Local Morrey and Campanato spaces on quasimetric measure spaces. (English) Zbl 1305.46023
Summary: We define and investigate generalized local Morrey spaces and generalized local Campanato spaces, within a context of a general quasimetric measure space. The locality is manifested here by a restriction to a subfamily of involved balls. The structural properties of these spaces and the maximal operators associated to them are studied. In numerous remarks, we relate the developed theory, mostly in the “global” case, to the cases existing in the literature. We also suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets of \(\mathbb R^n\).
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
References:
| [1] | K. Stempak, “On quasi-metric measure spaces,” preprint. In press. · Zbl 1403.42023 |
| [2] | E. Nakai, “The Campanato, Morrey and Hölder spaces on spaces of homogeneous type,” Studia Mathematica, vol. 176, no. 1, pp. 1-19, 2006. · Zbl 1121.46031 · doi:10.4064/sm176-1-1 |
| [3] | Y. Sawano and H. Tanaka, “Morrey spaces for non-doubling measures,” Acta Mathematica Sinica. English Series, vol. 21, no. 6, pp. 1535-1544, 2005. · Zbl 1129.42403 · doi:10.1007/s10114-005-0660-z |
| [4] | C.-C. Lin, K. Stempak, and Y.-S. Wang, “Local maximal operators on measure metric spaces,” Publicacions Matemàtiques, vol. 57, no. 1, pp. 239-264, 2013. · Zbl 1291.42015 · doi:10.5565/PUBLMAT_57113_09 |
| [5] | D. Yang, D. Yang, and Y. Zhou, “Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators,” Nagoya Mathematical Journal, vol. 198, pp. 77-119, 2010. · Zbl 1214.46019 · doi:10.1215/00277630-2009-008 |
| [6] | L. Liu, Y. Sawano, and D. Yang, “Morrey-type spaces on Gauss measure spaces and boundedness of singular integrals,” Journal of Geometric Analysis, vol. 24, no. 2, pp. 1007-1051, 2014. · Zbl 1306.42042 · doi:10.1007/s12220-012-9362-9 |
| [7] | T. Hytönen, “A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa,” Publicacions Matemàtiques, vol. 54, no. 2, pp. 485-504, 2010. · Zbl 1246.30087 · doi:10.5565/PUBLMAT_54210_10 |
| [8] | X. Tolsa, “BMO, H1, and Calderón-Zygmund operators for non doubling measures,” Mathematische Annalen, vol. 319, no. 1, pp. 89-149, 2001. · Zbl 0974.42014 · doi:10.1007/s002080000144 |
| [9] | F. Nazarov, S. Treil, and A. Volberg, “Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces,” International Mathematics Research Notices, no. 9, pp. 463-487, 1998. · Zbl 0918.42009 · doi:10.1155/S1073792898000312 |
| [10] | W. Chen and E. Sawyer, “A note on commutators of fractional integrals with RBMO (\mu ) functions,” Illinois Journal of Mathematics, vol. 46, no. 4, pp. 1287-1298, 2002. · Zbl 1033.42008 |
| [11] | T. Heikkinen, J. Lehrbäck, J. Nuutinen, and H. Tuominen, “Fractional maximal functions in metric measure spaces,” Analysis and Geometry in Metric Spaces, vol. 1, pp. 147-162, 2012. · Zbl 1275.42032 · doi:10.2478/agms-2013-0002 |
| [12] | T. Heikkinen, J. Kinnunen, J. Korvenpää, and H. Tuominen, “Regularity of the local fractional maximal function,” http://arxiv.org/abs/1310.4298. · Zbl 1316.42019 |
| [13] | J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York, NY, USA, 2001. · Zbl 0985.46008 · doi:10.1007/978-1-4613-0131-8 |
| [14] | Y. Sawano, “Sharp estimates of the modified Hardy-Littlewood maximal operator on the nonhomogeneous space via covering lemmas,” Hokkaido Mathematical Journal, vol. 34, no. 2, pp. 435-458, 2005. · Zbl 1088.42010 · doi:10.14492/hokmj/1285766231 |
| [15] | Y. Terasawa, “Outer measures and weak type (1,1) estimates of Hardy-Littlewood maximal operators,” Journal of Inequalities and Applications, vol. 2006, Article ID 15063, 13 pages, 2006. · Zbl 1090.42012 · doi:10.1155/JIA/2006/15063 |
| [16] | J. Petree, “On the theory of Lp,\lambda spaces,” Journal of Functional Analysis, vol. 4, pp. 71-87, 1969. |
| [17] | Y. Sawano, “Generalized Morrey spaces for non-doubling measures,” Nonlinear Differential Equations and Applications, vol. 15, no. 4-5, pp. 413-425, 2008. · Zbl 1173.42317 · doi:10.1007/s00030-008-6032-5 |
| [18] | E. Nakai, “A characterization of pointwise multipliers on the Morrey spaces,” Scientiae Mathematicae, vol. 3, no. 3, pp. 445-454, 2000. · Zbl 0980.42005 |
| [19] | J. Mateu, P. Mattila, A. Nicolau, and J. Orobitg, “BMO for nondoubling measures,” Duke Mathematical Journal, vol. 102, no. 3, pp. 533-565, 2000. · Zbl 0964.42009 · doi:10.1215/S0012-7094-00-10238-4 |
| [20] | P. W. Jones, “Extension theorems for BMO,” Indiana University Mathematics Journal, vol. 29, no. 1, pp. 41-66, 1980. · Zbl 0432.42017 · doi:10.1512/iumj.1980.29.29005 |
| [21] | C. T. Zorko, “Morrey space,” Proceedings of the American Mathematical Society, vol. 98, no. 4, pp. 586-592, 1986. · Zbl 0612.43003 · doi:10.2307/2045731 |
| [22] | 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 |
| [23] | S. Campanato, “Proprietà di una famiglia di spazi funzionali,” Annali della Scuola Normale Superiore di Pisa, vol. 18, pp. 137-160, 1964. · Zbl 0133.06801 |
| [24] | C.-C. Lin and K. Stempak, “Atomic Hp spaces and their duals on open subsets of Rn,” Forum Mathematicum, 2013. · Zbl 1318.42028 · doi:10.1515/forum-2013-0053 |
| [25] | H. Luiro, “On the regularity of the Hardy-Littlewood maximal operator on subdomains of Rn,” Proceedings of the Edinburgh Mathematical Society, vol. 53, no. 1, pp. 211-237, 2010. · Zbl 1183.42025 · doi:10.1017/S0013091507000867 |
| [26] | Eridani, H. Gunawan, E. Nakai, and Y. Sawano, “Characterizations for the generalized fractional integral operators on Morrey spaces,” Mathematical Inequalities & Applications. In press. · Zbl 1298.42018 |
| [27] | Y. Sawano, S. Sugano, and H. Tanaka, “Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces,” Transactions of the American Mathematical Society, vol. 363, no. 12, pp. 6481-6503, 2011. · Zbl 1229.42024 · doi:10.1090/S0002-9947-2011-05294-3 |
| [28] | W. Yuan, W. Sickel, and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, vol. 2005 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010. · Zbl 1207.46002 · doi:10.1007/978-3-642-14606-0 |
| [29] | I. Sihwaningrum and Y. Sawano, “Weak and strong type estimates for fractional integral operators on Morrey spaces over metric measure spaces,” Eurasian Mathematical Journal, vol. 4, no. 1, pp. 76-81, 2013. · Zbl 1277.42019 |
| [30] | Y. Shi and X. Tao, “Some multi-sublinear operators on generalized Morrey spaces with non-doubling measures,” Journal of the Korean Mathematical Society, vol. 49, no. 5, pp. 907-925, 2012. · Zbl 1257.42026 · doi:10.4134/JKMS.2012.49.5.907 |
| [31] | F. Chiarenza and M. Frasca, “Morrey spaces and Hardy-Littlewood maximal function,” Rendiconti di Matematica e delle sue Applicazioni. Serie VII, vol. 7, no. 3-4, pp. 273-279, 1987. · Zbl 0717.42023 |
| [32] | E. Nakai, “Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces,” Mathematische Nachrichten, vol. 166, pp. 95-103, 1994. · Zbl 0837.42008 · doi:10.1002/mana.19941660108 |
| [33] | J. García-Cuerva and A. E. Gatto, “Boundedness properties of fractional integral operators associated to non-doubling measures,” Studia Mathematica, vol. 162, no. 3, pp. 245-261, 2004. · Zbl 1045.42006 · doi:10.4064/sm162-3-5 |
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.
