[Objective]Homogeneous spaces naturally contain Euclidean spaces Rn,smooth tight Riemann manifolds,and boundaries of Lipschitz regions,etc.It is imperative to establish on homogeneous spaces the Tb theorem that singular integral operators are bounded on weighted Besov spaces and Triebel-Lizorkin spaces.[Method]Plancherel-Pôlya feature characterizations of weighted Besov spaces and weighted Triebel-Lizorkin spaces were established by means of discrete Calderón regeneration formulas and almost orthogonal estimation to ensure that the number of paradigms in the function space was chosen independent of the constant approximation.[Result]Sufficient conditions are obtained for Calderón-Zygmund singular integral operators on homogeneous spaces to be bounded on weighted Besov spaces as well as on Triebel-Lizorkin spaces.[Conclusion]Extending the Calderón-Zygmund theory of singular integrals on Euclidean spaces to a wider range of homogeneous spaces provides a method for determining that singular integral operators are bounded on function spaces.
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