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The aim of this paper is to show that the integral and derivative operators defined by local regularities are homeomorphisms for generalized Besov and Triebel-Lizorkin spaces with local regularities. The underlying geometry is that of homogeneous type spaces and the functions defining local regularities belong to a larger class of growth functions than the potentials t, related to classical fractional integral and derivative operators and Besov and 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 $1\le p,q<\infty$, 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,...

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 $\phi \left(t\right)=t^{\alpha }$, 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...