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\(T1\) theorems for Besov and Triebel-Lizorkin spaces. (English) Zbl 0779.42010

The authors give \(T1\) theorems for singular integrals in the case of Besov and Triebel-Lizorkin spaces under weaker conditions on the kernels. These are refinements of Lemarié’s result on Besov spaces \(\dot B_ q^{\alpha,p}\) and Frazier, Jarberth, Han and Weiss’ one on Triebel- Lizorkin spaces \(\dot F_ q^{\alpha,p}\) \((0<\alpha<1)\). The proof method is a direct one, decomposing the operator \(T\) as \(T=\int_ 0^ \infty\int_ 0^ \infty Q_ s^ 2TQ_ t^ 2{dt\over t}{ds\over s}\) and estimating the kernels of \(Q_ sTQ_ t\). Here \(Q_ t\) is a convolution operator defined by radial smooth kernel \(t^{-n}\psi(x/t)\).
Reviewer: K.Yabuta (Nara)

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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