In this paper we extend the definition of the Besov and the Triebel-Lizorkin spaces in the context of spaces of homo g eneous-type g iven by Han and Sawyer in [HS]. We consider, as a control of the 'local re g ularity ', fu nctions 'IjJ(t) more g en­ eral than the potentials tet used in their case. We also state Tl-type theorems in these spaces. Our approach yields some new results for kernels satisfyin g inte g ral re g ularity conditions. 1 Introduction In the context of spaces of homogeneous

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... type, G. David, J.L. Journee and S. Semmes , in [DJSj , showed how to construct an appropiate family of operators {DdkeZl whose kernels satisfy certain size, smoothness and moment conditions and the nondegeneracy condition :Lkez D k = I on L 2. In [HSj , Han and E. Sawyer introduced a class of distributions on spaces of h,omogeneous type and then established a Calderon-type reproducing formula associated to that family of operators for this class. This formula allowed �hem to define the Besov spaces il;,q, 1 :::; p, q < 00 and the Triebel-Lizorkin spaces F;,q, 1· < p, q < 00 and to show that those spaces are independent of tlw family of operators {Dd ke Zl involved in their definition and, in this way, to develop. Littlewood-Paley characterizations of them. By considering more general functions 'lj; (t) than the potential functions tet as a measure of the local regularity, in this paper we define the Besov spaces ilt,q, 1 :::; p, q < 00 and Triebel-Lizorkin spaces i'/ ,q, 1 < p, q < 00 on spaces of homogeneous­ type. We also state Tl-theQrems of boundedness of generalized Calderon-Zygmund operators on these spaces for kernels satisfying integral conditions of size and smooth­ ness. In the context of IRn , Y. Han and S. Hofmann in [HHj prove Tl-theorems on the Besov spaces il; ,q, 1 :::; p, q :::; 00 and the Triebel-LizoIkin spaces i'; ,q(w), 1 < p, q < ·Supported by UNL and IMAL-CONICET tSupported by UNL and IMAL-CONICET