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integral and derivative operators of functional order; fractional integral operator; fractional derivative operator; spaces of homogeneous type; Besov spaces; Triebel-Lizorkin spaces
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.
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