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".

In this paper, we investigate the strong summability of multiple Fourier series by the Bochner-Riesz spherical means at criticnl index. We prove that the localization theorem of the strosng summability holds for any posityve degree, that is, if f(x) EL(Qn) and f(x)=0(1x-x01), then lim R-∞ 1/R R0|S #-1/2(f; x0)|qdu=0 (for any q>0). We also give certain results on the strong summability which improves Boehner-Ohand sekharan's theorem.

Let f Е Hp (R) or Hp (T) (0<p<1), again let Srj(..) and σ δRf(x) (δ=1/P-1) bethe Riesz means of f at the critical index on T anci R respectively. In this paper, by usr ofgrand maximal functions and the atom decomposition method, the fundamental estimares on approximation tor SδR f and σδRf on the respective real Hardy spaces Hp(T) ant Hp (R) ar estanloshed: ‖f-Sσδf‖H(P T)≤Cpω(f; 1/R) Hp(T), ‖f-Sσδf‖H(PR)≤Cpω(f; 1/R) Hp(R), where H (P, ∞) are Lorentz-Hardy spaces and ω(f; h)HP=sup ‖f(·+u)-f(·)‖HP.