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The boundedness of sublinear operators on Morrey-Herz spaces over the homogeneous type space. (English) Zbl 1299.42081

In the 1960’s, Herz spaces and weak Herz spaces were introduced by A. Beurling [Ann. Inst. Fourier 14, No. 2, 1–32 (1964; Zbl 0133.07501)] and C. Herz [J. Math. Mech. 18, 283–323 (1968; Zbl 0177.15701)]. In the paper under review, the authors introduce the Morrey-Herz spaces over the homogeneous type basic space under certain weak size conditions, which are the generalizations of the classical Herz type spaces.
The authors prove the boundedness of some sublinear operators on the homogeneous (weak) Morrey-Herz spaces. These sublinear operators satisfy certain size conditions. Many operators in harmonic analysis, such as the Calderón-Zygmund operator, fractional integral and so on satisfy these size conditions.

MSC:

42B35 Function spaces arising in harmonic analysis
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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[1] A. Beurling, Construction and analysis of some convolution algebras, Ann. Inst. Fourier Grenoble, 14(1964), 1–32. · Zbl 0133.07501
[2] C. Herz, Lipschhitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech., 18(1968), 283–324. · Zbl 0177.15701
[3] D. Deng and Y. Han, Harmonic analysis on spaces of homogeneous type, Springer (Berlin-Heidelberg, 2008).
[4] S. Lu and L. Xu, Boundedness of rough singular integral operators on homogeneous Morrey-Herz spaces, Hokkaido Math. J., 34(2005), 299–314. · Zbl 1081.42012
[5] V. Kokilashvili and A. Kufnerc, Fractional integrals on spaces of homogeneous type, Comment. Math. Univ. Carolin., 30(1989), 511–523. · Zbl 0686.42013
[6] W. Pan, Weighted norm inequalities for fractional integrals and maximal functions on spaces of homogeneous type, Acta Sci. Natur. Univ. Pekinensis, 26(1990), 543–553. · Zbl 0729.42008
[7] X. Li and D. Yang, Boundedness of some sublinear operators on Herz spaces, Illinois J. Math., 40(1996), 484–501. · Zbl 0956.46025
[8] X-X. Tao, X. Yu and S-Y. Zhang, Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces, J. Funct. Spaces Appl., 8(2010), 1–16. · Zbl 1191.42007
[9] Y-L. Shi and X-X. Tao, Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey spaces, Hokkaido Math. J., 38(2009), 635–662. · Zbl 1187.42012
[10] Y-L. Shi and X-X. Tao, Boundedness for multilinear fractional integral operators on Herz type spaces, Appl. Math. J. Chinese Univ. Ser. B, 4(2008), 437–446. · Zbl 1199.42079
[11] Y-L. Shi, X-X. Tao, and T-T. Zheng, Multilinear Riesz Potential on Morrey-Herz spaces with non-doubling measures, J. Inequal. Appl., 2010(2010), Article ID 731016, 21 pages. · Zbl 1203.42026
[12] Z. Liu and Y. Zeng, Herz spaces over spaces of homogeneous type and their application, Acta Math. Sci. Ser. A Chin., 19(1999), 270–277. · Zbl 0941.42010
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