On the composition of the integral and derivative operators of functional order

summary: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]. 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