Some inequalities for homogeneous $B_n$-potential type integrals on $H^{p}_{\Delta_{\nu}}$ Hardy spaces

Abstract

We prove the norm inequalities for potential operators and fractional integrals related to generalized shift operator defined on spaces of homogeneous type. We show that these operators are bounded from $H^{p}_{\Delta_{\nu}}$ to $H^{q}_{\Delta_{\nu}}$, for $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{Q}$, provided $0<\alpha<\frac{1}{2}$, and $\alpha<\beta\leq 1$ and $\frac{Q}{Q+\beta}<p\leq\frac{Q}{Q+\alpha}$. By applying atomic-molecular decomposition of $H^{p}_{\Delta_{\nu}}$ Hardy space, we obtain the boundedness of homogeneous fractional type integrals which extends the Stein-Weiss and Taibleson-Weiss's results for the boundedness of the $B_n$-Riesz potential operator on $H^{p}_{\Delta_{\nu}}$ Hardy space.