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

In this paper we describe the Euler semigroup $$\{e^{-t\mathbb {E}^{*}\mathbb {E}}\}_{t>0}$$ { e - t E ∗ 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.

Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators

In this paper we study the homogeneous Wick differential equation associated to the quantum white noise (𝚀𝚆𝙽) Euler operator

Nonclassical convolutions and their uses

A survey of three types of convolutions, depending on arbitrary linear functionals is made. They are convolutions for right inverse operators of the differentiation operator, the Euler operator and the square of the differentiation operator. Three lines of applications of these convolutions are outlined: characterizing their multipliers, the commutants and direct construction of operational calculi.

QWN-EULER OPERATOR AND ASSOCIATED CAUCHY PROBLEM

In this paper the quantum white noise (QWN)-Euler operator [Formula: see text] is defined as the sum [Formula: see text], where [Formula: see text] and NQ stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that [Formula: see text] has an integral representation in terms of the QWN-derivatives [Formula: see text] as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.