Fractional integrals on spaces of homogeneous type

Abstract

We study fractional integrals on spaces of homogeneous type defined byI _αf(x)=∫_Xf(y)|B(x,d(x,y))|^ga−1dμ(y), 0<α<1. If$1< p\frac{1}{\alpha },\frac{1}{q} = \frac{1}{p} - \alpha$, we show that I_αf is of strong type (p,q) and is of weak type ($\left( {1,\frac{1}{{1 - \alpha }}} \right)$). We also consider the necessary and sufficient conditions on two weights for which I_αf is of weak type (p,q) with respect to (w,v).