### On the Composition of the Integral and Derivative Operators of Functional Order

#### Abstract

The Integral, I_phi, and Derivative, D_phi, operators of order phi, with phi a function of positive lower type and upper type lower 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 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 Calderón-Zygmund operator bounded on the generalized Besov, ˙B^{psi,q}_p 1 leq p, q < infty, and Triebel-Lizorkin spaces, ˙F^{psi,q}_p , 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.

In this work we show that the composition T_phi = D_phi 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 Calderón-Zygmund operator bounded on the generalized Besov, ˙B^{psi,q}_p 1 leq p, q < infty, and Triebel-Lizorkin spaces, ˙F^{psi,q}_p , 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.