×

On commutators of Marcinkiewicz integrals with rough kernel. (English) Zbl 1019.42009

Let \(h(t)\) be a bounded function on \(\mathbb{R}^+\) and \(b\) a function in the \(\text{BMO}(\mathbb{R}^n)\) space. In this paper, the authors consider the higher-order commutator \(\mu^m_{\Omega,b}(f)\) of the Marcinkiewicz integral defined by \[ \mu^m_{\Omega,b}(f)(x)= \Biggl\{\int^\infty_0|F^m_{t, b}(x)|^2 t^{-3} dt\Biggr\}^{1/2}, \] where \[ F^m_{t,b}(x)= \int_{|x-y|< t} h(|x-y|) \Omega(x-y)|x-y|^{-n+1} \{b(x)- b(y)\}^m f(y) dy \] and \(\Omega\) is an \(L^q\), \(q> 1\), function on the sphere \(S^{n-1}\) that satisfies the following conditions: \[ \Omega(tx)= \Omega(x),\quad\text{for any }t\in \mathbb{R}^+;\;\int_{S^{n-1}}\Omega(x) d\sigma(x)= 0. \] The authors also consider its related operators \(\mu^{*,m}_{\Omega, \lambda,b}\) that is corresponding to the Stein \(g^*_\lambda\)-function and \(\mu^m_{\Omega, S,b}\) that is corresponding to the Lusin area integral.
The authors establish various boundedness properties of these operators on the weighted spaces \(L^p(\mathbb{R}^n,\omega)\), where the weigh \(\omega\) depends on \(p\) and \(q\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alvarez, J.; Bagby, R.; Kurtz, D.; Pérez, C., Weighted estimates for commutators of linear operators, Studia Math., 104, 195-209 (1993) · Zbl 0809.42006
[2] Benedek, A.; Calderón, A.; Panzones, R., Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. USA, 48, 356-365 (1962) · Zbl 0103.33402
[3] Coifman, R.; Rochberg, R.; Weiss, G., Factorization theorem for Hardy spaces in several variables, Ann. of Math., 103, 611-635 (1976) · Zbl 0326.32011
[4] L. Colzani, Hardy spaces on sphere, Ph.D. thesis, Washington University, St. Louis (1982); L. Colzani, Hardy spaces on sphere, Ph.D. thesis, Washington University, St. Louis (1982) · Zbl 0505.46030
[5] Colzani, L.; Taibleson, M.; Weiss, G., Maximal estimates for Cesàro and Riesz means on sphere, Indiana Univ. Math. J., 33, 873-889 (1984) · Zbl 0545.42017
[6] Ding, Y.; Fan, D.; Pan, Y., \(L^p\)-boundedness of Marcinkiewicz integrals with Hardy space function kernel, Acta. Math. Sinica (English Ser.), 16, 593-600 (2000) · Zbl 0966.42008
[7] Ding, Y.; Fan, D.; Pan, Y., Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana Univ. Math. J., 48, 1037-1055 (1999) · Zbl 0949.42014
[8] Garcia-Cuerva, J.; Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topics (1985), North-Holland: North-Holland Amsterdam · Zbl 0578.46046
[9] Sakamoto, M.; Yabuta, K., Boundedness of Marcinkiewicz functions, Studia Math., 135, 103-142 (1999) · Zbl 0930.42009
[10] Stein, E. M., On the function of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc., 88, 430-466 (1958) · Zbl 0105.05104
[11] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (1993), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0821.42001
[12] Stein, E. M.; Weiss, G., Interpolation of operators with change of measures, Trans. Amer. Math. Soc., 87, 159-172 (1958) · Zbl 0083.34301
[13] Torchinsky, A.; Wang, S., A note on the On the Marcinkiewicz integral, Coll. Math., 61-62, 235-243 (1990) · Zbl 0731.42019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.