The linear symmetric systems associated with the modified Cherednik operators and applications
Annales mathématiques Blaise Pascal, Volume 19 (2012) no. 1, pp. 213-245.

We introduce and study the linear symmetric systems associated with the modified Cherednik operators. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite propagation speed property of it.

Nous présentons et étudions les systèmes symétriques linéaires associés aux opérateurs de Cherednik modifiés. Nous prouvons que le problème de Cauchy pour ces systèmes sont bien posé. Finalement nous en décrivons le principe de vitesse finie.

DOI: 10.5802/ambp.311
Classification: 35L05,  22E30
Keywords: Modified Cherednik operators, modified Cherednik symmetric systems, energy estimates, finite speed of propagation, generalized wave equations with variable coefficients
Hatem Mejjaoli 1

1 Department of Mathematics College of Sciences King Faisal University Ahsaa, Kingdom of Saudi Arabia
@article{AMBP_2012__19_1_213_0,
     author = {Hatem Mejjaoli},
     title = {The linear symmetric systems associated with the modified {Cherednik} operators and applications},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {213--245},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {19},
     number = {1},
     year = {2012},
     doi = {10.5802/ambp.311},
     mrnumber = {2978320},
     zbl = {1248.35116},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.311/}
}
TY  - JOUR
AU  - Hatem Mejjaoli
TI  - The linear symmetric systems associated with the modified Cherednik operators and applications
JO  - Annales mathématiques Blaise Pascal
PY  - 2012
DA  - 2012///
SP  - 213
EP  - 245
VL  - 19
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.311/
UR  - https://www.ams.org/mathscinet-getitem?mr=2978320
UR  - https://zbmath.org/?q=an%3A1248.35116
UR  - https://doi.org/10.5802/ambp.311
DO  - 10.5802/ambp.311
LA  - en
ID  - AMBP_2012__19_1_213_0
ER  - 
%0 Journal Article
%A Hatem Mejjaoli
%T The linear symmetric systems associated with the modified Cherednik operators and applications
%J Annales mathématiques Blaise Pascal
%D 2012
%P 213-245
%V 19
%N 1
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.311
%R 10.5802/ambp.311
%G en
%F AMBP_2012__19_1_213_0
Hatem Mejjaoli. The linear symmetric systems associated with the modified Cherednik operators and applications. Annales mathématiques Blaise Pascal, Volume 19 (2012) no. 1, pp. 213-245. doi : 10.5802/ambp.311. https://ambp.centre-mersenne.org/articles/10.5802/ambp.311/

[1] N. Ben Salem; A. Ould Ahmed Salem Convolution structure associated with the Jacobi-Dunkl operator on , Ramanujan J., Volume 12 (2006), pp. 359-378 | DOI | MR

[2] J. Chazarain; A. Piriou Introduction to the theory of linear partial differential equations, North-Holland Publishing Company, 1982 | MR | Zbl

[3] I. Cherednik A unification of Knizhnik-Zamolodchnikove equations and Dunkl operators via affine Hecke algebras, Invent. Math., Volume 106 (1991), pp. 411-432 | DOI | MR | Zbl

[4] F. Chouchene; M. Mili; K. Trimèche Positivity of the intertwining operator and harmonic analysis associated with the Jacobi-Dunkl operator on , Anal. and Appl., Volume 1 (2003), pp. 387-412 | DOI | MR

[5] R. Courant; K. Friedrichs; H. Lewy Über die partiellen Differenzengleichungen der mathematischen Physik", Math. Ann., Volume 100 (1928), pp. 32-74 | DOI | MR

[6] K. Friedrichs Symmetric hyperbolic linear differential equations, Comm. Pure and Appl. Math., Volume 7 (1954), pp. 345-392 | DOI | MR | Zbl

[7] K. Friedrichs Symmetric positive linear hyperbolic differential equations, Comm. Pure and Appl. Math., Volume 11 (1958), pp. 333-418 | DOI | MR | Zbl

[8] K. Friedrichs; P. D. Lax On symmetrizable differential operators, Proc. Symp. Pure Math., Volume 9 (1967), pp. 128-137 | DOI | MR | Zbl

[9] P. D. Lax On Cauchy’s problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure and Appl. Math., Volume 8 (1955), pp. 615-633 | DOI | MR | Zbl

[10] P. D. Lax; R. S. Phillips Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure and Appl. Math., Volume 13 (1960), pp. 427-455 | DOI | MR | Zbl

[11] G. Lebeau Majeure d’équations aux dérivées partielles, Prépublication de l’école polytechniques, Volume 28 (1990), pp. 43-62

[12] Eric M. Opdam Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., Volume 175 (1995), pp. 75-121 | DOI | MR | Zbl

[13] Eric M. Opdam Lecture notes on Dunkl operators for real and complex reflection groups, MSJ Memoirs, 8, Mathematical Society of Japan, Tokyo, 2000 (With a preface by Toshio Oshima) | MR

[14] J. Rauch L 2 is a continuable initial condition for Kreiss’ mixed problems, Comm. Pure. and Appl. Math., Volume 15 (1972), pp. 265-285 | DOI | MR | Zbl

[15] B. Schapira Contributions to the hypergeometric function theory of Heckman and Opdam: sharpe stimates, Schwartz spaces, heat kernel, Geom. Funct. Anal., Volume 18 (2008), pp. 222-250 | DOI | MR

[16] T. Shirota On the propagation speed of a hyperbolic operator with mixed boundary conditions, J. Fac. Sci. Hokkaido Univ., Volume 22 (1972), pp. 25-31 | MR | Zbl

[17] H. Triebel Interpolation theory, functions spaces differential operators, North Holland, Amesterdam, 1978 | Zbl

[18] H. Weber Die partiellen Differentialgleichungen der mathematischen Physik nach Riemann’s Vorlesungen in 4-ter Auflage neu bearbeitet, Braunschweig, Friederich Vieweg, Volume 1 (1900), pp. 1-390

[19] S. Zaremba Sopra un theorema d’unicità relativo alla equazione delle onde sferiche, Rend. Accad. Naz. Lincei, Volume 24 (1915), pp. 904-908

Cited by Sources: