玛欣凯维奇函数与泊松核的一个新微分性质

汪成咏, 渠刚荣

数学学报 ›› 2016, Vol. 59 ›› Issue (1) : 1-10.

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PDF(454 KB)
数学学报 ›› 2016, Vol. 59 ›› Issue (1) : 1-10. DOI: 10.12386/A2016sxxb0001
论文

玛欣凯维奇函数与泊松核的一个新微分性质

    汪成咏, 渠刚荣
作者信息 +

On the Functions of Marcinkiewicz and Some New Differential Properties of Poisson Kernel

    Cheng Yong WANG, Gang Rong QU
Author information +
文章历史 +

摘要

将Stein [On the functions of Littlewood-Paley, Lusin,and Marcinkiewicz,Trans. Amer. Math. Soc.,1958, 88:430-466]中的玛欣凯维奇函数的逆向不等式推广到一般情形.主要结果是对于n-维欧几里得空间k-阶球面调和函数空间的任意一基底,得到玛欣凯维奇函数的一般性的逆向不等式, 即存在不依赖于函数f正常数 Cp, 使得 ||f||p CpjN=1 ||μj(f)||p,其中{μj(f)}jN =1 是f的由这些球面调和函数生成的玛欣凯维奇函数. 此外, 对于任意的n-变元的k-阶调和多项式 Q(x) 以及泊松核 Pt(x), 有Q(D)Pt(x) =Cn,k((tQ(x))/((|x|2+t2)/(n+2k+1/2)).

Abstract

We generalize the inverse inequality of Marcinkiewicz function in Stein [On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc., 1958, 88: 430-466] to general cases. The main result of this paper is that, for any basis of spherical harmonic functions of order k in n-dimensional Euclidean space, we obtain the general inverse inequalities for Marcinkiewicz function, i.e. there exist a constant Cp does not depend on f such that ||f||p CpjN=1 ||μj(f)||p, where {μj(f)}jN =1 are the Marcinkiewicz functions of f generated by these spherical harmonic functions. Moreover, for any n-variable homogeneous harmonic polynomial Q(x) of order k, and the Poisson kernel Pt(x), we have Q(D)Pt(x) =Cn,k((tQ(x))/((|x|2+t2)/(n+2k+1/2)).

关键词

玛欣凯维奇函数 / 泊松核 / 向量值奇异积分算子

Key words

Marcinkiewicz function / Poisson kernel / vector-valued singular integral operator

引用本文

导出引用
汪成咏, 渠刚荣. 玛欣凯维奇函数与泊松核的一个新微分性质. 数学学报, 2016, 59(1): 1-10 https://doi.org/10.12386/A2016sxxb0001
Cheng Yong WANG, Gang Rong QU. On the Functions of Marcinkiewicz and Some New Differential Properties of Poisson Kernel. Acta Mathematica Sinica, Chinese Series, 2016, 59(1): 1-10 https://doi.org/10.12386/A2016sxxb0001

参考文献

[1] Calderon A. P., On the theorem of Marcinkiewicz and Zygmund, Trans. Amer. Math. Soc., 1950, 8: 55-61.
[2] Calderon A. P., Zygmund A., On the existence of certaun singular integrals, Acta Math., 1952, 88: 85-139.
[3] Grafakos L., Classical and Modern Fourier Analysis, China Machine Press, Beijing, 2004.
[4] Müller C., Spherical Harmonics, Lecture Notes in Math. 17, Springer-Verlag, Berlin, 1966.
[5] Stein E. M., On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc., 1958, 88: 430-466.
[6] Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, New Jersey, 1970.
[7] Stein E. M., Weiss G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, New Jersey, 1968.
[8] Zygmund A., On certain integrals, Trans. Amer. Math. Soc., 1944, 55: 170-204.

基金

国家自然科学基金资助项目(61271012)

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