Mapping properties for operators on Herz-Morrey and Herz-Morrey-Hardy spaces. (English) Zbl 1505.42018
Summary: In this paper, we build the extrapolation theory for Herz-Morrey spaces, from which the mapping properties of the parametric Marcinkiewicz integrals, the geometric maximal operator, the minimal operator and the Rubio de Francia inequalities on Herz-Morrey spaces are deduced. By extending the extrapolation theory to Herz-Morrey-Hardy spaces, we also obtain the boundedness of the singular integral operators and the maximal Bochner-Riesz means on Herz-Morrey-Hardy spaces.
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B35 | Function spaces arising in harmonic analysis |
Keywords:
Herz-Morrey space; boundedness; extrapolation; dyadic Herz block space; Herz-Morrey-Hardy spaceReferences:
| [1] | Abdalmonem, A.; Scapellato, A., Intrinsic square functions and commutators on Morrey-Herz spaces with variable exponents, Math. Methods Appl. Sci., 44, 17, 12408-12425 (2021) · Zbl 1481.42013 · doi:10.1002/mma.7487 |
| [2] | Almeida, A.; Drihem, D., Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl., 394, 2, 781-795 (2012) · Zbl 1250.42077 · doi:10.1016/j.jmaa.2012.04.043 |
| [3] | Chiarenza, F.; Frasca, M., Morrey spaces and Hardy-Littlewood maximal function, Rendiconti Lincei-Matematica e Applicazioni, 7, 7, 273-279 (1987) · Zbl 0717.42023 |
| [4] | Cruz-Uribe, D.; Fiorenza, A.; Martell, JM; Pérez, C., The boundedness of classical operators on variable \({L^p}\) spaces, Annales-Academiae Scientiarum Fennicae Mathematica, 31, 1, 239-264 (2006) · Zbl 1100.42012 |
| [5] | Cruz-Uribe, D.; Martell, JM; Pérez, C., Weights, extrapolation and the theory of Rubio de Francia (2011), Berlin: Springer Science & Business Media, Berlin · Zbl 1234.46003 · doi:10.1007/978-3-0348-0072-3 |
| [6] | Cruz-Uribe, D.; Martell Berrocal, JM; Perez Moreno, C., Extrapolation from \({A_{\infty }}\) weights and applications, J. Funct. Anal., 213, 2, 412-439 (2004) · Zbl 1052.42016 · doi:10.1016/j.jfa.2003.09.002 |
| [7] | Cruz-Uribe, D.; Neugebauer, CJ, Weighted norm inequalities for the geometric maximal operator, Publicacions Matemátiques, 42, 239-263 (1998) · Zbl 0919.42014 · doi:10.5565/PUBLMAT_42198_13 |
| [8] | Dong, B.; Xu, J., Herz-Morrey type Besov and Triebel-Lizorkin spaces with variable exponents, Banach J. Math. Anal., 9, 1, 75-101 (2015) · Zbl 1333.46026 · doi:10.15352/bjma/09-1-7 |
| [9] | Drihem, D.; Seghiri, F., Notes on the Herz-type Hardy spaces of variable smoothness and integrability, Math. Inequal. Appl., 19, 1, 145-165 (2016) · Zbl 1334.42047 |
| [10] | Duoandikoetxea, J.; Rosenthal, M., Extension and boundedness of operators on Morrey spaces from extrapolation techniques and embeddings, J. Geom. Anal., 28, 4, 3081-3108 (2018) · Zbl 1405.42045 · doi:10.1007/s12220-017-9946-5 |
| [11] | Duoandikoetxea, J.; Rosenthal, M., Muckenhoupt-type conditions on weighted Morrey spaces, J. Fourier Anal. Appl., 27, 2, 32 (2021) · Zbl 1464.42013 · doi:10.1007/s00041-021-09839-w |
| [12] | Fefferman, C.; Stein, EM, \(H^p\) spaces of several variables, Acta Math., 129, 1, 137-193 (1972) · Zbl 0257.46078 · doi:10.1007/BF02392215 |
| [13] | Garcia-Cuerva, J., Weighted \(H^p\) spaces, Diss. Math., 162, 1, 1-63 (1979) · Zbl 0434.42023 |
| [14] | Grafakos, L., Classical Fourier analysis, voleme 249 (2008), New York: Springer, New York · Zbl 1220.42001 |
| [15] | Guliyev, VS; Omarova, MN; Ragusa, MA; Scapellato, A., Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces, Anal. Math. Phys., 11, 1, 1-20 (2021) · Zbl 1467.35078 · doi:10.1007/s13324-020-00433-9 |
| [16] | Hart, J.; Oliveira, L., Hardy space estimates for limited ranges of Muckenhoupt weights, Adv. Math., 313, 803-838 (2017) · Zbl 1372.42009 · doi:10.1016/j.aim.2017.04.018 |
| [17] | Hästö, PA, Local-to-global results in variable exponent spaces, Math. Res. Lett., 16, 2, 263-278 (2009) · Zbl 1184.46033 · doi:10.4310/MRL.2009.v16.n2.a5 |
| [18] | Hernández, E.; Yang, D., Interpolation of Herz spaces and applications, Math. Nachr., 205, 1, 69-87 (1999) · Zbl 0936.41001 · doi:10.1002/mana.3212050104 |
| [19] | Herz, CS, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech., 18, 4, 283-323 (1968) · Zbl 0177.15701 |
| [20] | Ho, K-P, Extrapolation, John-Nirenberg inequalities and characterizations of BMO in terms of Morrey type spaces, Revista Matemática Complutense, 30, 3, 487-505 (2017) · Zbl 1385.42012 · doi:10.1007/s13163-016-0216-z |
| [21] | Ho, K-P, Intrinsic square functions on Morrey and block spaces with variable exponents, Bulletin Malaysian Math. Sci. Soc., 40, 3, 995-1010 (2017) · Zbl 1372.42014 · doi:10.1007/s40840-016-0330-6 |
| [22] | Ho, K-P, Sublinear operators on weighted Hardy spaces with variable exponents, Forum Math., 31, 3, 607-617 (2019) · Zbl 1417.42024 · doi:10.1515/forum-2018-0142 |
| [23] | Ho, K-P, Extrapolation to Herz spaces with variable exponents and applications, Revista Matemática Complutense, 33, 2, 437-463 (2020) · Zbl 1465.42015 · doi:10.1007/s13163-019-00320-3 |
| [24] | Ho, K-P, Sublinear operators on block type space, Sci. China Math., 63, 6, 1107-1124 (2020) · Zbl 1446.42014 · doi:10.1007/s11425-018-9441-9 |
| [25] | Ho, K-P, Nonlinear commutators on Morrey-Banach spaces, J. Pseudo-Diff. Oper. Appl., 12, 3, 1-12 (2021) · Zbl 1476.42009 |
| [26] | Ho, K-P, Operators on Orlicz-slice spaces and Orlicz-slice Hardy spaces, J. Math. Anal. Appl., 503, 1, 125279 (2021) · Zbl 1468.42013 · doi:10.1016/j.jmaa.2021.125279 |
| [27] | Ho, K-P, Singular integral operators and sublinear operators on Hardy local Morrey spaces with variable exponents, Bulletin des Sciences Mathématiques, 171, 103033 (2021) · Zbl 1475.42025 · doi:10.1016/j.bulsci.2021.103033 |
| [28] | Ho, K-P, Spherical maximal function, maximal Bochner-Riesz mean and geometrical maximal function on Herz spaces with variable exponents, Rendiconti del Circolo Matematico di Palermo Series, 2, 70, 559-574 (2021) · Zbl 1462.42030 · doi:10.1007/s12215-020-00511-8 |
| [29] | Ho, K-P, Boundedness of operators and inequalities on Morrey-Banach spaces, Publ. Res. Inst. Math. Sci., 58, 3, 551-577 (2022) · Zbl 1502.42011 · doi:10.4171/PRIMS/58-3-4 |
| [30] | Ho, K-P, Sublinear operators on Herz-Hardy spaces with variable exponents, Math. Nachr., 295, 5, 876-889 (2022) · Zbl 1523.42024 · doi:10.1002/mana.202000161 |
| [31] | Hörmander, L., Estimates for translation invariant operators in \(L^p\) spaces, Acta Math., 104, 1-2, 93-140 (1960) · Zbl 0093.11402 · doi:10.1007/BF02547187 |
| [32] | Izuki, M., Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J., 40, 3, 343-355 (2010) · Zbl 1217.42034 · doi:10.32917/hmj/1291818849 |
| [33] | Izuki, M.; Noi, T., Two weighted Herz spaces with variable exponents, Bulletin Malaysian Math. Sci. Soc., 43, 1, 169-200 (2020) · Zbl 1433.42022 · doi:10.1007/s40840-018-0671-4 |
| [34] | Kokilashvili, V., Meskhi, A., Ragusa, M.A.: Weighted extrapolation in grand Morrey spaces and applications to partial differential equations. In: Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni, vol. 30(1), pp. 67-92 (2019) · Zbl 1422.42018 |
| [35] | Komori, Y., Notes on singular integrals on some inhomogeneous Herz spaces, Taiwan J. Math., 8, 3, 547-556 (2004) · Zbl 1090.42008 · doi:10.11650/twjm/1500407672 |
| [36] | Lee, M., Weighted norm inequalities of Bochner-Riesz means, J. Math. Anal. Appl., 324, 2, 1274-1281 (2006) · Zbl 1103.42013 · doi:10.1016/j.jmaa.2005.07.085 |
| [37] | Li, X.; Yang, D., Boundedness of some sublinear operators on Herz spaces, Ill. J. Math., 40, 3, 484-501 (1996) · Zbl 0956.46025 |
| [38] | Liu, J.; Weisz, F.; Yang, D.; Yuan, W., Variable anisotropic Hardy spaces and their applications, Taiwan J. Math., 22, 5, 1173-1216 (2018) · Zbl 1401.42024 · doi:10.11650/tjm/171101 |
| [39] | Lu, S.; Xu, L., Boundedness of rough singular integral operatorson the homogeneous Morrey-Herz spaces, Hokkaido Math. J., 34, 2, 299-314 (2005) · Zbl 1081.42012 · doi:10.14492/hokmj/1285766224 |
| [40] | Lu, S.; Yang, D., The decomposition of weighted Herz space on \({\mathbb{R} }^n\) and its applications, Sci. China Math., 38, 2, 147-158 (1995) · Zbl 0830.42014 |
| [41] | Lu, S.; Yang, D., Hardy-Littlewood-Sobolev theorems of fractional integration on Herz-type spaces and its applications, Can. J. Math., 48, 2, 363-380 (1996) · Zbl 0893.42005 · doi:10.4153/CJM-1996-020-4 |
| [42] | Lu, S.; Yang, D.; Hu, G., Herz type spaces and their applications (2008), Beijing: Science press, Beijing |
| [43] | Lu, Y.; Zhu, Y., Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-herz spaces with variable exponents, Acta Mathematica Sinica English Series, 30, 7, 1180-1194 (2014) · Zbl 1304.42042 · doi:10.1007/s10114-014-3410-2 |
| [44] | Mastylo, M.; Sawano, Y.; Tanaka, H., Morrey-type space and its Köthe dual space, Bulletin Malaysian Math. Sci. Soc., 41, 3, 1181-1198 (2018) · Zbl 1393.42007 · doi:10.1007/s40840-016-0382-7 |
| [45] | Miyachi, A., Remarks on Herz-type Hardy spaces, Acta Math. Sinica, 17, 2, 339-360 (2001) · Zbl 0993.42010 · doi:10.1007/s101140100104 |
| [46] | Morrey, CB, On the solutions of quasi-linear elliptic partial differential equations, Trans. Am. Math. Soc., 43, 1, 126-166 (1938) · Zbl 0018.40501 · doi:10.1090/S0002-9947-1938-1501936-8 |
| [47] | Nogayama, T., Boundedness of commutators of fractional integral operators on mixed Morrey spaces, Integral Transform. Spec. Funct., 30, 10, 790-816 (2019) · Zbl 1422.42029 · doi:10.1080/10652469.2019.1619718 |
| [48] | Ragusa, MA, Homogeneous Herz spaces and regularity results, Nonlinear Anal. Theory Methods Appl., 71, 12, 1909-1914 (2009) · Zbl 1238.35021 · doi:10.1016/j.na.2009.02.075 |
| [49] | Rosenthal, M.; Schmeisser, H-J, The boundedness of operators in Muckenhoupt weighted Morrey spaces via extrapolation techniques and duality, Revista Matemática Complutense, 29, 3, 623-657 (2016) · Zbl 1350.42042 · doi:10.1007/s13163-016-0208-z |
| [50] | Rubio de Francia, J., Factorization and extrapolation of weights, Bull. Am. Math. Soc., 7, 2, 393-395 (1982) · Zbl 0491.42020 · doi:10.1090/S0273-0979-1982-15047-9 |
| [51] | Rubio de Francia, J., Factorization theory and \(A_p\) weights, Am. J. Math., 106, 3, 533-547 (1984) · Zbl 0558.42012 · doi:10.2307/2374284 |
| [52] | Rubio de Francia, J., A Littlewood-Paley inequality for arbitrary intervals, Revista Matemática Iberoamericana, 1, 2, 1-14 (1985) · Zbl 0611.42005 · doi:10.4171/RMI/7 |
| [53] | Sato, S., Weak type estimates for some maximal operators on the weighted Hardy spaces, Ark. Mat., 33, 2, 377-384 (1995) · Zbl 0852.47016 · doi:10.1007/BF02559715 |
| [54] | Sato, S., Remarks on square functions in the Littlewood-Paley theory, Bull. Aust. Math. Soc., 58, 2, 199-211 (1998) · Zbl 0914.42012 · doi:10.1017/S0004972700032172 |
| [55] | Sawano, Y.; Ho, K-P; Yang, D.; Yang, S., Hardy spaces for ball quasi-Banach function spaces, Diss. Math., 525, 1, 1-102 (2017) · Zbl 1392.42021 |
| [56] | Sawano, Y.; Shimomura, T., Predual spaces of generalized grand Morrey spaces over non-doubling measure spaces, Georgian Math. J., 27, 3, 433-439 (2020) · Zbl 1448.42033 · doi:10.1515/gmj-2018-0068 |
| [57] | Sawano, Y.; Sugano, S.; Tanaka, H., Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Am. Math. Soc., 363, 12, 6481-6503 (2011) · Zbl 1229.42024 · doi:10.1090/S0002-9947-2011-05294-3 |
| [58] | Sawano, Y.; Tanaka, H., Predual spaces of Morrey spaces with non-doubling measures, Tokyo J. Math., 32, 2, 471-486 (2009) · Zbl 1193.42094 · doi:10.3836/tjm/1264170244 |
| [59] | Sawano, Y.; Tanaka, H., The Fatou property of block spaces, J. Math. Sci. Univ. Tokyo, 22, 3, 663-683 (2015) · Zbl 1334.42051 |
| [60] | Shi, X., Two inequalities related to geometric mean operators, J. Zhejiang Teach. College, 1, 21-25 (1980) |
| [61] | Shi, Y.; Tao, X., Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey spaces, Hokkaido Math. J., 38, 4, 635-662 (2009) · Zbl 1187.42012 · doi:10.14492/hokmj/1258554238 |
| [62] | Stein, EM, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Am. Math. Soc., 88, 2, 430-466 (1958) · Zbl 0105.05104 · doi:10.1090/S0002-9947-1958-0112932-2 |
| [63] | Tao, J.; Yang, D.; Yang, D., Beurling-Ahlfors commutators on weighted Morrey spaces and applications to Beltrami equations, Potential Anal., 53, 4, 1467-1491 (2020) · Zbl 1448.42025 · doi:10.1007/s11118-019-09814-7 |
| [64] | Tao, J.; Yang, D.; Yuan, W., A bridge connecting Lebesgue and Morrey spaces via Riesz norms, Banach J. Math. Anal., 15, 1, 1-29 (2021) · Zbl 1455.42022 · doi:10.1007/s43037-020-00106-6 |
| [65] | Tao, X.; Zhang, H., On the boundedness of multilinear operators on weighted Herz-Morrey spaces, Taiwan J. Math., 15, 4, 1527-1543 (2011) · Zbl 1245.42015 · doi:10.11650/twjm/1500406362 |
| [66] | Torres, R.H.: Boundedness results for operators with singular kernels on distribution spaces. Memories of the American Mathematical Society, 90(442):viii+172, (1991) · Zbl 0737.47041 |
| [67] | Wang, F.; Yang, D.; Yang, S., Applications of Hardy spaces associated with ball quasi-Banach function spaces, RM, 75, 1, 26 (2020) · Zbl 1431.42040 |
| [68] | Wang, S.; Yang, D.; Yuan, W.; Zhang, Y., Weak Hardy-type spaces associated with ball quasi-Banach function spaces II: Littlewood-Paley characterizations and real interpolation, J. Geom. Anal., 31, 1, 631-696 (2021) · Zbl 1460.42033 · doi:10.1007/s12220-019-00293-1 |
| [69] | Xu, J.; Yang, X., Herz-Morrey-Hardy spaces with variable exponents and their applications, J. Funct. Spaces., 2015 (2015) · Zbl 1343.46032 |
| [70] | Xu, J.; Yang, X., The molecular decomposition of Herz-Morrey-Hardy spaces with variable exponents and its application, J. Math. Inequal., 10, 4, 977-1008 (2016) · Zbl 1370.46020 · doi:10.7153/jmi-10-79 |
| [71] | Yan, X.; Yang, D.; Yuan, W.; Zhuo, C., Variable weak Hardy spaces and their applications, J. Funct. Anal., 271, 10, 2822-2887 (2016) · Zbl 1440.42100 · doi:10.1016/j.jfa.2016.07.006 |
| [72] | Yang, M.; Fu, Z.; Sun, J., Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces, J. Diff. Equ., 266, 9, 5867-5894 (2019) · Zbl 1412.35351 · doi:10.1016/j.jde.2018.10.050 |
| [73] | Yang, Y.; Tao, S., Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents, Open Math., 16, 1, 326-345 (2018) · Zbl 1393.42017 · doi:10.1515/math-2018-0036 |
| [74] | Yee, TL; Ho, K-P, Hardy’s inequalities and integral operators on Herz-Morrey spaces, Open Math., 18, 1, 106-121 (2020) · Zbl 1475.42039 · doi:10.1515/math-2020-0008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
