On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line. (English) Zbl 1162.42006

Guliyev, Vagif S.; Mammadov, Yagub Y.

J. Math. Anal. Appl. 353, No. 1, 449-459 (2009).

The Dunkl operators are differential-difference operators associated with the reflection group \(Z_2\) on \(\mathbb{R}^1\). F. Soltani [J. Funct. Anal. 221, No. 1, 205–225 (2005; Zbl 1071.42008)] considered the maximal operator associated with the Dunkl operators. The authors study the fractional maximal function and the fractional integral associated with the Dunkl operators.
Let \(\alpha > -1/2\) and \(\mu_{\alpha}\) be the weighted Lebesgue measure on \(\mathbb{R}^1\) given by \( d\mu_{\alpha}(x) = (2^{\alpha + 1} \Gamma (\alpha + 1))^{-1} | x |^{2 \alpha +1}dx, \) and \(L_{p, \alpha}(\mathbb{R}^1)\) is the space of functions \(f\) such that \( \| f \|_{p, \alpha}= \left( \int_{\mathbb{R}^1} | f(x) |^{p} d \mu_{\alpha}(x) \right)^{1/p}< \infty. \) The operators \(\tau_x, x \in \mathbb{R}^1\) are Dunkl translation operators, and the Dunkl-type fractional integral is defined by \[ I_{\beta}f(x) = \int_{\mathbb{R}^1} \tau_x | y |^{\beta - 2\alpha -2}f(y) d \mu_{\alpha}(y) \quad 0 < \beta < 2\alpha +2. \] They prove the following: If \(1<p< \frac{2\alpha +2}{\beta}\), then the condition \(\frac{1}{p} = \frac{1}{q}- \frac{\beta}{2\alpha +2}\) is the necessary and sufficient for the boundedness of \(I_{\beta}\) from \(L_{p, \alpha}\) to \(L_{q, \alpha}\).
They also consider the estimates on BMO and the Besov spaces.

Reviewer: Yasuo Komori (Numazu)

MSC:

42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25	Maximal functions, Littlewood-Paley theory
42-02	Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces

Keywords:

Dunkl operators; Dunkl-type fractional integral operator; Dunkl-type BMO; Dunkl-type Besov spaces

Citations:

Zbl 1071.42008

References:

[1]	Abdelkefi, C.; Sifi, M., Dunkl translation and uncentered maximal operator on the real line, Int. J. Math. Math. Sci., 2007 (2007), Article ID 87808, 9 pp · Zbl 1201.42012
[2]	Abdelkefi, C.; Sifi, M., On the uniform convergence of partial Dunkl integrals in Besov-Dunkl spaces, Fract. Calc. Appl. Anal., 9, 1, 43-56 (2006) · Zbl 1116.44002
[3]	Abdelkefi, C.; Sifi, M., Characterization of Besov spaces for the Dunkl operator on the real line, JIPAM. J. Inequal. Pure Appl. Math., 8, 3, 1-21 (2007), Article 73 · Zbl 1159.42304
[4]	Bouguila, R.; Lazhari, M. N.; Assal, M., Besov spaces associated with Dunkl’s operator, Integral Transforms Spec. Funct., 18, 8, 545-557 (2007) · Zbl 1143.42023
[5]	Guliyev, V. S., On maximal function and fractional integral, associated with the Bessel differential operator, Math. Inequal. Appl., 6, 2, 317-330 (2003) · Zbl 1047.47035
[6]	Guliyev, V. S.; Mammadov, Y. Y., Function spaces and integral operators for the Dunkl operator on the real line, Khazar J. Math., 2, 4, 17-42 (2006)
[7]	Dunkl, C. F., Differential-difference operators associated with reflections groups, Trans. Amer. Math. Soc., 311, 167-183 (1989) · Zbl 0652.33004
[8]	Mammadov, Y. Y., On maximal operator associated with the Dunkl operator on \(R\), Khazar J. Math., 2, 4, 59-70 (2006)
[9]	Mammadov, Y. Y., On the boundedness of the maximal operator associated with the Dunkl operator on \(R\) on the Besov-Dunkl spaces, Proc. of NAS of Azerbaijan. Embedding Theorems. Proc. of NAS of Azerbaijan. Embedding Theorems, Harmonic Analysis, 26 (34), XIII, 244-257 (2007)
[10]	Kamoun, L., Besov-type spaces for the Dunkl operator on the real line, J. Comput. Appl. Math., 199, 56-67 (2007) · Zbl 1106.44001
[11]	Sifi, M.; Soltani, F., Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line, J. Math. Anal. Appl., 270, 92-106 (2002) · Zbl 1012.46033
[12]	Soltani, F., \(L^p\)-Fourier multipliers for the Dunkl operator on the real line, J. Funct. Anal., 209, 16-35 (2004) · Zbl 1045.43003
[13]	Soltani, F., Littlewood-Paley operators associated with the Dunkl operator on \(R\), J. Funct. Anal., 221, 205-225 (2005) · Zbl 1071.42008
[14]	Rösler, M., Bessel-type signed hypergroups on \(R\), (Heyer, H.; Mukherjea, A., Probability Measures on Groups and Related Structures, XI. Probability Measures on Groups and Related Structures, XI, Oberwolfach, 1994 (1995), World Scientific: World Scientific River Edge, NJ, USA), 292-304 · Zbl 0908.43005
[15]	Trimeche, K., Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transforms Spec. Funct., 13, 17-38 (2002) · Zbl 1030.44004
[16]	Watson, G. N., A Treatise on Theory of Bessel Functions (1966), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0174.36202