On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type

We introduce Sobolev spaces $L_{α}^{p}$ for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in $L^{p}$ with fractional derivative of order α, $D^{α}f$, as introduced in [2], in $L^{p}$. We show that for small α, $L_{α}^{p}$ coincides with the continuous version of the Triebel-Lizorkin space $F_p^{α,2}$ as defined by Y. S. Han and E. T. Sawyer in [4]. To prove this result we give a more general definition of ε-families of operators on spaces of homogeneous type, in which the identity operator is replaced by an invertible operator. Then we show that the family $t^{α} D^{α} q(x,y,t)$ is an ε-family of operators in this new sense, where $q(x,y,t) = t ∂/∂t s(x,y,t)$, and s(x,y,t) is a Coifman type approximation to the identity