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Frontiers of Mathematics in China

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ISSN 1673-3576(Online)

CN 11-5739/O1

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Front. Math. China    2021, Vol. 16 Issue (1) : 1-12    https://doi.org/10.1007/s11464-021-0894-9
SURVEY ARTICLE
Function characterizations via commutators of Hardy operator
Shanzhen LU()
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
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Abstract

This paper is a summary of the research on the characterizations of central function spaces by the author and his collaborators in the past ten years. More precisely, the author gives some characterizations of central Campanato spaces via the boundedness and compactness of commutators of Hardy operator.

Keywords Hardy operator      commutator      central function space     
Corresponding Author(s): Shanzhen LU   
Issue Date: 26 March 2021
 Cite this article:   
Shanzhen LU. Function characterizations via commutators of Hardy operator[J]. Front. Math. China, 2021, 16(1): 1-12.
 URL:  
https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0894-9
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I1/1
1 D R Adams, J Xiao. Morrey spaces in harmonic analysis. Ark Mat, 2012, 50: 201–230
https://doi.org/10.1007/s11512-010-0134-0
2 D R Adams, J Xiao. Regularity of Morrey commutators. Trans Amer Math Soc, 2012, 364: 4801–4818
https://doi.org/10.1090/S0002-9947-2012-05595-4
3 J Alvarez, M Guzmán-Partida, J Lakey. Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures. Collect Math, 2000, 51: 1–47
4 K Anderson, B Muckenhoupt. Weighted weak type Hardy inequalities with application to Hilbert transforms and maximal functions. Studia Math, 1982, 72: 9–26
https://doi.org/10.4064/sm-72-1-9-26
5 F Beatrous, S Y Li, On the boundedness and compactness of operators of Hankel type. J Funct Anal, 1993, 111: 350–379
https://doi.org/10.1006/jfan.1993.1017
6 M Branmanti, M Cerutti.Wp1,2-solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients. Comm Partial Differential Equations. 1993, 18: 1735–1763
7 S Campanato. Proprietàdi Hölderianità di alcune classi di funzioni. Ann Sc Norm Super Pisa, 1963, 17: 173–188
8 S Chanillo. A note on commutators. Indiana Univ Math J, 1982, 31: 7–16
https://doi.org/10.1512/iumj.1982.31.31002
9 Y P Chen, Y Ding. Compactness of the commutators of parabolic singular integrals. Sci China Math, 2010, 53: 2633–2648
https://doi.org/10.1007/s11425-010-4004-9
10 Y P Chen, Y Ding. Compactness of commutators of Riesz potential on Morrey spaces. Potential Anal, 2009, 30: 301–313
https://doi.org/10.1007/s11118-008-9114-4
11 Y P Chen, Y Ding. Compactness of commutators for singular integrals on Morrey spaces. Canad J Math, 2012, 64: 257–281
https://doi.org/10.4153/CJM-2011-043-1
12 F Chiarenza, M Frasca, P Longo. W2,p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans Amer Math Soc, 1993, 336: 841–853
13 M Christ, L Grafakos. Best constants for two nonconvolution inequalities. Proc Amer Math Soc, 1995, 123: 1687–1693
https://doi.org/10.1090/S0002-9939-1995-1239796-6
14 R R Coifman, R Rochberg, G Weiss. Factorization theorems for Hardy spaces in several variables. Ann of Math, 1976, 103: 611–635
https://doi.org/10.2307/1970954
15 D Cruz, C J Neugebauer. The structure of the reverse Hölder classes. Trans Amer Math Soc, 1995, 345: 2941–2960
https://doi.org/10.1090/S0002-9947-1995-1308005-6
16 D G Deng, X T Duong, L X Yan. A characterization of Morrey-Campanato spaces. Math Z, 2005, 250: 641–655
https://doi.org/10.1007/s00209-005-0769-x
17 Y Ding. A characterization of BMO via commutators for some operators. Northeast Math, 1997, 13: 422–432
18 Y Ding, T Mei. Boundedness and compactness for the commutators of bilinear operators on Morrey spaces. Potential Anal, 2015, 42: 717–748
https://doi.org/10.1007/s11118-014-9455-0
19 X T Duong, J Xiao, L X Yan. Old and new Morrey spaces with heat kernel bounds. J Fourier Anal Appl, 2007, 13: 87–111
https://doi.org/10.1007/s00041-006-6057-2
20 D S Fan, S Z Lu, D C Yang. Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with V MO coefficients. Georgian Math J, 1998, 5: 425–440
https://doi.org/10.1023/B:GEOR.0000008114.52420.af
21 W Faris. Weak Lebesgue spaces and quantum mechanical binding. Duke Math J, 1976, 43: 365–373
https://doi.org/10.1215/S0012-7094-76-04332-5
22 G D Fazio, M A Ragusa. Interior estimates in Morrey spaces for strongly solutions to nondivergence form equations with discontinuous coefficients. J Funct Anal, 1993, 112: 241–256
https://doi.org/10.1006/jfan.1993.1032
23 C Fefferman. The uncertainty principle. Bull Amer Math Soc, 1983, 9: 129–206
https://doi.org/10.1090/S0273-0979-1983-15154-6
24 Z W Fu, Z G Liu, S Z Lu, H B Wang. Characterization for commutators of n-dimensional fractional Hardy operators. Sci China Ser A, 2007, 50: 1418–1426
https://doi.org/10.1007/s11425-007-0094-4
25 Z W Fu, S Z Lu. Commutators of generalized Hardy operators. Math Nachr, 2009, 282: 832–845
https://doi.org/10.1002/mana.200610775
26 Z W Fu, Q Y Wu, S Z Lu. Sharp estimates of p-adic Hardy and Hardy-Littlewood-Pólya operators. Acta Math Sin (Engl Ser), 2013, 29: 137–150
https://doi.org/10.1007/s10114-012-0695-x
27 J García-Cuerva. Hardy spaces and Beurling algebras. J Lond Math Soc, 1989, 39: 499–513
https://doi.org/10.1112/jlms/s2-39.3.499
28 D Gilbarg, N Trudinger. Elliptic Partial Differential Equations of Second Order. Grundlehren Math Wiss, Vol 224. Berlin: Springer-Verlag, 1983,
29 B Golubov. Boundedness of the Hardy and the Hardy-Littlewood operators in the spaces Re H1 and BMO. Mat Sb, 1997, 188: 93–106
30 G H Hardy, J E Littlewood, G Pólya. Inequalities. London: Cambridge Univ Press, 1934
31 E Harboure, O Salinas, B Viviani. Reverse Hölder classes in the Orlicz-spaces setting. Studia Math, 1998, 130: 245–261
32 T Iwaniec, C Sbordone. Riesz transforms and elliptic PDEs with VMO coefficients. J Anal Math, 1998, 74: 183–212
https://doi.org/10.1007/BF02819450
33 S Janson. Mean oscillation and commutators of singular integral operators. Ark Mat, 1978, 16: 263–270
https://doi.org/10.1007/BF02386000
34 Y Komori. Notes on commutators of Hardy operators. Int J Pure Appl Math, 2003, 7: 329–334
35 S Krantz, S Y Li. Boundedness and compactness of integral operators on spaces of homogeneous type and applications II. J Math Anal Appl, 2001, 258: 642–657
https://doi.org/10.1006/jmaa.2000.7403
36 P Lemarié-Rieusset. The Navier-Stokes equations in the critical Morrey-Campanato space. Rev Mat Iberoam, 2007, 23: 897–930
https://doi.org/10.4171/RMI/518
37 G Z Lu. Embedding theorems on Campanato-Morrey spaces for degenerate vector fields and applications. C R Acad Sci Paris Sér I Math, 1995, 320: 429–434
38 S Z Lu, D C Yang. The central BMO spaces and Littlewood-Paley operators. Approx Theory Appl, 1995, 11: 72–94
39 S Z Lu, D Y Yan, F Y Zhao. Sharp bounds for Hardy type operators on higherdimensional product spaces. J Inequal Appl, 2013, 148: 1–11
https://doi.org/10.1186/1029-242X-2013-148
40 S C Long, J Wang. Commutators of Hardy operators. J Math Anal Appl, 2002, 274: 626–644
https://doi.org/10.1016/S0022-247X(02)00321-9
41 C Morrey. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43: 126–166
https://doi.org/10.1090/S0002-9947-1938-1501936-8
42 D Palagachev, L Softova. Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s. Potential Anal, 2004, 20: 237–263
https://doi.org/10.1023/B:POTA.0000010664.71807.f6
43 M Paluszynski. Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ Math J, 1995, 44: 1–17
https://doi.org/10.1512/iumj.1995.44.1976
44 E Sawyer. Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans Amer Math Soc, 1984, 1: 329–337
https://doi.org/10.1090/S0002-9947-1984-0719673-4
45 S G Shi, Z W Fu, S Z Lu. On the compactness of commutators of Hardy operators. Pacific J Math, 2020, 307: 239–256
https://doi.org/10.2140/pjm.2020.307.239
46 S G Shi, S Z Lu. Some characterizations of Campanato spaces via commutators on Morrey spaces. Pacific J Math, 2013, 264: 221–234
https://doi.org/10.2140/pjm.2013.264.221
47 S G Shi, S Z Lu. A characterization of Campanato space via commutator of fractional integral. J Math Anal Appl, 2014, 419: 123–137
https://doi.org/10.1016/j.jmaa.2014.04.040
48 S G Shi, S Z Lu. Characterization of the central Campanato space via the commutator operator of Hardy type. J Math Anal Appl, 2015, 429: 713–732
https://doi.org/10.1016/j.jmaa.2015.03.083
49 E M Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Monographs in Harmonic Analysis, III. Princeton Math Ser, 43. Princeton: Princeton Univ Press, 1993
https://doi.org/10.1515/9781400883929
50 E M Stein, G Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Monographs in Harmonic Analysis, I. Princeton Math Ser, 32. Princeton: Princeton Univ Press, 1971
https://doi.org/10.1515/9781400883899
51 A Uchiyama. On the compactness of operators of Hankel type. Tohoku Math J, 1978, 30: 163–171
https://doi.org/10.2748/tmj/1178230105
52 S M Wang, S Z Lu, D Y Yan, Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces. Sci China Math, 2012, 55(12): 2469–2480
https://doi.org/10.1007/s11425-012-4453-4
53 Q Y Wu, Z W Fu. Weighted p-adic Hardy operators and their commutators on p-adic central Morrey spaces. Bull Malays Math Sci Soc, 2017, 40: 635–654
https://doi.org/10.1007/s40840-017-0444-5
54 Q Y Wu, L Mi, Z W Fu. Boundedness of p-adic Hardy operators and their commutators on p-adic central Morrey and BMO spaces. J Funct Spaces, 2013, 2013: 1–10
https://doi.org/10.1155/2013/359193
55 D C Yang, D Y Yang, Y Zhou. Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators. Nagoya Math J, 2010, 198: 77–119
https://doi.org/10.1017/S0027763000009946
56 W Yuan, W Sickel, D C Yang. Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Math, Vol 2005. Berlin: Springer-Verlag, 2010
https://doi.org/10.1007/978-3-642-14606-0
57 F Y Zhao, S Z Lu. A characterization of λ-central BMO space. Front Math China, 2013, 8: 229–238
https://doi.org/10.1007/s11464-012-0251-0
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