By introducing a kind of maximal operator of the fractional order associated with the mean Luxemburg norm and using the technique of the sharp function, the weak type Llog Lestimates for the commutators of the fractional integral operator and the related maximal operator are established.

In this paper we study the bounded property of the Rice means for the cigcnfunction expansio~ foe elliptic operators on the Hardy spaces. Our result generalizes the classical result due to Sjolin and Stein-Taibleson-Weiss on the Boehner-Riesz means.

本文内容为两部分。第一部分中，研究了多重福里哀级数Rmz球形平均（临界指数）的几乎处处收敛性，将古典的关于福里哀级数几乎处处收敛的salem定理在多维情形中的拓广问题给予解决。第二部分中，研究了多重福里哀级数Ralem球形平均（临界指数）的一致收敛，将古典的关于一致收敛的salem-CTeчkиH定理推广到多维情形中去，并且改进了иъ.гOJIyбOB新近的相应结果。这两方面结果的获得均与本文所提出的球形积分这一新概念有着紧密的关系。

In a recent paper [6] the last two authors of this article introduced a class of function spaces associated with the one-dimensional torus T. They showed that the Fourier series of each function f that belongs to one of these spaces converges to f (x) a.e. Moreover, they indicated that the motion of entropy is closely related to the study of these spaces. These features are not restricted to the one-dimenslonal case. In this paper, in fact, we show how these ideas can be used to study the convergence of Bochner-Riesz means of multiple Fourier series at the "critical index".