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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 $\bbfR\sp n$, 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 $f(x)= \sum\sb k \psi\sb k* \widetilde\psi\sb k* f(x)$, where $\psi,\widetilde\psi\in {\cal S} (\bbfR\sp n)$ are such that $\sum\sb k\widehat\psi(k\xi)\widehat{\widetilde\psi}(k\xi)= 1$, and $$\text{supp }\widehat\psi\cup\text{supp }\widehat{\widetilde\psi}\subset\{\xi: \textstyle{{1\over 2}}\le \vert\xi\vert\le 2\},\ \vert\widehat\psi(\xi)\vert\ge c,\ \vert\widehat{\widetilde\psi}(\xi)\vert\le c,\text{ on }\{\xi:\textstyle{{3\over 5}}\le \vert\xi\vert\le\textstyle{{5\over 3}}\}.$$ The new formular in the paper is $f= \sum\sb k \widetilde D\sb k M\sb b D\sb k M\sb b$, where $M\sb b$ is the multiplication operator defined by $b(x)$, a given paraaccretive function, and $D\sb k$, $\widetilde D\sb k$ are integral operators with nice kernels $S\sb k(x,y)$, $\widetilde S\sb k(x,y)$.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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