## Calderón-type reproducing formula and the $$Tb$$ theorem.(English)Zbl 0797.42009

Using the Calderón-Zygmund operator theory, the paper presents a Calderón-type reproducing formula associated with a paraaccretive function, which is not of convolution type and hence can be applied in the setting other than $$\mathbb{R}^ n$$, say Coifman-Weiss’ spaces of homogeneous type. As an application of the Calderón-type reproducing formula obtained in the paper, a kind of $$T(b)$$ theorem for some new classes of Besov spaces and of Triebel-Lizorkin spaces, and for some singular integral operators having strong smoothness condition is given. The original Calderón reproducing formula the paper means is $$f(x)= \sum_ k \psi_ k* \widetilde\psi_ k* f(x)$$, where $$\psi,\widetilde\psi\in {\mathcal S} (\mathbb{R}^ n)$$ are such that $$\sum_ k\widehat\psi(k\xi)\widehat{\widetilde\psi}(k\xi)= 1$$, and $\text{supp }\widehat\psi\cup\text{supp }\widehat{\widetilde\psi}\subset\{\xi: \textstyle{{1\over 2}}\leq |\xi|\leq 2\},\;|\widehat\psi(\xi)|\geq c,\;|\widehat{\widetilde\psi}(\xi)|\leq c,\text{ on }\{\xi:\textstyle{{3\over 5}}\leq |\xi|\leq\textstyle{{5\over 3}}\}.$ The new formular in the paper is $$f= \sum_ k \widetilde D_ k M_ b D_ k M_ b$$, where $$M_ b$$ is the multiplication operator defined by $$b(x)$$, a given paraaccretive function, and $$D_ k$$, $$\widetilde D_ k$$ are integral operators with nice kernels $$S_ k(x,y)$$, $$\widetilde S_ k(x,y)$$.

### MSC:

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