source Cambridge University Press

One of the main uses of harmonic analysis on the sphere is to discover new theorems about series of ultraspherical (Gegenbauer) polynomials. In this paper, we will construct singular integral operators from scalar functions on the sphere to vector functions. These operators when restricted to zonal functions give Lp-bounded (1 < p < ∞ ) operators on ultraspherical series.We will use [7, Chapter 9] as our main reference. Let G denote a compact group, with identity e, and Ĝ its dual, the set of equivalence classes of continuous irreducible unitary representations of G. 
Dunkl, Charles F. Singular Integrals on Ultraspherical Series. Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 60-71. doi: 10.4153/CJM-1972-008-1
@article{10_4153_CJM_1972_008_1,
     author = {Dunkl, Charles F.},
     title = {Singular {Integrals} on {Ultraspherical} {Series}},
     journal = {Canadian journal of mathematics},
     pages = {60--71},
     year = {1972},
     volume = {24},
     number = {1},
     doi = {10.4153/CJM-1972-008-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1972-008-1/}
}
TY  - JOUR
AU  - Dunkl, Charles F.
TI  - Singular Integrals on Ultraspherical Series
JO  - Canadian journal of mathematics
PY  - 1972
SP  - 60
EP  - 71
VL  - 24
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1972-008-1/
DO  - 10.4153/CJM-1972-008-1
ID  - 10_4153_CJM_1972_008_1
ER  - 
%0 Journal Article
%A Dunkl, Charles F.
%T Singular Integrals on Ultraspherical Series
%J Canadian journal of mathematics
%D 1972
%P 60-71
%V 24
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1972-008-1/
%R 10.4153/CJM-1972-008-1
%F 10_4153_CJM_1972_008_1

[1] 1. Abramowitz, M. and Stegun, I., Handbook of mathematical functions, National Bureau of Standards (Dover, New York, 1964). Google Scholar

[2] 2. Askey, R. and Wainger, S., A transplantation theorem between ultraspherical series, Illinois J. Math. 10 (1966), 322–344. Google Scholar

[3] 3. Askey, R. and Wainger, S., On the behaviour of special classes of ultraspherical expansions. I, J. d'Anal. Math. 15 (1965), 193–220. Google Scholar

[4] 4. Boerner, H., Representations of groups (North-Holland, Amsterdam, 1963). Google Scholar

[5] 5. Calderón, A. P. and Zygmund, A., On singular integrals, Amer. J. Math. 78 (1956), 289–309. Google Scholar

[6] 6. Dunkl, C., Linear functionals on homogeneous polynomials, Can. Math. Bull. 11 (1968), 465–468. Google Scholar

[7] 7. Dunkl, C. and Ramirez, D., Topics in harmonic analysis (Appleton-Century-Crofts, New-York, 1971). Google Scholar

[8] 8. Hua, L. K., Harmonic analysis of functions of several complex variables in the classical domains, Amer. Math. Soc. Translation (Amer. Math. Soc, Providence, R.I., 1963). Google Scholar

[9] 9. Levine, D., Systems of singular integral operators on spheres, Trans. Amer. Math. Soc. 144 (1969), 493–522. Google Scholar

[10] 10. Muckenhoupt, B. and Stein, E. M., Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92. Google Scholar

[11] 11. Seeley, R. T., Singular integrals on compact manifolds, Amer. J. Math. 81 (1959), 658–690. Google Scholar

[12] 12. Seeley, R. T., Integro-differential operators on vector bundles, Trans. Amer. Math. Soc. 117 (1965), 167–204. Google Scholar

Cité par Sources :