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

In this paper we describe the Euler semigroup {e-tE∗E}t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e^{-t\mathbb {E}^{*}\mathbb {E}}\}_{t>0}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}$$\end{document}. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and |·|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\cdot |$$\end{document}-radial weighted Hardy–Sobolev type inequality are established.


Introduction
In this paper we continue the research from [20] devoted to properties of Euler operators on homogeneous groups, their consequences, and related analysis. In turn, 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.
Communicated by Jerome A. Goldstein.

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Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type… Recall the following Sobolev type (improved Hardy) inequality: Let 1 < p < ∞ . Then we have for all f ∈ C ∞ 0 (ℝ n ) , where ∇ is the standard gradient in ℝ n . In [15,Proposition 1.4] the authors showed that the Sobolev type inequality (1.1) is equivalent to the following Hardy inequality when p = 2 and n ≥ 3: for all f , g ∈ C ∞ 0 (ℝ n �{0}) , where ‖x‖ = � x 2 1 + ⋯ + x 2 n . We note that the analysis of the remainder terms in such inequalities has a long history, for example, we refer to [4] for Sobolev inequalities, to [6,Section 4] and [5] for Hardy inequalities and for many others, and a more recent literature review to [10].
In this paper we are interested, among other things, in obtaining (1.1) with remainder terms, that is, for example, in [1,Corollary 4.4] the authors obtained the following: Let n ≥ 3 and 0 ≤ < n 2 ∕4 . Then there exists a positive constant C such that holds for all f ∈ C ∞ 0 (ℝ n ) , where F(x) ∶= M(f ) = 1 n−1 ∫ n−1 f (r, y)dy is the integral mean of f over the unit sphere n−1 , and 2 * = 2n∕(n − 2) , r = |x| . They derived (1.3) from the following Gagliardo-Nirenberg inequality [1,Theorem 4.1] using the explicit representation of the semigroup e −t(x⋅∇) * (x⋅∇) : Let 1 ≤ p < q < ∞ and let −i( ∕ r)f ∈ L p (ℝ × n−1 ) and f ∈ B p∕(p−q) (ℝ n ) . Then there exists a positive constant C = C(p, q) such that where R ∶= d d|x| is the radial derivative, and ‖ ⋅ ‖ B is the Besov type norm associated with the semigroup.
We show analogues of (1.3) and (1.4), and calculate the semigroup e −t * on homogeneous (Lie) groups, where is the Euler operator (see Sect. 2 for more information). We can also obtain (1.3) with sharp constant without using (1.4) and semigroup e −t * . We will also explain that the obtained homogeneous group results are not only analogues of the known Euclidean results, but also they give new inequalities even in Abelian cases with arbitrary quasi-norms (see Remark 4.13). Since we do not have sub-Laplacian and "horizontal" gradients on general homogeneous groups, we will work with radial derivative ( R ∶= d dr ) and Euler ( ∶= |x|R ) operators. Some other discussions on this idea will be provided in the final section. We refer to [19] for horizontal versions of these inequalities.
For the convenience of the reader we briefly summarise the obtained results. Let be a homogeneous group of homogeneous dimension Q and let us fix any homogeneous quasi-norm | ⋅ | . Then we have the following results: -Let x = ry with r = |x| and |y| = 1 . Then the semigroup e −t * is given by where = |x|R is the Euler operator and R ∶= d d|x| is the radial derivative. -Let 1 ≤ p < q < ∞ and let f be such that Rf ∈ L p (ℝ × ) and f ∈ B p∕(p−q) (ℝ × ) . Then there exists a positive constant C = C(p, q) such that where R ∶= d d|x| is the radial derivative, is the unit sphere in and ‖ ⋅ ‖ B is the Besov type norm defined by (4.1).
all p > 1 and all f ∈ C 1 0 ( ) we have In the Abelian case with the standard Euclidean distance ‖x‖ = � x 2 1 + … + x 2 n it implies the "critical" Hardy inequality where ∇ is the standard gradient on ℝ n .
In the Abelian case with the standard Euclidean distance ‖x‖ = � x 2 1 + … + x 2 n it implies the "improved" Heisenberg-Pauli-Weyl uncertainty principle where ∇ is the standard gradient on ℝ n .
Indeed the same inequality with the constant 2 is known as the Heisenberg-Pauli-Weyl uncertainty principle (see, e.g. [17, Remark 2.10]), that is, for all f ∈ C 1 0 (ℝ n ). Obviously, since 2 n−2 ≥ 2 n , n ≥ 3, inequality (1.10) is an improved version of (1.11). Note that equality case in (1.10) holds for the func- The organisation of the paper is as follows. In Sect. 2 we briefly recall the necessary concepts of homogeneous Lie groups and fix the notation. The operator semigroup {e −t * } t>0 is determined in Sect. 3. In Sect. 4 we establish Gagliardo-Nirenberg type inequalities, and obtain sharp remainder terms of the Sobolev type inequality. The maximal Hardy inequality is established in Sect. 5. In Sect. 6 we give some further discussions on (critical) Hardy-Sobolev type inequalities on homogeneous groups.

Preliminaries
In this section we briefly recall the necessary notions and fix the notation for homogeneous groups. We refer to [8] for a detailed discussion of the appearing objects. Let us recall the notion of a family of dilations of a Lie algebra , that is, a family of linear mappings of the following form where A is a diagonalisable linear operator on with positive eigenvalues. We also recall that D is a morphism of if it is a linear mapping from to itself satisfying the property where [X, Y] ∶= XY − YX is the Lie bracket. Then, a homogeneous group is a connected simply connected Lie group whose Lie algebra is equipped with a morphism family of dilations. It induces the dilation structure on which we denote by D x or just by x.
Let dx be the Haar measure on and let |S| denote the volume of a measurable subset S of . The homogeneous dimension of is defined by The quasi-ball centred at x ∈ with radius R > 0 can be defined by The polar decomposition on homogeneous groups can be formulated as follows: there is a (unique) positive Borel measure on the unit sphere such that holds for all f ∈ L 1 ( ).
We refer to Folland and Stein [9] for more information (see also [8, Section 3.1.7] for a detailed discussion).
If we fix a basis {X 1 , … , X n } of such that we have for each k, then the matrix A can be taken to be The decomposition of exp −1 (x) in defines the vector by the formula On the other hand, we also have the following equality Using homogeneity and denoting x = ry, y ∈ , we obtain By a direct calculation, we get which implies the following equality for each homogeneous quasi-norm |x| on a homogeneous group , where we have denoted e(x) = e(ry) = (r 1 e 1 (y), … , r n e n (y)).
We will also use the Euler operator Let us recall the following property of the Euler operator . Now we give useful inequality and identity for Euler operator.

Remark 2.3
In most cases, this lemma allows that instead of proving an inequality for non-radial functions, it is enough to prove it for radial functions (see e.g. in the proof of Theorem 4.12 and [21, in the proof of Theorem 3.1]).
Proof Using representation (2.8), we obtain which is (2.9). Now we prove (2.10). First let us prove the following by induction for any k ∈ ℕ . So, we need to check this for k = 1 . Using Hölder's inequality, we get Now assuming that holds for ∈ ℕ , we prove Then, using (2.14) we calculate

Euler semigroup e −t *
In this section we introduce the operator semigroup {e −t * } t>0 associated with the Euler operator on homogeneous groups. First, let us prepare some preliminary results. We define the map , We note that the map M = F•F is the Mellin transformation, where F is the Fourier transform on ℝ . The map M has an explicit representation using the group structure of ℝ + under multiplication: it is the Fourier transform on this group. Now we are ready to give the representation of the operator semigroup {e −t * } t>0 on homogeneous groups. Theorem 3.1 Let be a homogeneous group of homogeneous dimension Q. Let x = ry and r = |x| with y ∈ . Then the semigroup e −t * is given by Now using (3.9) we get from above that which implies (3.11) after setting t = 0. Now we prove that To obtain (3.13), we write On the other hand, by iteration we have from (3.11) that Putting this in (3.14), we obtain Thus, we have obtained (3.13). Then, it implies that s −Q∕2 f (sy)s Q−1 ds.
. Since we have e −t * = e −tQ 2 ∕4 e −tA 2 by (3.6), we obtain from above that yielding (3.10). ◻ Now let us give the following representation for e −tA 2 , which is useful to obtain Gagliardo-Nirenberg type inequalities (see Sect. 4):

Hardy-Sobolev and Gagliardo-Nirenberg type inequalities
In this section we establish a class of the Hardy-Sobolev and Gagliardo-Nirenberg type inequalities on homogeneous groups. Moreover, sharp remainder terms of the Sobolev type inequality are also obtained. We define the Besov type space B (ℝ × ) as the space of all tempered distributions f on ℝ × with the norm We will also use the one-dimensional case of the following result: Now we state the Gagliardo-Nirenberg type inequalities:

Theorem 4.2 Let be a homogeneous group of homogeneous dimension Q. Let 1 ≤ p < q < ∞ and let f be such that Rf ∈ L p (ℝ × ) and f ∈ B p∕(p−q) (ℝ × ) . Then there exists a positive constant C = C(p, q) such that
Proof Using Theorem 4.1 with n = 1 and Corollary 3.2, we obtain for any y ∈ , in view of (4.1). One obtains (4.3) after integrating the above inequality with respect to y over . (4.10)

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Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type… Let us give the proof of Corollary 4.6.

and g ∶= M(h) . Then there exists a positive constant C such that
By substituting (4.16) and h = f ∕|x| in (4.17), we obtain

Corollary 4.11 Let be a homogeneous group of homogeneous dimension Q. Let
Then there exists a positive constant C such that where Now we prove (4.12) with sharp constant, which can be viewed as an analogue of Stubbe's inequality [23] on homogeneous groups.

Theorem 4.12
Let be a homogeneous group of homogeneous dimension Q > 2 . Let | ⋅ | be a homogeneous quasi-norm. Then we have for all f ∈ C ∞ 0 ( ) and 0 ≤ < Q 2 4 , the inequality with sharp constant, where and Remark 4. 13 In the Abelian case = (ℝ n , +) and Q = n , the inequality (4.19) gives that An interesting observation is that the constant in the above inequality is sharp for any quasi-norm | ⋅ | , that is, it does not depend on the quasi-norm | ⋅ | . Therefore, this inequality is new already in the Euclidean setting of ℝ n . When ‖x‖ = � x 2 1 + x 2 2 + ⋯ + x 2 n is the Euclidean distance, the inequality (4.21) was investigated in ℝ n in [1,Corollary 4.4] and in [24, Theorem 1.1].
Before starting the proof of this result, let us recall the following result [3]:  Taking into account (4.25) and (4.26), we rewrite (4.24) as , then recalling g(|x|) =f (|x|) we see that (4.28) implies that since (4.20). Thus, we have obtained (4.19) with sharp constant for all radial functions f ∈ C ∞ 0 ( ). Finally, using Lemma 2.2, and taking into account g(|x|) =f (|x|) in (4.29), we obtain (4.19) for non-radial functions. The constant in (4.19) is sharp, since this constant is sharp for radial functions by Lemma 4.14. ◻

Maximal Hardy inequality
In this section we discuss a weighted exponential inequality.
where | | = s and |x| = r . Here, we continue our calculation by changing the variables rs = t to get Here, choosing the following radial function for a fixed R > 0, we compute since is the cut-off function. Now, taking into account this and plugging (5.8) into the left hand side of (5.7) we get (5.7) � W 3 (x) exp 1 |B(0, |x|)| � |z|≤|x| log(u(z))dz dx ≤ C � u(x)dx.
In the Euclidean (Abelian) case by using this functional, H.-M. Nguyen and M. Squassina (see, e.g. [13] and [14]) obtained nonlocal verions of the classical Hardy-Sobolev type inequality. Note that from their inequalities in the singular limit ↘ 0 one recovers the classical Hardy-Sobolev type results since in the Euclidean case the functional I converges to the Dirichlet energy up to a normalisation constant. Their main inequalities are gradient free ones. Therefore, those are extendable to the homogeneous (Lie) groups. We believe that such ideas of proofs of (nonlocal) gradient free inequalities can be generalised to the homogeneous groups. Below we demonstrate this idea in a special case. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.