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Necessary and sufficient conditions for the boundedness of Dunkl-type fractional maximal operator in the Dunkl-type Morrey spaces. (English) Zbl 1210.42035

The authors consider Morrey type spaces (\(\equiv\) Dunkl-type Morrey spaces) \(L_{p,\lambda,\alpha},\) associated with the Dunkl operator, defined by the norm \[ \|f\|_{p,\lambda,\alpha}:= \sup\limits_{x\in\mathbb{R},r>0}\left(\frac{1}{r^\lambda} \int_{|y|<r}\tau_x|f(y)|^p d\mu_\alpha(y)\right)^\frac{1}{p}, \;\;d\mu_\alpha(y)= \text{const} |y|^{2\alpha+1}dy, \] where \(\tau_x\) is the Dunkl translation operator, and the corresponding weak Dunkl-type Morrey spaces.
The main result is the theorem which provides condition for the boundedness of the maximal fractional operator \(M_\beta\) (introduced in similar terms via \(\tau_x\)) from \(L_{p,\lambda,\alpha}\) to \(L_{q,\lambda,\alpha}\), when \(1\leq p\leq\frac{2\alpha+2-\lambda}{\beta}\) with the weak version of the space \(L_{q,\lambda,\alpha}\) when \(p=1.\) In the case \(1\leq p<\frac{2\alpha+2-\lambda}{\beta}\) the obtained condition \[ \frac{1}{p}-\frac{1}{q}=\frac{\beta}{2\alpha+2-\lambda} \] is necessary and sufficient.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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References:

[1] C. F. Dunkl, “Differential-difference operators associated to reflection groups,” Transactions of the American Mathematical Society, vol. 311, no. 1, pp. 167-183, 1989. · Zbl 0652.33004 · doi:10.2307/2001022
[2] C. B. Morrey Jr., “On the solutions of quasi-linear elliptic partial differential equations,” Transactions of the American Mathematical Society, vol. 43, no. 1, pp. 126-166, 1938. · Zbl 0018.40501 · doi:10.2307/1989904
[3] C. Abdelkefi and M. Sifi, “Dunkl translation and uncentered maximal operator on the real line,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 87808, 9 pages, 2007. · Zbl 1201.42012 · doi:10.1155/2007/87808
[4] Y. Y. Mammadov, “On maximal operator associated with the Dunkl operator on \Bbb R,” Khazar Journal of Mathematics, vol. 2, no. 4, pp. 59-70, 2006.
[5] F. Soltani, “Littlewood-Paley operators associated with the Dunkl operator on \Bbb R,” Journal of Functional Analysis, vol. 221, no. 1, pp. 205-225, 2005. · Zbl 1071.42008 · doi:10.1016/j.jfa.2004.10.001
[6] V. S. Guliyev and Y. Y. Mammadov, “(Lp,Lq) boundedness of the fractional maximal operator associated with the Dunkl operator on the real line,” Integral Transforms and Special Functions, pp. 1-11, 2010. · Zbl 1229.47049 · doi:10.1080/10652460903507248
[7] E. V. Guliyev and Y. Y. Mammadov, “Some embeddings into the Morrey spaces associated with the Dunkl operator,” Abstract and Applied Analysis, vol. 2010, Article ID 291345, 10 pages, 2010. · Zbl 1198.46027 · doi:10.1155/2010/291345
[8] M. Rösler, “Bessel-type signed hypergroups on \Bbb R,” in Probability Measures on Groups and Related Structures, XI (Oberwolfach, 1994), pp. 292-304, World Scientific, River edge, NJ, USA, 1995. · Zbl 0908.43005
[9] F. Soltani, “Lp-Fourier multipliers for the Dunkl operator on the real line,” Journal of Functional Analysis, vol. 209, no. 1, pp. 16-35, 2004. · Zbl 1045.43003 · doi:10.1016/j.jfa.2003.11.009
[10] V. S. Guliyev and Y. Y. Mammadov, “On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 449-459, 2009. · Zbl 1162.42006 · doi:10.1016/j.jmaa.2008.11.083
[11] C. Abdelkefi and M. Sifi, “Characterization of Besov spaces for the Dunkl operator on the real line,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 3, article no. 73, p. 11, 2007. · Zbl 1139.46028
[12] R. Bouguila, M. N. Lazhari, and M. Assal, “Besov spaces associated with Dunkl’s operator,” Integral Transforms and Special Functions, vol. 18, no. 7-8, pp. 545-557, 2007. · Zbl 1143.42023 · doi:10.1080/10652460701359065
[13] L. Kamoun, “Besov-type spaces for the Dunkl operator on the real line,” Journal of Computational and Applied Mathematics, vol. 199, no. 1, pp. 56-67, 2007. · Zbl 1106.44001 · doi:10.1016/j.cam.2005.06.054
[14] G. Pradolini and O. Salinas, “Maximal operators on spaces of homogeneous type,” Proceedings of the American Mathematical Society, vol. 132, no. 2, pp. 435-441, 2004. · Zbl 1044.42021 · doi:10.1090/S0002-9939-03-07079-5
[15] N. Samko, “Weighted Hardy and singular operators in Morrey spaces,” Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 56-72, 2009. · Zbl 1155.42005 · doi:10.1016/j.jmaa.2008.09.021
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