非齐型空间中一类满足Hörmander条件的Marcinkiewicz交换子估计

李亮, 江寅生

数学学报 ›› 2010, Vol. 53 ›› Issue (1) : 87-96.

PDF(518 KB)
中国科学院数学与系统科学研究院期刊网
PDF(518 KB)
数学学报 ›› 2010, Vol. 53 ›› Issue (1) : 87-96. DOI: 10.12386/A2010sxxb0013
论文

非齐型空间中一类满足Hörmander条件的Marcinkiewicz交换子估计

    李亮, 江寅生
作者信息 +

Estimates for Commtators of Marcinkiewicz Integrals with Hörmander-Type Condition in Non-Homogeneous Spaces

    Liang LI, Yin Sheng JIANG
Author information +
文章历史 +

摘要

记μ为上的非负Radon测度,且仅满足对固定的C0>0和n∈(0,d],及所有的和r>0, μ(B(x,r))≤C0 rn.作者建立了一类核函数满足Hörmander条件的Marcinkiewicz积分与Lipβ(μ)(0<β)函数生成的交换子由Lp(μ)到Lq(μ),由Lp(μ) 到Lipβ-n/p(μ)及Ln/β(μ)到RBMO(μ)有界.部分结论对经典 Marcinkiewicz积分也是新的.  

Abstract

Let μ be a positive Radon measure on which may be non doubling. The only condition that μ must satisfy is μ(B(x,r))≤C0 rn for all , r>0 and some fixed n∈(0,d]. In this paper, the authors establish the boundedness of the commutator generated by the Lipβ(μ)(0<β) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger Hörmander-type condition, respectively, from Lp(μ) to Lq(μ) with 1<p≤n/β and 1/q=1/p-β/n, from the space Lp(μ) to Lipβ-n/p(μ) and from the space Ln/β(μ) to RBMO(μ). Some of the results are also new even for the classical Marcinkiewicz integral.  

关键词

非倍测度 / Lipβ(μ)函数 / Marcinkiewicz积分

Key words

Marcinkiewicz integral / Lipβ(μ) / doubling measure

引用本文

EndNote

Ris (Procite)

Bibtex

导出引用
李亮, 江寅生. 非齐型空间中一类满足Hörmander条件的Marcinkiewicz交换子估计. 数学学报, 2010, 53(1): 87-96 https://doi.org/10.12386/A2010sxxb0013
Liang LI, Yin Sheng JIANG. Estimates for Commtators of Marcinkiewicz Integrals with Hörmander-Type Condition in Non-Homogeneous Spaces. Acta Mathematica Sinica, Chinese Series, 2010, 53(1): 87-96 https://doi.org/10.12386/A2010sxxb0013

参考文献



[1] Marcinkiewicz J., Sur quelques intégrales du type de Dini, Ann. Soc. Polon. Math., 1938, 17: 42--50.



[2] Zygmund A., Trigonometric Series, 3rd Edition, Cambridge: Cambridge University Press, 2002.



[3] Stein E. M., On the functions of Littlewood--Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc., 1958, 88: 430--466.



[4] Al-Salman A., Al-Qassem H., Cheng L. C., Pan Y., Lp bounds for the function of Marcinkiewicz, Math. Res. Lett., 2002, 9: 697--700.



[5] Fan D., Sato S., Weak type (1,1) estimates for Marcinkiewicz integrals with rough kernels, T?hoku Math. J., 2001, 53: 265--284.



[6] Ding Y., Lu S., Xue Q., Marcinkiewicz integral on Hardy spaces, Integral Equations Operator Theory, 2002, 42: 174--182.



[7] Ding Y., Lu S., Xue Q., On Marcinkiewicz integral with homogeneous kernels, J. Math. Anal. Appl., 2000, 245: 471--488.



[8] Lu S. Z., On Marcinkiewicz integral with rough kernels, Front. Math. China, 2008, 3(1): 1--14.



[9] Torchinsky A., Wang S., A note on the Marcinkiewicz integral, Colloq. Math., 1990, 60/61: 235--243.



[10] Hu G., Lp(Rn) boundedness for a class of g-functions and applications, Hokkaido Math. J., 2003, 32: 497--521.



[11] Hu G., Yan D., On the commutator of the Marcinkiewicz integral, J. Math. Anal. Appl., 2003, 283: 351--361.



[12] Ding Y., Lu S., Zhang P., Weighted weak type estimates for commutators of the Marcinkiewicz integrals, Sci. China Ser. A., 2004, 47: 83--95.



[13] Mo H., Lu S., Boudedness of Generalized higher Commutators of Marcinkiewicz integrals, Acta Mathematica Scientia, 2007, 27B(4): 852--866 (in Chinese).



[14] Hu G., Lin H., Yang D., Marcinkiewicz integrals with non-doubling measures, Integral Equations and Operator Theory, 2007, 58: 205--238.



[15] Tolsa X., BMO, H1 and Calderón--Zygmund operators for non doubling measures, Math. Ann., 2001, 319: 89--149.



[16] Tolsa X., Littlewood--Paley theory and the T(1) theorem with non-doubling measures, Adv. Math., 2001, 164: 57--116.



[17] Tolsa X., A proof of weak (1,1) inequality for singular integrals with non doubling measures based on a Calderón--Zygmund decomposition, Publ. Mat., 2001, 45: 163--174.



[18] Tolsa X., The space H1 for nondoubling measures in terms of a grand maximal operator, Trans. Amer. Math. Soc., 2003, 355: 315--348.



[19] Hu G., Meng Y., Yang D., Multilinear commutators of singular integrals with non doubling measures, Integral Equations Operator Theory, 2005, 51: 235--255.



[20] Meng Y., Yang D., Boundedness of commutators with Lipschitz functions in non-homogeneous spaces, Taiwanese J. Math., 2006, 10: 1443--1464.



[21] Li L., Jiang Y. S., Estimates for maximal multilinear commutators on non-homogeneous spaces, J. Math. Anal. Appl., 2009, 355: 243--257.



[22] Tolsa X., Painlevé's problem and the semiadditivity of analytic capacity, Acta Math., 2003, 190: 105--149.



[23] García-Cuerva J., Gatto A. E., Boundedness properties of fractional integral operators associated to non-doubling measures, Studia Math., 2004, 162: 245--261.



[24] García-Cuerva J., Gatto A. E., Lipschitz spaces and Calderón--Zygmund operators associated to non-doubling measures, Publ. Mat., 2005, 49(2): 285--296.



[25] Lu S. Z., Ding Y., Yan D. Y., Singular Integral and Related Topics, Singapore: World Scientific Publishing Company, 2007.

基金

国家自然科学基金(10861010);新疆高校科研计划(XJEDU2008S58);伊犁师范学院科研项目

PDF(518 KB)

227

Accesses

0

Citation

Detail
段落导航
相关文章

/

〈 〉