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

Abstract

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.