Geodesic
Weighted inequalities for Dunkl--Riesz transforms and Dunkl gradient
Čebyševskij sbornik, Tome 21 (2020) no. 4, pp. 97-106.

Voir la notice de l'article provenant de la source Math-Net.Ru

Over the past 30 years a meaningful harmonic analysis has been constructed in the spaces with Dunkl weights of power type on $\mathbb{R}^d$. The classical Fourier analysis on Euclidean space corresponds to the weightless case. The Dunkl–Riesz potential and the Dunkl–Riesz transforms defined by Thangavelu and Xu play an important role in the Dunkl harmonic analysis. In particular, they allow one to prove the Sobolev inequalities for the Dunkl gradient. Particular results were obtained here by Amri and Sifi, Abdelkefi and Rachdi, Veliku. Based on the weighted inequalities for the Dunkl–Riesz potential and the Dunkl–Riesz transforms, we prove the general $(L^q,L^p)$ Sobolev inequalities for the Dunkl gradient with radial power weights. The weighted inequalities for the Dunkl–Riesz potential were established earlier.The $L^p$-inequalities for the Dunkl–Riesz transforms with radial power weights are established in this paper. A weightless version of these inequalities was proved by Amri and Sifi.
Keywords: Dunkl–Riesz potential, Dunkl–Riesz transforms, Dunkl gradient, Sobolev inequality. 
@article{CHEB_2020_21_4_a9,
     author = {V. I. Ivanov},
     title = {Weighted inequalities for {Dunkl--Riesz} transforms and {Dunkl} gradient},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {97--106},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2020_21_4_a9/}
}
TY  - JOUR
AU  - V. I. Ivanov
TI  - Weighted inequalities for Dunkl--Riesz transforms and Dunkl gradient
JO  - Čebyševskij sbornik
PY  - 2020
SP  - 97
EP  - 106
VL  - 21
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2020_21_4_a9/
LA  - ru
ID  - CHEB_2020_21_4_a9
ER  - 
%0 Journal Article
%A V. I. Ivanov
%T Weighted inequalities for Dunkl--Riesz transforms and Dunkl gradient
%J Čebyševskij sbornik
%D 2020
%P 97-106
%V 21
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2020_21_4_a9/
%G ru
%F CHEB_2020_21_4_a9
V. I. Ivanov. Weighted inequalities for Dunkl--Riesz transforms and Dunkl gradient. Čebyševskij sbornik, Tome 21 (2020) no. 4, pp. 97-106. http://geodesic.mathdoc.fr/item/CHEB_2020_21_4_a9/

[1] Rösler M., “Dunkl operators. Theory and applications”, Orthogonal Polynomials and Special Functions, Lecture Notes in Math., 1817, Springer-Verlag, 2002, 93–135 | DOI | MR

[2] Thangavelu S., Xu Y., “Riesz transform and Riesz potentials for Dunkl transform”, J. Comput. Appl. Math., 199 (2007), 181–195 | DOI | MR | Zbl

[3] Amri B., Sifi M., “Riesz transforms for Dunkl transform”, Annales mathématiques Blaise Pascal, 19:1 (2012), 147–162 | DOI | MR

[4] Abdelkefi C., Rachdi M., “Some properties of the Riesz potentials in Dunkl analysis”, Ricerche di Matematica, 64:1 (2015), 195–215 | DOI | MR | Zbl

[5] Gorbachev D. V., Ivanov V. I., Tikhonov S. Yu., “Riesz Potential and Maximal Function for Dunkl transform”, Potential Analysis, 2020 | DOI | MR

[6] Hassani S., Mustapha S., Sifi M., “Riesz potentials and fractional maximal function for the Dunkl transform”, J. Lie Theory, 19:4 (2009), 725–734 | MR | Zbl

[7] Velicu A., Hardy-type inequalities for Dunkl operators, 2019, 20 pp., arXiv: 1901.08866 | MR

[8] Velicu A., “Hardy-type inequalities for Dunkl operators with applications to many-particle Hardy inequalities”, Communications in Contemporary Mathematics, 2020 | DOI | MR | Zbl

[9] Stein E. M., “Note on Singular Integrals”, Proc. Amer. Math. Soc., 8:2 (1957), 250–254 | DOI | MR | Zbl

[10] Gorbachev D. V., Ivanov V. I., “Weighted inequalities for Dunkl-Riesz potential”, Chebyshevskii Sbornik, 20:1 (2019), 131–147 (In Russian) | DOI | MR | Zbl