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].