Euler semigroup, Hardy–Sobolev and Gagliardo–Nirenberg type inequalities on homogeneous groups
Euler semigroup, Hardy–Sobolev and Gagliardo–Nirenberg type inequalities on homogeneous groups
Abstract In this paper we describe the Euler semigroup $$\{e^{-t\mathbb {E}^{*}\mathbb {E}}\}_{t>0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mo>{</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>t</mml:mi><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mrow /><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mi>E</mml:mi></mml:mrow></mml:msup><mml:mo>}</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> on homogeneous Lie groups, which allows us to obtain various types of the Hardy–Sobolev and Gagliardo–Nirenberg type inequalities for the Euler operator $$\mathbb {E}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi></mml:math> . Moreover, the sharp remainder terms of the Sobolev …