Regional Function Spaces And T (1) Theorem

The decomposition of function spaces plays an important role in harmonic analysis.The aim of the decomposition is to divide the complex function spaces into the linear combination of sample functions.Because of such decomposition,one can further understand the structure of function spaces. Just as the case of Hardy space,the atomic decomposition and the molecular decomposition are obtained successively. The decomposition and the characterization of many similar function spaces is studied in the same way.Generally speaking.the molecular decomposition is after the atomic de-composition.Because the atoms of the function spaces is of compact supports.it is necessary to study the moleculars which have no compact supports sometimes.When we get the atomic and molecular decompositions,the boundedness of Calderon-Zygmund operators is obtained easily.Besov spaces and Triebel-Lizorkin spaces are two kinds of spaces which are studied frequently in the field.Th.ere are two reasons.On one hand,when the parameters are of special values.one can get some classical spaces,for example Hardy space, Sobolev spaces,Lipschitz spaces, and so on.They are all special cases of Triebel-Lizorkin spaces.If an assertion is vaild on Besov spaces and Triebel-Lizorkin spaces,it is also vaild on other spaces.On the other hand,the study of partial differential equations often depends on the estimate of many operators on Besov spaces and Triebel-Lizorkin spaces.In a word,it is necessary to study the decomposition of Besov spaces and Triebel-Lizorkin spaces .There are four parts in this article.In the first part,the author defined the molecular of Besov spaces on domins and discussed the molecular decomposition of these Besov spaces.In the second part,the author proved a T(l) type theorem.This indicates that Calderon-Zygmund operators are bounded in the interior of these Besov spaces.In the third part.the author defined some Triebel-Lizorkin spaces on domains and proved that there exist the atomic and molecular decomposition on these Triebel-Lizorkin spaces.Accordingly.the author got the Calderon-Zygmund operators are also bounded in the interior of these Triebel-Lizorkin spaces.In the last part,the author discussed the interpolation of these Besov spaces and Triebel-Lizorkin spaces .