Frontiers of Mathematics in China >
Marcinkiewicz integrals with rough kernels in
Received date: 13 Oct 2020
Accepted date: 25 Jan 2021
Copyright
This paper is devoted to studying the Marcinkiewicz integral operators associated to polynomial compound curves. Some new bounds for the above operators on the Lebesgue, Triebel-Lizorkin, and Besov spaces are established by assuming that their rough kernels are given by andfor some: It should be pointed out that the bounds are independent of and the coefficients of the polynomials in the definition of the operators.
Key words: Marcinkiewicz integral; polynomial curve; ; Triebel-Lizorkin space
Daiqing ZHANG , Feng LIU . Marcinkiewicz integrals with rough kernels in [J]. Frontiers of Mathematics in China, 2021 , 16(4) : 1163 -1189 . DOI: 10.1007/s11464-021-0913-x
1 |
Al-Salman A. Marcinkiewicz functions along at surfaces with Hardy space kernels. J Integral Equations Appl, 2005, 17: 357–373
|
2 |
Al-Salman A. Rough maximal functions supported by subvarieties. J Operator Theory, 2008, 59: 263–275
|
3 |
Al-Salman A. A note on parabolic Marcinkiewicz integrals along surfaces. Proc A Razmadze Math Inst, 2010, 154: 21–36
|
4 |
Al-Salman A. On the L2 boundedness of parametric Marcinkiewicz integral operator. J Math Anal Appl, 2011, 375: 745–752
|
5 |
Al-Salman A, Al-Qassume H, Cheng L C, Pan Y B. Lp bounds for the function of Marcinkiewicz. Math Res Lett, 2002, 9: 697–700
|
6 |
Al-Salman A, Pan Y B. On certain estimates for Marcinkiewicz integrals and extrapolation. Collect Math, 2009, 60: 123–145
|
7 |
Benedek A, Calderón A, Panzone R. Convolution operators on Banach space valued functions. Proc Natl Acad Sci USA, 1962, 48: 356–365
|
8 |
Chen J C, Fan D S, Pan Y B. A note on a Marcinkiewicz integral operator. Math Nachr, 2001, 227: 33–42
|
9 |
Colzani L. Hardy spaces on spheres. Ph D Thesis. St Louis: Washington Univ, 1982
|
10 |
Colzani L, Taibleson M, Weiss G. Maximal estimates for Cesàro and Riesz means on sphere. Indiana Univ Math J, 1984, 33: 873–889
|
11 |
Ding Y, Fan D S, Pan Y B. Lp-boundedness of Marcinkiewicz integrals with Hardy space function kernel. Acta Math Sin (Engl Ser), 2000, 16: 593–600
|
12 |
Ding Y, Fan D S, Pan Y B. On the Lp boundedness of Marcinkiewicz integrals. Michigan Math J, 2002, 50: 17–26
|
13 |
Ding Y, Lu S Z, Yabuta Y. A problem on rough parametric Marcinkiewicz functions. J Aust Math Soc, 2002, 72: 13–21
|
14 |
Ding Y, Xue Q Y, Yabuta K. Boundedness of the Marcinkiewicz integrals with rough kernel associated to surfaces. Tohoku Math J, 2010, 62: 233–262
|
15 |
Ding Y, Xue Q Y, Yabuta K. A remark to the L2 boundedness of parametric Marcinkiewicz integral. J Math Anal Appl, 2012, 387: 691–697
|
16 |
Fan D S, Pan Y B. A singular integral operator with rough kernel. Proc Amer Math Soc, 1997, 125: 3695–3703
|
17 |
Frazier M, Jawerth B, Weiss G. Littlewood-Paley Theory and the Study of Function Spaces. CBMS Reg Conf Ser Math, No 79. Providence: Amer Math Soc, 1991
|
18 |
Grafakos L, Stefanov A. Lp bounds for singular integrals and maximal singular integrals with rough kernels. Indiana Univ Math J, 1998, 47: 455–469
|
19 |
Hörmander L. Estimates for translation invariant operators in Lp spaces. Acta Math, 1960, 104: 93–104
|
20 |
Jiang Y S, Lu S Z. Lp boundedness of a class of maximal singular integral operators. Acta Math Sinica (Chin Ser), 1992, 35: 63–72
|
21 |
Liu F. A note on Marcinkiewicz integrals associated to surfaces of revolution. J Aust Math Soc, 2018, 104: 380–402
|
22 |
Liu F, Wu H X. On Marcinkiewicz integrals associated to compound mappings with rough kernels. Acta Math Sin (Engl Ser), 2014, 30: 1210–1230
|
23 |
Liu F, Wu H X.Lp bounds for Marcinkiewicz integrals associated to homogeneous mappings. Monatsh Math, 2016, 181: 875–906
|
24 |
Liu F, Wu H X. On the regularity of maximal operators supported by submanifolds. J Math Anal Appl, 2017, 453: 144–158
|
25 |
Liu F, Zhang D Q. Parametric Marcinkiewicz integrals associated to surfaces with rough kernels and extrapolation. Bull Korean Math Soc, 2015, 52: 771–788
|
26 |
Ricci F, Stein E M. Harmonic analysis on nilpotent groups and singular integrals I: Oscillatory integrals. J Funct Anal, 1987, 73: 179–194
|
27 |
Sakamoto M, Yabuta K. Boundedness of Marcinkiewicz functions. Studia Math, 1999, 135: 103–142
|
28 |
Sato S. Estimates for singular integrals and extrapolation. Studia Math, 2009, 192: 219–233
|
29 |
Stein E M. On the function of Littlewood-Paley, Lusin and Marcinkiewicz. Trans Amer Math Soc, 1958, 88: 430–466
|
30 |
Taibleson M H, Weiss G. Certain function spaces connected with almost everywhere convergence of Fourier series. In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, I, II (Chicago, Ill, 1981). Wadsworth Math Ser. Belmont: Wadsworth, 1983, 95C–113
|
31 |
Triebel H. Theory of Function Spaces. Monogr Math, Vol 78. Basel: Birkhäser Verlag, 1983
|
32 |
Wu H X. On Marcinkiewicz integral operators with rough kernels. Integral Equations Operator Theory, 2005, 52: 285–298
|
33 |
Yabuta K. Triebel-Lizorkin space boundedness of Marcinkiewicz integrals associated to surfaces. Appl Math J Chinese Univ, 2015, 30: 418–446
|
34 |
Zhang C J, Chen J C. Boundedness of g-functions on Triebel-Lizorkin spaces. Taiwanese J Math, 2009, 13: 973–981
|
35 |
Zhang C J, Chen J C. Boundedness of Marcinkiewicz integral on Triebel-Lizorkin spaces. Appl Math J Chinese Univ, 2010, 25: 48–54
|
/
〈 | 〉 |