## $$T1$$ 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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