Some inequalities for homogeneous Bn-potential type integrals on HpΔν 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 HpΔν to HqΔν , for 1q= 1p− α Q, provided 0 < α < 1/ 2 , and α < β ≤ 1 and Q Q+β < p ≤ Q Q+α. By applying atomic-molecular decomposition of HpΔν 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 Bn-Riesz potential operator on HpΔ ν Hardy space.