Abstract:The Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi $, with $\phi $ a function of positive lower type and upper type less than $1$, were defined in [HV2] in the setting of spaces of homogeneous-type. These definitions generalize those of the fractional integral and derivative operators of order $\alpha $, where $\phi (t)=t^{\alpha }$, given in [GSV]. \par In this work we show that the composition $T_{\phi }= D_{\phi }\circ I_{\phi }$ is a singular integral operator. This result in addition with the results obtained in [HV2] of boundedness of $I_{\phi }$ and $D_{\phi }$ or the $T1$-theorems proved in [HV1] yield the fact that $T_{\phi }$ is a Calder\'on-Zygmund operator bounded on the generalized Besov, $\dot {B}_{p}^{\psi ,q}$, $1 \le p,q < \infty $, and Triebel-Lizorkin spaces, $\dot {F}_{p}^{\psi ,q}$, $1< p, q < \infty $, of order $\psi = \psi _1/\psi _2$, where $\psi _1$ and $\psi _2$ are two quasi-increasing functions of adequate upper types $s_1$ and $s_2$, respectively.
Keywords: fractional integral operators, fractional derivative operators, spaces of homogeneous type, Besov spaces, Triebel-Lizorkin spaces
AMS Subject Classification: 26A33