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Function characterizations via commutators of Hardy operator
Shanzhen LU
Function characterizations via commutators of Hardy operator
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.
Hardy operator / commutator / central function space
[1] |
Adams D R, Xiao J. Morrey spaces in harmonic analysis. Ark Mat, 2012, 50: 201–230
CrossRef
Google scholar
|
[2] |
Adams D R, Xiao J. Regularity of Morrey commutators. Trans Amer Math Soc, 2012, 364: 4801–4818
CrossRef
Google scholar
|
[3] |
Alvarez J, Guzmán-Partida M, Lakey J. Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures. Collect Math, 2000, 51: 1–47
|
[4] |
Anderson K, Muckenhoupt B. Weighted weak type Hardy inequalities with application to Hilbert transforms and maximal functions. Studia Math, 1982, 72: 9–26
CrossRef
Google scholar
|
[5] |
Beatrous F, Li S Y, On the boundedness and compactness of operators of Hankel type. J Funct Anal, 1993, 111: 350–379
CrossRef
Google scholar
|
[6] |
Branmanti M, Cerutti M.Wp1,2-solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients. Comm Partial Differential Equations. 1993, 18: 1735–1763
|
[7] |
Campanato S. Proprietàdi Hölderianità di alcune classi di funzioni. Ann Sc Norm Super Pisa, 1963, 17: 173–188
|
[8] |
Chanillo S. A note on commutators. Indiana Univ Math J, 1982, 31: 7–16
CrossRef
Google scholar
|
[9] |
Chen Y P, Ding Y. Compactness of the commutators of parabolic singular integrals. Sci China Math, 2010, 53: 2633–2648
CrossRef
Google scholar
|
[10] |
Chen Y P, Ding Y. Compactness of commutators of Riesz potential on Morrey spaces. Potential Anal, 2009, 30: 301–313
CrossRef
Google scholar
|
[11] |
Chen Y P, Ding Y. Compactness of commutators for singular integrals on Morrey spaces. Canad J Math, 2012, 64: 257–281
CrossRef
Google scholar
|
[12] |
Chiarenza F, Frasca M, Longo P. W2,p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans Amer Math Soc, 1993, 336: 841–853
|
[13] |
Christ M, Grafakos L. Best constants for two nonconvolution inequalities. Proc Amer Math Soc, 1995, 123: 1687–1693
CrossRef
Google scholar
|
[14] |
Coifman R R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann of Math, 1976, 103: 611–635
CrossRef
Google scholar
|
[15] |
Cruz D, Neugebauer C J. The structure of the reverse Hölder classes. Trans Amer Math Soc, 1995, 345: 2941–2960
CrossRef
Google scholar
|
[16] |
Deng D G, Duong X T, Yan L X. A characterization of Morrey-Campanato spaces. Math Z, 2005, 250: 641–655
CrossRef
Google scholar
|
[17] |
Ding Y. A characterization of BMO via commutators for some operators. Northeast Math, 1997, 13: 422–432
|
[18] |
Ding Y, Mei T. Boundedness and compactness for the commutators of bilinear operators on Morrey spaces. Potential Anal, 2015, 42: 717–748
CrossRef
Google scholar
|
[19] |
Duong X T, Xiao J, Yan L X. Old and new Morrey spaces with heat kernel bounds. J Fourier Anal Appl, 2007, 13: 87–111
CrossRef
Google scholar
|
[20] |
Fan D S, Lu S Z, Yang D C. Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with V MO coefficients. Georgian Math J, 1998, 5: 425–440
CrossRef
Google scholar
|
[21] |
Faris W. Weak Lebesgue spaces and quantum mechanical binding. Duke Math J, 1976, 43: 365–373
CrossRef
Google scholar
|
[22] |
Fazio G D, Ragusa M A. Interior estimates in Morrey spaces for strongly solutions to nondivergence form equations with discontinuous coefficients. J Funct Anal, 1993, 112: 241–256
CrossRef
Google scholar
|
[23] |
Fefferman C. The uncertainty principle. Bull Amer Math Soc, 1983, 9: 129–206
CrossRef
Google scholar
|
[24] |
Fu Z W, Liu Z G, Lu S Z, Wang H B. Characterization for commutators of n-dimensional fractional Hardy operators. Sci China Ser A, 2007, 50: 1418–1426
CrossRef
Google scholar
|
[25] |
Fu Z W, Lu S Z. Commutators of generalized Hardy operators. Math Nachr, 2009, 282: 832–845
CrossRef
Google scholar
|
[26] |
Fu Z W, Wu Q Y, Lu S Z. Sharp estimates of p-adic Hardy and Hardy-Littlewood-Pólya operators. Acta Math Sin (Engl Ser), 2013, 29: 137–150
CrossRef
Google scholar
|
[27] |
García-Cuerva J. Hardy spaces and Beurling algebras. J Lond Math Soc, 1989, 39: 499–513
CrossRef
Google scholar
|
[28] |
Gilbarg D, Trudinger N. Elliptic Partial Differential Equations of Second Order. Grundlehren Math Wiss, Vol 224. Berlin: Springer-Verlag, 1983,
|
[29] |
Golubov B. Boundedness of the Hardy and the Hardy-Littlewood operators in the spaces Re H1 and BMO. Mat Sb, 1997, 188: 93–106
|
[30] |
Hardy G H, Littlewood J E, Pólya G. Inequalities. London: Cambridge Univ Press, 1934
|
[31] |
Harboure E, Salinas O, Viviani B. Reverse Hölder classes in the Orlicz-spaces setting. Studia Math, 1998, 130: 245–261
|
[32] |
Iwaniec T, Sbordone C. Riesz transforms and elliptic PDEs with VMO coefficients. J Anal Math, 1998, 74: 183–212
CrossRef
Google scholar
|
[33] |
Janson S. Mean oscillation and commutators of singular integral operators. Ark Mat, 1978, 16: 263–270
CrossRef
Google scholar
|
[34] |
Komori Y. Notes on commutators of Hardy operators. Int J Pure Appl Math, 2003, 7: 329–334
|
[35] |
Krantz S, Li S Y. Boundedness and compactness of integral operators on spaces of homogeneous type and applications II. J Math Anal Appl, 2001, 258: 642–657
CrossRef
Google scholar
|
[36] |
Lemarié-Rieusset P. The Navier-Stokes equations in the critical Morrey-Campanato space. Rev Mat Iberoam, 2007, 23: 897–930
CrossRef
Google scholar
|
[37] |
Lu G Z. 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] |
Lu S Z, Yang D C. The central BMO spaces and Littlewood-Paley operators. Approx Theory Appl, 1995, 11: 72–94
|
[39] |
Lu S Z, Yan D Y, Zhao F Y. Sharp bounds for Hardy type operators on higherdimensional product spaces. J Inequal Appl, 2013, 148: 1–11
CrossRef
Google scholar
|
[40] |
Long S C, Wang J. Commutators of Hardy operators. J Math Anal Appl, 2002, 274: 626–644
CrossRef
Google scholar
|
[41] |
Morrey C. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43: 126–166
CrossRef
Google scholar
|
[42] |
Palagachev D, Softova L. Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s. Potential Anal, 2004, 20: 237–263
CrossRef
Google scholar
|
[43] |
Paluszynski M. Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ Math J, 1995, 44: 1–17
CrossRef
Google scholar
|
[44] |
Sawyer E. Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans Amer Math Soc, 1984, 1: 329–337
CrossRef
Google scholar
|
[45] |
Shi S G, Fu Z W, Lu S Z. On the compactness of commutators of Hardy operators. Pacific J Math, 2020, 307: 239–256
CrossRef
Google scholar
|
[46] |
Shi S G, Lu S Z. Some characterizations of Campanato spaces via commutators on Morrey spaces. Pacific J Math, 2013, 264: 221–234
CrossRef
Google scholar
|
[47] |
Shi S G, Lu S Z. A characterization of Campanato space via commutator of fractional integral. J Math Anal Appl, 2014, 419: 123–137
CrossRef
Google scholar
|
[48] |
Shi S G, Lu S Z. Characterization of the central Campanato space via the commutator operator of Hardy type. J Math Anal Appl, 2015, 429: 713–732
CrossRef
Google scholar
|
[49] |
Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Monographs in Harmonic Analysis, III. Princeton Math Ser, 43. Princeton: Princeton Univ Press, 1993
CrossRef
Google scholar
|
[50] |
Stein E M, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Monographs in Harmonic Analysis, I. Princeton Math Ser, 32. Princeton: Princeton Univ Press, 1971
CrossRef
Google scholar
|
[51] |
Uchiyama A. On the compactness of operators of Hankel type. Tohoku Math J, 1978, 30: 163–171
CrossRef
Google scholar
|
[52] |
Wang S M, Lu S Z, Yan D Y, Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces. Sci China Math, 2012, 55(12): 2469–2480
CrossRef
Google scholar
|
[53] |
Wu Q Y, Fu Z W. Weighted p-adic Hardy operators and their commutators on p-adic central Morrey spaces. Bull Malays Math Sci Soc, 2017, 40: 635–654
CrossRef
Google scholar
|
[54] |
Wu Q Y, Mi L, Fu Z W. Boundedness of p-adic Hardy operators and their commutators on p-adic central Morrey and BMO spaces. J Funct Spaces, 2013, 2013: 1–10
CrossRef
Google scholar
|
[55] |
Yang D C, Yang D Y, Zhou Y. Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators. Nagoya Math J, 2010, 198: 77–119
CrossRef
Google scholar
|
[56] |
Yuan W, Sickel W, Yang D C. Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Math, Vol 2005. Berlin: Springer-Verlag, 2010
CrossRef
Google scholar
|
[57] |
Zhao F Y, Lu S Z. A characterization of λ-central BMO space. Front Math China, 2013, 8: 229–238
CrossRef
Google scholar
|
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