An all-round answer is given to the following question: 'Find a necessary and sufficient size condition for an integrable compactly supported function on Rn with mean value zero to be in the Hardy space H1.' Stefanov answers it only for n = 1. Equivalent answer is also given for n = 1.
WANG Silei Department of Mathematics, Zhejiang University, Hangzhou 310028, China
The singular integral operatorTα,βf（x）=p.v.∫R^n[e^i｜y｜^-βΩ（y’）]/[｜y｜^n＋α]f（x-y）dy,defined for all test functions f is studied, where Ω（y＇） is a distribution on the unit sphere S^n-1 satisfying certain cancellation condition. It is proved that Tα,β is a bounded operator from the Triebel-Lizorkin space Fp^s,q to the Triebel-Lizorkin space Fp^s＋γ,q, provided that Ω（y＇） is a distribution in the Hardy space H^r（S^n-1） with r = （n - 1）/（n - 1 ＋ γ）.
Ye Xiaofeng Dept.of Math.,Zhejiang Univ.,Hangzhou 310027,China
