玛欣凯维奇函数与泊松核的一个新微分性质
On the Functions of Marcinkiewicz and Some New Differential Properties of Poisson Kernel
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 ≤ Cp∑jN=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)).
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 ≤ Cp∑jN=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)).
玛欣凯维奇函数 / 泊松核 / 向量值奇异积分算子 {{custom_keyword}} /
Marcinkiewicz function / Poisson kernel / vector-valued singular integral operator {{custom_keyword}} /
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国家自然科学基金资助项目(61271012)
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