Marcinkiewicz integral with rough kernels

Abstract

In this summary paper, we will introduce some recent progress in the theory of Marcinkiewicz integral and will pay more attention to the case of rough kernels.

References

1. Al-Qassem H RoughMarcinkiewicz integral operators on product spacesCollect Math 2005 56275297
2. Al-Salman A Marcinkiewiczfunction along .at surfaces with Hardy space kernelsJ Integral Equations Appl 2005 17357373. doi:10.1216/jiea/1181075348
3. Al-Salman A Al-Qassem H Cheng L et al.L^p bounds for the function of MarcinkiewiczMath Res Letter 2002 9697700
4. Al-Salman A Al-Qassem H Cheng L et al.Marcinkiewicz integrals on product spacesStudia Math 2005 167227234
5. Benedek A Calderón AP Panzone R Convolution operators on Banach value functionsProc Nat Acad Sci 1962 48256365
6. Calderón A P Zygmund A On singular integrals withvariable kernelsApplicable Anal 1977/78 7221238. doi:10.1080/00036817808839193
7. Chen J Ding Y Fan D L^p boundedness of rough Marcinkiewicz integral on product domainsChinese J Contem Math 2000 49481490
8. Chen J Fan D Ying Y Rough Marcinkiewicz integrals with L(log⁺L)² kernels on product spacesAdv Math (China) 2001 30179181
9. Chen J Fan D Ying Y The method of rotation and Marcinkiewicz integrals on productdomainsStudia Math 2002 1534158
10. Ding Y Fan D Pan Y Weighted boundedness for a class of Marcinkiewicz integralsIndiana Univ Math J 1999 4810371055. doi:10.1512/iumj.1999.48.1696
11. Ding Y Fan D Pan Y L^p boundedness of Marcinkiewicz with Hardy space function kernelsActa Math Sinica (English Ser) 2000 16593600. doi:10.1007/s101140000015
12. Ding Y Fan D Pan Y On Littlewood-Paley functions and singular integralsHokkaido Math J 2000 29537552
13. Ding Y Lin C Shao S On the Marcinkiewicz integral with variable kernelsIndiana Univ Math J 2004 53(3)805821. doi:10.1512/iumj.2004.53.2406
14. Ding Y Lu S Xue Q On Marcinkiewicz integral with homogeneous kernelJ Math Anal Appl 2000 245471488. doi:10.1006/jmaa.2000.6765
15. Ding Y Lu S Xue Q Marcinkiewicz integral on Hardy spacesInteg Equa Oper Theory 2002 42174182. doi:10.1007/BF01275514
16. Ding Y Lu S Yabuta K A problem on rough parametric Marcinkiewicz functionsJ Austral Math Soc 2002 721321
17. Ding Y Lu S Yabuta K On commutators of Marcinkiewicz integrals with rough kernelJ Math Anal Appl 2002 2756068. doi:10.1016/S0022‐247X(02)00230‐5
18. Ding Y Lu S Zhang P Weighted weak type estimates for commutators of the MarcinkiewiczintegralsSci China (Ser A-Math) 2004 478395. doi:10.1360/03ys0084
19. Duoandikoetxea J Seijo E Weighted inequalities for roughsquare functions through extrapolationStudiaMath 2002 149239252
20. Fan D Sato S Weak type (1,1) estimates forMarcinkiewicz integrals with rough kernelsTôhoku Math J 2001 53265284. doi:10.2748/tmj/1178207481
21. Fefferman R Soria F The space weak H¹Studia Math 1987 85116
22. Han Y Someproperties of the S-function and Marcinkiewicz integralsJ Peking Univ 1987 52134
23. Hu G Lu S Yang D Boundedness of rough singular integral operators on homogeneousHerz spaceJ Austral Math Soc (Ser A) 1999 66201223
24. Hu G Lu S Yan D L^p(R^m × Rⁿ) boundedness for the Marcinkiewicz integral on product spacesSci China (Ser A-Math) 2003 467582. doi:10.1360/03ys9008
25. Hörmander L Translationinvariant operatorsActa Math 1960 10493139. doi:10.1007/BF02547187
26. Jiang Y Lu S A class of singular integraloperators with rough kernel on product domainsHokkaido Math J 1995 2417
27. Li J Boundednessof some operators and their commutators. Ph D ThesisBeijingBeijing Normal University 2005
28. Li R ParameterizedMarcinkiewicz integral with variable kernels. Master D DissertationBeijingBeijingNormal University 2006
29. Lu S Mo H The Boundedness of commutatorsfor the Marcinkiewicz IntegralsActa MathSinica (Chinese Ser) 2006 49481490
30. Lu S Mo H Boundedness of higher orderMarcinkiewicz type integralsInter J MathMath Sci, Article Id 2006 31705121
31. Marcinkiewicz J Surquelques integrales de type de DiniAnnalesde la Société Polon 1938 174250
32. Qiu S Boundednessof Littlewood-Paley operators and Marcinkiewicz integralsJ Math Res Exposition 1992 124250
33. Sato S Remarkson square functions in the Littlewood-Paley TheoryBull Austral Math Soc 1998 58199211
34. Stein E M Onthe function of Littlewood-Paley, Lusin and MarcinkiewiczTrans Amer Math Soc 1958 88430466. doi:10.2307/1993226
35. Tao G Boundednessof some operators on Lesbesgue spaces and Herz spaces. Master D DissertationBeijingBeijingNormal University 2002
36. Torchinsky A Wang S A note on the MarcinkiewiczintegralColloq Math 1990 60/61235243
37. Wang S Someproperties of Littlewood-Paley's g-functionsSci China (Ser A-Math) 1985 3252262
38. Walsh T Onthe function of MarcinkiewiczStudia Math 1972 44203217
39. Yabuta K Boundednessof Littlewood operatorsMath Japonica 1996 43143150
40. Zygmund A Oncertain integralsTrans Amer Math Soc 1944 55170204. doi:10.2307/1990189