Fractional Hardy-type inequalities on homogeneous Lie groups in the case Q<sp

In this paper, we obtain a fractional Hardy inequality in the case Q<sp on homogeneous Lie groups, and as an application we show the corresponding uncertainty principle. Also, we show a fractional Hardy–Sobolev-type inequality on homogeneous Lie groups. In addition, we prove fractional logarithmic Hardy–Sobolev and fractional Nash-type inequalities on homogeneous Lie groups. We note that the case Q>sp was extensively studied in the literature, while here we are dealing with the complementary range Q<sp.