On fractional differentiation and integration on spaces of homogeneous type

Abstract

In this paper we define derivatives of fractional order on spaces of homogeneous type by generalizing a classical formula for the fractional powers of the Laplacean [S1], [S2], [SZ] and introducing suitable quasidistances related to an approximation of the identity. We define integration of fractional order as in [GV] but using quasidistances related to the approximation of the identity mentioned before.We show that these operators act on Lipschitz spaces as in the classical cases. We prove that the composition T_\alpha of a fractional integral I_\alpha and a fractional derivative D_\alpha of the same order and its transpose (a fractional derivative composed with a fractional integral of the same order) are Calderón-Zygmund operators. We also prove that for small order \alpha a, T_\alpha is an invertible operator in L^2 . In order to prove that T_\alpha is invertible we obtain Nahmod type representations for I_\alpha and D_\alpha and then we follow the method of her thesis [N1], [N2].