### Integral and Derivative Operators of Functional Order on Generalized Besov and Triebel-Lizorkin Spaces in the Setting of Spaces of Homogeneous TypeE

#### Abstract

In this work we define the Integral, I_phi, and Derivative, D_phi, operators of order phi, in the setting of spaces of homogeneous-type, where phi is a function of positive lower type and upper type lower than 1.

We show that I_phi and D_phi are bounded from Lipschitz spaces Lambda^{xi} to Lambda^{xi phi} and Lambda^{xi/phi} respectively, with suitable restrictions on the quasi-increasing function xi for each case.

We also prove that I_phi and D_phi are bounded from the generalized Besov ˙B^{psi ,q}_p , with 1 leq p, q < infty, and Triebel-Lizorkin spaces ˙F^{psi ,q}_p , with 1 < p, q < infty, of order psi to those of order phi psi and psi /phi respectively, where psi is the quotient of two quasi-increasing functions of adequate upper types.

We show that I_phi and D_phi are bounded from Lipschitz spaces Lambda^{xi} to Lambda^{xi phi} and Lambda^{xi/phi} respectively, with suitable restrictions on the quasi-increasing function xi for each case.

We also prove that I_phi and D_phi are bounded from the generalized Besov ˙B^{psi ,q}_p , with 1 leq p, q < infty, and Triebel-Lizorkin spaces ˙F^{psi ,q}_p , with 1 < p, q < infty, of order psi to those of order phi psi and psi /phi respectively, where psi is the quotient of two quasi-increasing functions of adequate upper types.