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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 \(\mathbb{R}^ n\), 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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