Fractional integral operator on spaces of homogeneous type (Q1764391)

A space of homogeneous type \((X,d,\mu)\) is a set \(X\) with a quasi-metric \(d\) and a positive measure \(\mu\) such that \(\mu(B(x,r))<\infty\) for all \(x\in X\) and all \(r>0\). Here \[ B(x,r)=\{y\in X:\, d(x,y)<r\}. \] The measure \(\mu\) satisfies the doubling property: there exists a universal constant \(A>0\) such that \[ \mu(B(x,2r))\leq A \mu(B(x,r)) \] for \(x\in X\) and all \(r>0\). For \(f\in L_{loc}(X)\), one defines the fractional integral operator as follows: \[ I_\alpha(f)(x)=\int_X \frac {1}{\rho(x,y)^{1-\alpha}}f(y)dy,\quad 0<\alpha<1, \] where \[ \rho(x,y)=\inf\{\mu(B):\, B\,\, {\text{is a ball containing \(x\) and \(y\)}}\}. \] In this paper, the authors prove that if \(\max \Big\{\frac{1}{1+\varepsilon}, \frac{1}{1+\varepsilon+\alpha} \Big\}<p,q<\infty\), then \[ \| I_\alpha(f)\| _{\dot F_p^{\alpha,q}} \leq C\| f\| _{\dot F_p^{0,q}}, \] where \(0<\alpha<\varepsilon\). Here \(\dot F_p^{\alpha,q}\) is a Triebel-Lizorkin type space.