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.

A new kind of funcion space generated by somooth blocks is introduced. The new spaces are used to investigate the relation between the smoothness imposed on blocks nad the rate of convergence of the Bochner-Riesz eans at the critical index.

Let 0<p≤1<q<∞and α=n (1/p-1-q). We introduce some new Hardy spaces HKα,q p (Rn) which are the local versions of Hp(Rn) spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizatins in M. Frazier and B. Jawerth's sense. We also prove an interpolation theorem for operators on HKα,qp(Rn) and discuss the HKα,qp (Rn)-boundedness of Calderon-Zygmund operators. similar results can also be obtined for the non-homogeneous Hardy spaces HKα,qp (Rn).