Geodesic
Dunkl Translations, Dunkl-Type BMO Space, and Riesz Transforms for the Dunkl Transform on $L^\infty$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 4, pp. 63-77.

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

In this paper we will give some results on the support of Dunkl translations on compactly supported functions. Then we will define the Dunkl-type BMO space and Riesz transforms for the Dunkl transform on $L^\infty$ and prove the boundedness of the Riesz transforms from $L^\infty$ to the Dunkl-type BMO space under the assumption of the uniform boundedness of Dunkl translations. The proof and the definition in the Dunkl setting will be harder than in the classical case for the lack of some properties of Dunkl translations similar to those of classical translations. We will also extend Gallardo and Rejeb's precise description of the support of Dunkl translations on characteristic functions to all nonnegative radial functions in $L^2(m_k)$.
Keywords: Dunkl translations, Riesz transforms, Dunkl-type BMO space. 
@article{FAA_2021_55_4_a4,
     author = {W. Teng},
     title = {Dunkl {Translations,} {Dunkl-Type} {BMO} {Space,} and {Riesz} {Transforms} for the {Dunkl} {Transform} on $L^\infty$},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {63--77},
     publisher = {mathdoc},
     volume = {55},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2021_55_4_a4/}
}
TY  - JOUR
AU  - W. Teng
TI  - Dunkl Translations, Dunkl-Type BMO Space, and Riesz Transforms for the Dunkl Transform on $L^\infty$
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2021
SP  - 63
EP  - 77
VL  - 55
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2021_55_4_a4/
LA  - ru
ID  - FAA_2021_55_4_a4
ER  - 
%0 Journal Article
%A W. Teng
%T Dunkl Translations, Dunkl-Type BMO Space, and Riesz Transforms for the Dunkl Transform on $L^\infty$
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2021
%P 63-77
%V 55
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2021_55_4_a4/
%G ru
%F FAA_2021_55_4_a4
W. Teng. Dunkl Translations, Dunkl-Type BMO Space, and Riesz Transforms for the Dunkl Transform on $L^\infty$. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 4, pp. 63-77. http://geodesic.mathdoc.fr/item/FAA_2021_55_4_a4/

[1] B. Amri, J.-P. Anker, M. Sifi, “Three results in Dunkl analysis”, Colloq. Math., 118:1 (2010), 299–312 | DOI | MR | Zbl

[2] B. Amri, M. Sifi, “Riesz transforms for the Dunkl transform”, Ann. Math. Blaise Pascal, 19:1 (2012), 247–262 | DOI | MR | Zbl

[3] J.-P. Anker, J. Dziubański, A. Hejna, “Harmonic functions, conjugate harmonic functions and the Hardy space $H^1$ in the rational Dunkl setting”, J. Fourier Anal. Appl., 25:5 (2019), 2356–2418 | DOI | MR | Zbl

[4] F. Dai, H. Wang, “A transference theorem for the Dunkl transform and its applications”, J. Funct. Anal., 258:12 (2010), 4052–4074 | DOI | MR | Zbl

[5] C. F. Dunkl, “Differential-difference operators associated to reflection groups”, Trans. Amer. Math. Soc., 311:1 (1989), 167–183 | DOI | MR | Zbl

[6] J. Dziubański, A. Hejna, “Remark on atomic decompositions for the Hardy space $H^1$ in the rational Dunkl setting”, Studia Math., 251:1 (2020), 89–110 | DOI | MR | Zbl

[7] J. Dziubański, A. Hejna, “Hörmander's multiplier theorem for the Dunkl transform”, J. Funct. Anal, 277:7 (2019), 2133–2159 | DOI | MR | Zbl

[8] V. I. Ivanov, “Vesovye neravenstva dlya preobrazovanii Danklya–Rissa i gradienta Danklya”, Chebyshevskii sb., 21:4 (2020), 97–106 | DOI | MR | Zbl

[9] L. Gallardo, C. Rejeb, “Support properties of the intertwining and the mean value operators in Dunkl theory”, Proc. Amer. Math. Soc., 146:1 (2017), 145–152 | DOI | MR

[10] L. Gallardo, C. Rejeb, “A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications”, Trans. Amer. Math. Soc., 368:5 (2016), 3727–3753 | DOI | MR | Zbl

[11] D. V. Gorbachev, V. I. Ivanov, S. Y. Tikhonov, “Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications”, Constr. Approx., 49:3 (2019), 555–605 | DOI | MR | Zbl

[12] V. S. Guliyev, Y. Y. Mammadov, “On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line”, J. Math. Anal. Appl., 353:1 (2009), 449–459 | DOI | MR | Zbl

[13] M. Rösler, “A positive radial product formula for the Dunkl kernel”, Trans. Amer. Math. Soc., 355:6 (2003), 2413–2438 | DOI | MR | Zbl

[14] S. Thangavelu, Y. Xu, “Riesz transform and Riesz potentials for Dunkl transform”, J. Comput. Appl. Math., 199:1 (2007), 181–195 | DOI | MR | Zbl