Abstract
In this paper, we bulid 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.
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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)
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)
Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rendiconti Lincei-Matematica e Applicazioni 7(7), 273–279 (1987)
Cruz-Uribe, D., Fiorenza, A., Martell, J.M., Pérez, C.: The boundedness of classical operators on variable \({L^{p}}\) spaces. Annales-Academiae Scientiarum Fennicae Mathematica 31(1), 239–264 (2006)
Cruz-Uribe, D., Martell, J.M., Pérez, C.: Weights, extrapolation and the theory of Rubio de Francia, vol. 215. Springer Science & Business Media, Berlin (2011)
Cruz-Uribe, D., Martell Berrocal, J.M., Perez Moreno, C.: Extrapolation from \({A_{\infty }}\) weights and applications. J. Funct. Anal. 213(2), 412–439 (2004)
Cruz-Uribe, D., Neugebauer, C.J.: Weighted norm inequalities for the geometric maximal operator. Publicacions Matemátiques 42, 239–263 (1998)
Dong, B., Xu, J.: Herz-Morrey type Besov and Triebel-Lizorkin spaces with variable exponents. Banach J. Math. Anal. 9(1), 75–101 (2015)
Drihem, D., Seghiri, F.: Notes on the Herz-type Hardy spaces of variable smoothness and integrability. Math. Inequal. Appl. 19(1), 145–165 (2016)
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)
Duoandikoetxea, J., Rosenthal, M.: Muckenhoupt-type conditions on weighted Morrey spaces. J. Fourier Anal. Appl. 27(2), 32 (2021)
Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129(1), 137–193 (1972)
Garcia-Cuerva, J.: Weighted \(H^p\) spaces. Diss. Math. 162(1), 1–63 (1979)
Grafakos, L.: Classical Fourier analysis, voleme 249. Springer, New York (2008)
Guliyev, V.S., Omarova, M.N., Ragusa, M.A., 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)
Hart, J., Oliveira, L.: Hardy space estimates for limited ranges of Muckenhoupt weights. Adv. Math. 313, 803–838 (2017)
Hästö, P.A.: Local-to-global results in variable exponent spaces. Math. Res. Lett. 16(2), 263–278 (2009)
Hernández, E., Yang, D.: Interpolation of Herz spaces and applications. Math. Nachr. 205(1), 69–87 (1999)
Herz, C.S.: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech. 18(4), 283–323 (1968)
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)
Ho, K.-P.: Intrinsic square functions on Morrey and block spaces with variable exponents. Bulletin Malaysian Math. Sci. Soc. 40(3), 995–1010 (2017)
Ho, K.-P.: Sublinear operators on weighted Hardy spaces with variable exponents. Forum Math. 31(3), 607–617 (2019)
Ho, K.-P.: Extrapolation to Herz spaces with variable exponents and applications. Revista Matemática Complutense 33(2), 437–463 (2020)
Ho, K.-P.: Sublinear operators on block type space. Sci. China Math. 63(6), 1107–1124 (2020)
Ho, K.-P.: Nonlinear commutators on Morrey-Banach spaces. J. Pseudo-Diff. Oper. Appl. 12(3), 1–12 (2021)
Ho, K.-P.: Operators on Orlicz-slice spaces and Orlicz-slice Hardy spaces. J. Math. Anal. Appl. 503(1), 125279 (2021)
Ho, K.-P.: Singular integral operators and sublinear operators on Hardy local Morrey spaces with variable exponents. Bulletin des Sciences Mathématiques 171, 103033 (2021)
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)
Ho, K.-P.: Boundedness of operators and inequalities on Morrey-Banach spaces. Publ. Res. Inst. Math. Sci. 58(3), 551–577 (2022)
Ho, K.-P.: Sublinear operators on Herz-Hardy spaces with variable exponents. Math. Nachr. 295(5), 876–889 (2022)
Hörmander, L.: Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104(1–2), 93–140 (1960)
Izuki, M.: Fractional integrals on Herz-Morrey spaces with variable exponent. Hiroshima Math. J. 40(3), 343–355 (2010)
Izuki, M., Noi, T.: Two weighted Herz spaces with variable exponents. Bulletin Malaysian Math. Sci. Soc. 43(1), 169–200 (2020)
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)
Komori, Y.: Notes on singular integrals on some inhomogeneous Herz spaces. Taiwan J. Math. 8(3), 547–556 (2004)
Lee, M.: Weighted norm inequalities of Bochner-Riesz means. J. Math. Anal. Appl. 324(2), 1274–1281 (2006)
Li, X., Yang, D.: Boundedness of some sublinear operators on Herz spaces. Ill. J. Math. 40(3), 484–501 (1996)
Liu, J., Weisz, F., Yang, D., Yuan, W.: Variable anisotropic Hardy spaces and their applications. Taiwan J. Math. 22(5), 1173–1216 (2018)
Lu, S., Xu, L.: Boundedness of rough singular integral operatorson the homogeneous Morrey-Herz spaces. Hokkaido Math. J. 34(2), 299–314 (2005)
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)
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)
Lu, S., Yang, D., Hu, G.: Herz type spaces and their applications. Science press, Beijing (2008)
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)
Mastylo, M., Sawano, Y., Tanaka, H.: Morrey-type space and its Köthe dual space. Bulletin Malaysian Math. Sci. Soc. 41(3), 1181–1198 (2018)
Miyachi, A.: Remarks on Herz-type Hardy spaces. Acta Math. Sinica 17(2), 339–360 (2001)
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43(1), 126–166 (1938)
Nogayama, T.: Boundedness of commutators of fractional integral operators on mixed Morrey spaces. Integral Transform. Spec. Funct. 30(10), 790–816 (2019)
Ragusa, M.A.: Homogeneous Herz spaces and regularity results. Nonlinear Anal. Theory Methods Appl. 71(12), 1909–1914 (2009)
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)
Rubio de Francia, J.: Factorization and extrapolation of weights. Bull. Am. Math. Soc. 7(2), 393–395 (1982)
Rubio de Francia, J.: Factorization theory and \(A_{p}\) weights. Am. J. Math. 106(3), 533–547 (1984)
Rubio de Francia, J.: A Littlewood-Paley inequality for arbitrary intervals. Revista Matemática Iberoamericana 1(2), 1–14 (1985)
Sato, S.: Weak type estimates for some maximal operators on the weighted Hardy spaces. Ark. Mat. 33(2), 377–384 (1995)
Sato, S.: Remarks on square functions in the Littlewood-Paley theory. Bull. Aust. Math. Soc. 58(2), 199–211 (1998)
Sawano, Y., Ho, K.-P., Yang, D., Yang, S.: Hardy spaces for ball quasi-Banach function spaces. Diss. Math. 525(1), 1–102 (2017)
Sawano, Y., Shimomura, T.: Predual spaces of generalized grand Morrey spaces over non-doubling measure spaces. Georgian Math. J. 27(3), 433–439 (2020)
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)
Sawano, Y., Tanaka, H.: Predual spaces of Morrey spaces with non-doubling measures. Tokyo J. Math. 32(2), 471–486 (2009)
Sawano, Y., Tanaka, H.: The Fatou property of block spaces. J. Math. Sci. Univ. Tokyo 22(3), 663–683 (2015)
Shi, X.: Two inequalities related to geometric mean operators. J. Zhejiang Teach. College 1, 21–25 (1980)
Shi, Y., Tao, X.: Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey spaces. Hokkaido Math. J. 38(4), 635–662 (2009)
Stein, E.M.: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans. Am. Math. Soc. 88(2), 430–466 (1958)
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)
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)
Tao, X., Zhang, H.: On the boundedness of multilinear operators on weighted Herz-Morrey spaces. Taiwan J. Math. 15(4), 1527–1543 (2011)
Torres, R.H.: Boundedness results for operators with singular kernels on distribution spaces. Memories of the American Mathematical Society, 90(442):viii+172, (1991)
Wang, F., Yang, D., Yang, S.: Applications of Hardy spaces associated with ball quasi-Banach function spaces. RM 75(1), 26 (2020)
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)
Xu, J., Yang, X.: Herz-Morrey-Hardy spaces with variable exponents and their applications. J. Funct. Spaces. 2015, 160635 (2015)
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)
Yan, X., Yang, D., Yuan, W., Zhuo, C.: Variable weak Hardy spaces and their applications. J. Funct. Anal. 271(10), 2822–2887 (2016)
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)
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)
Yee, T.L., Ho, K.-P.: Hardy’s inequalities and integral operators on Herz-Morrey spaces. Open Math. 18(1), 106–121 (2020)
Acknowledgements
The author would like to express his deep gratitude to the anonymous referees for their careful reading of the manuscript and their comments and suggestions. This work is supported by the Natural Science Foundation of Henan Province (No. 202300410338) and the Nanhu Scholar Program for Young Scholars of Xinyang Normal University.
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Wei, M. Mapping properties for operators on Herz-Morrey and Herz-Morrey-Hardy spaces. J. Pseudo-Differ. Oper. Appl. 14, 4 (2023). https://doi.org/10.1007/s11868-022-00501-7
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DOI: https://doi.org/10.1007/s11868-022-00501-7