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

We introduce Sobolev spaces ${\displaystyle L_{\alpha }^p}$ for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in ${\displaystyle L^p}$ with fractional derivative of order α, ${\displaystyle D^{\alpha }f}$, as introduced in [2], in ${\displaystyle L^p}$. We show that for small α, ${\displaystyle L_{\alpha }^p}$ coincides with the continuous version of the Triebel-Lizorkin space ${\displaystyle F_p^{\alpha ,2}}$ 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 ${\displaystyle t^{\alpha }{D}^{\alpha }q(x,y,t)}$ is an ε-family of operators in this new sense, where ${\displaystyle q(x,y,t)=t\frac{\partial }{\partial }ts(x,y,t)}$, and s(x,y,t) is a Coifman type approximation to the identity.