Аннотация
Abstract In this paper we describe the Euler semigroup $$\{e^{-t\mathbb {E}^{*}\mathbb {E}}\}_{t>0}$$ {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 $$\mathbb {E}$$ E . Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $$|\cdot |$$ |·| -radial weighted Hardy–Sobolev type inequality are established.