Oscillation and variation inequalities for singular integrals and commutators on weighted Morrey spaces

doi:10.1007/s11464-015-0462-2

Front. Math. China

2016, Vol. 11

Issue (2) : 423-447 https://doi.org/10.1007/s11464-015-0462-2

RESEARCH ARTICLE

Oscillation and variation inequalities for singular integrals and commutators on weighted Morrey spaces

Jing ZHANG^1,²,Huoxiong WU^1,^*(

)

1. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
2. School of Mathematics and Statistics, Yili Normal College, Yining 835000, China

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Abstract

This paper is devoted to investigating the bounded behaviors of the oscillation and variation operators for Calderón-Zygmund singular integrals and the corresponding commutators on the weighted Morrey spaces. We establish several criterions of boundedness, which are applied to obtain the corresponding bounds for the oscillation and variation operators of Hilbert transform, Hermitian Riesz transform and their commutators with BMO functions, or Lipschitz functions on weighted Morrey spaces.

Keywords Oscillation variation singular integrals commutators Morrey spaces weights

Corresponding Author(s): Huoxiong WU

Issue Date: 18 April 2016

Cite this article:

Jing ZHANG,Huoxiong WU. Oscillation and variation inequalities for singular integrals and commutators on weighted Morrey spaces[J]. Front. Math. China, 2016, 11(2): 423-447.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-015-0462-2
https://academic.hep.com.cn/fmc/EN/Y2016/V11/I2/423

1	Adams D R. A note on Riesz potentials. Duke Math J, 1975, 42: 765–778 https://doi.org/10.1215/S0012-7094-75-04265-9
2	Akcoglu M, Jones R L, Schwartz P. Variation in probability, ergodic theory and analysis. Illinois J Math, 1998, 42(1): 154–177
3	Bougain J. Pointwise ergodic theorem for arithmetic sets. Publ Math Inst Hautes Études Sci, 1989, 69: 5–45 https://doi.org/10.1007/BF02698838
4	Campbell J T, Jones R L, Reinhold K, Wierdl M. Oscillation and variation for Hilbert transform. Duke Math J, 2000, 105: 59–83<?Pub Caret?> https://doi.org/10.1215/S0012-7094-00-10513-3
5	Campbell J T, Jones R L, Reinhold K, Wierdl M. Oscillation and variation for singular integrals in higher dimension. Trans Amer Math Soc, 2003, 355: 2115–2137 https://doi.org/10.1090/S0002-9947-02-03189-6
6	Chen S, Wu H, Xue Q. A note on multilinear Muckenhoupt classes for multiple weights. Studia Math, 2014, 223(1): 1–18 https://doi.org/10.4064/sm223-1-1
7	Chiarenza F, Frasca M. Morrey spaces and Hardy-Littlewood maximal function. Rend Mat Appl, 1987, 7: 273–279
8	Crescimbeni R, Martin-Reyes F J, Torre A L, Torrea J L. The ρ-variation of the Hermitian Riesz transform. Acta Math Sin (Engl Ser), 2010, 26: 1827–1838 https://doi.org/10.1007/s10114-010-9122-3
9	Garćıa-Cuerva J. Weighted Hp space. Dissertations Math, 1979, 162: 1–63
10	Gillespie T A, Torrea J L. Dimension free estimates for the oscillation of Riesz transforms. Israel J Math, 2004, 141: 125–144 https://doi.org/10.1007/BF02772215
11	Grafakos L. Classical and Modern Fourier Analysis. Upper Saddle River: Pearson Education, Inc, 2004
12	Jones R L. Ergodic theory and connections with analysis and probability. New York J Math, 1997, 3A: 31–67
13	Jones R L. Variation inequalities for singular integrals and related operators. Contemp Math, 2006, 411: 89–121 https://doi.org/10.1090/conm/411/07750
14	Jones R L, Reinhold K. Oscillation and variation inequalities for convolution powers. Ergodic Theory Dynam Systems, 2001, 21: 1809–1829 https://doi.org/10.1017/S0143385701001869
15	Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282(2): 219–231 https://doi.org/10.1002/mana.200610733
16	Liu F, Wu H. A criterion on osillation and variation for the commutators of singular integral operators. Forum Math, 2015, 27: 77–97 https://doi.org/10.1515/forum-2012-0019
17	Morrey C B. 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
18	Muckenhoupt B, Wheeden R. Weighted bounded mean oscillation and the Hilbert transform. Studia Math, 1976, 54: 221–237
19	Stempark K, Torrea J L. Poisson integrals and Riesz transforms for Hermite function expansions with weights. J Funct Anal, 2003, 202: 443–472 https://doi.org/10.1016/S0022-1236(03)00083-1
20	Thangavelu S. Lectures on Hermite and Laguerre Expansions. Math Notes 42. Princeton: Princeton Univ Press, 1993
21	Torchinsky A. Real Variable Methods in Harmonic Analysis. New York: Academic Press, 1986
22	Zhang J, Wu H. Oscillation and variation inequalities for the commutators of singular integrals with Lipschitz functions. J Inequal Appl, 2015, 214, DOI: 10.1186/s13660-015-0737-x https://doi.org/10.1186/s13660-015-0737-x

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