玛欣凯维奇函数与泊松核的一个新微分性质

将SteinOn 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C_p,使得||f||_p≤C_pΣ_(j=1)~N=1||μ_j(f)||_p,其中{μ_j(f)}_(j=1)~N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P_t(x),有Q(D)P_t(x)=C_n k(tQ(x))/((|x|)~2+t~2~(n+2k+1)/2).

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

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 C_p does not depend on f such that ||f||_p ≤ C_p∑_j^N=1 ||μj(f)||_p, where {μj(f)}_j^N =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 P_t(x), we have Q(D)P_t(x) =C_n,k((tQ(x))/((|x|²+t²)/(n+2k+1/2)).