Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type inequalities on homogeneous groups

In this paper we describe the Euler semigroup {e−tE∗E}t>0 on homogeneous Lie groups, which allows us to obtain various types of the Hardy-Sobolev and Gagliardo-Nirenberg type inequalities for the Euler operator E. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and |⋅|-radial weighted Hardy-Sobolev type inequality are established