The three-line theorem on the octonions is obtained, which generalizes the result of J. Peetre and P. Sj?lin from the associative Clifford algebra to non-associative octonion algebra.
Let D be the unit disk in the complex plane with the weighted measure dμβ(z) = (β+1)--π(1-|z|2)βdm(z)(β>-1). Then L2(D,dμβ(z)) = ∞k=0(Aβk Aβk) is the orthogonal direct sum decomposition.In this paper, we define the Hankel and Toeplitz type operators, and study the boundedness, compactness and Sp-criteria for them.
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