Abstract
In this paper we describe the Euler semigroup {e^{-tmathbb {E}^{*}mathbb {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}. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and |cdot |-radial weighted Hardy–Sobolev type inequality are established.
Highlights
In this paper we continue the research from [20] devoted to properties of Euler operators on homogeneous groups, their consequences, and related analysis
We show analogues of (1.3) and (1.4), and calculate the semigroup e−t ∗ on homogeneous (Lie) groups, where is the Euler operator
We introduce the operator semigroup {e−t ∗ }t>0 associated with the Euler operator on homogeneous groups
Summary
In this paper we continue the research from [20] devoted to properties of Euler operators on homogeneous groups, their consequences, and related analysis. This continues the research direction initiated in [16] devoted to Hardy and other functional inequalities in the setting of Folland and Stein’s [9] homogeneous groups.
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