Abstract:In the setting of spaces of homogeneous-type, we define the Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi $, where $\phi $ is a function of positive lower type and upper type less than $1$, and 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 $ in each case. We also prove that $I_{\phi }$ and $D_{\phi }$ are bounded from the generalized Besov $\dot {B}_{p}^{\psi , q}$, with $1 \leq p, q < \infty $, and Triebel-Lizorkin spaces $\dot {F}_{p}^{\psi , q}$, 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.
Keywords: integral and derivative operators of functional order, fractional integral operator, fractional derivative operator, spaces of homogeneous type, Besov spaces, Triebel-Lizorkin spaces
AMS Subject Classification: 26A33