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

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


Introduction
In this paper we continue the research from [RSY18a] devoted to properties of Euler operators on homogeneous groups, their consequences, and related analysis. In turn, this continues the research direction initiated in [RS17a] devoted to Hardy and other functional inequalities in the setting of Folland and Stein's [FS82] homogeneous groups.
Recall the following Sobolev type (improved Hardy) inequality: Let 1 < p < ∞. Then we have In [OS09,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 [BL85] for Sobolev inequalities, to [BV97,Section 4] and [BM97] for Hardy inequalities and for many others, and a more recent literature review to [GM11].
In this paper we are interested, among other things, in obtaining (1.1) with remainder terms, that is, for example, in [BEHL08,Corollary 4.4] the authors obtained the following: Let n ≥ 3 and 0 ≤ δ < 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 S n−1 S n−1 f (r, y)dy is the integral mean of f over the unit sphere S n−1 , and 2 * = 2n/(n−2), r = |x|. They derived (1.3) from the following Gagliardo-Nirenberg inequality [BEHL08,Theorem 4.1] using the explicit representation of the semi-group e −t(x·∇) * (x·∇) : Let 1 ≤ p < q < ∞ and let −i(∂/∂r)f ∈ L p (R × S n−1 ) and f ∈ B p/(p−q) (R n ). Then there exists a positive constant C = C(p, q) such that (1.4) where R := d d|x| is the radial derivative, and · B α is the Besov type norm associated with the semi-group.
We show analogues of (1.3) and (1.4), and calculate the semi-group e −tE * E on homogeneous (Lie) groups, where E is the Euler operator (see Section 2 for more information). We can also obtain (1.3) with sharp constant without using (1.4) and semi-group e −tE * E . 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 (E := |x|R) operators. Some other discussions on this idea will be provided in the final section. We refer to [RSY17] for horizontal versions of these inequalities.
For the convenience of the reader we briefly summarise the obtained results. Let G 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 −tE * E is given by where E = |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 × S) and f ∈ B p/(p−q) (R × S). Then there exists a positive constant C = C(p, q) such that where R := d d|x| is the radial derivative, S is the unit sphere in G and · B α is the Besov type norm defined by (4.1).
• Let φ and ψ be positive functions defined on G. Then there exists a positive constant C such that holds for all positive f if and only if A(φ, ψ) < ∞, where A is given in (5.2) and (Mf )(x) = 1 |B(0,|x|)| B(0,|x|) f (z)dz. • Let Q ≥ 2. For any quasi-norm | · |, all differentiable | · |-radial functions φ, all p > 1 and all f ∈ C 1 0 (G) 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 .
• Let Q ≥ 2. For any quasi-norm | · |, all differentiable | · |-radial functions φ, all p > 1 and all f ∈ C 1 0 (G) we have 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 n−2 2 , n ≥ 3, (instead of 2 n 2 ) is known as the Heisenberg-Pauli-Weyl uncertainty principle (see, e.g. [RS17b, 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 The organisation of the paper is as follows. In Section 2 we briefly recall the necessary concepts of homogeneous Lie groups and fix the notation. The operator semigroup {e −tE * E } t>0 is determined in Section 3. In Section 4 we establish Gagliardo-Nirenberg type inequalities, and obtain sharp remainder terms of the Sobolev type inequality. The maximal Hardy inequality is established in Section 5. In Section 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 [FR16] for a detailed discussion of the appearing objects.
Let us recall the notion of a family of dilations of a Lie algebra g, that is, a family of linear mappings of the following form where A is a diagonalisable linear operator on g with positive eigenvalues. We also recall that D λ is a morphism of g, if it is a linear mapping from g to itself satisfying the property where [X, Y ] := XY − Y X is the Lie bracket. Then, a homogeneous group G is a connected simply connected Lie group whose Lie algebra is equipped with a morphism family of dilations. It induces the dilation structure on G which we denote by D λ x or just by λx.
Let dx be the Haar measure on G and let |S| denote the volume of a measurable subset S of G. The homogeneous dimension of G is defined by Then we have (2.1) A homogeneous quasi-norm on G is a continuous non-negative function satisfying the properties • |x −1 | = |x| for all x ∈ G, • |λx| = λ|x| for all x ∈ G and λ > 0, • |x| = 0 if and only if x = 0. The quasi-ball centred at x ∈ G 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 σ on the unit sphere holds for all f ∈ L 1 (G). We refer to Folland and Stein [FS82] for more information (see also [FR16, Section 3.1.7] for a detailed discussion).
If we fix a basis {X 1 , . . . , X n } of g such that we have for each k, then the matrix A can be taken to be A = diag(ν 1 , . . . , ν n ), where each X k is homogeneous of degree ν k and The decomposition of exp −1 G (x) in g defines the vector e(x) = (e 1 (x), . . . , e n (x)) by the formula where ∇ = (X 1 , . . . , X n ). On the other hand, we also have the following equality x = exp G (e 1 (x)X 1 + . . . + e n (x)X n ) .
By a direct calculation, we get which implies the following equality for each homogeneous quasi-norm |x| on a homogeneous group G, where we have denoted (2.5) We will also use the Euler operator Let us recall the following property of the Euler operator E.
where I and E * are the identity operator and the formal adjoint operator of E, respectively.
Now we give useful inequality and identity for Euler operator.
for any k ∈ N and all f ∈ L p loc (G) such that E k f ∈ L p loc (G) with sharp constants. The constants are attained when f = f . 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 [RSY18b, in the proof of Theorem 3.1]).
Proof of Lemma 2.2. 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
(2.14) holds for ℓ ∈ N, we prove Then, using (2.14) we calculate where we have used Hölder's inequality in the last line. Thus, we have proved (2.13). Now using (2.13), we obtain Thus, we have proved (2.10) for any k ∈ N. Similarly, one can obtain (2.11).
3. Euler semigroup e −tE * E In this section we introduce the operator semigroup {e −tE * E } t>0 associated with the Euler operator on homogeneous groups.
First, let us prepare some preliminary results. We define the map F : (3.1) for y ∈ S and s ∈ R, and its inverse F −1 : We note that F preserves the L 2 norm. We will also use the dilations U(t) : L 2 (G) → L 2 (G) given by These form a group of unitary operators with generator U(t) = e iAt , where A is given by as follows from the following equality with x = ry and ρ := e t r, where y ∈ S. Using Lemma 2.1 we obtain from (3.4) that which implies (3.6) By (3.1) and (3.3), we have (F f )(s, y) = (U(s)f )(y) for y ∈ S and s ∈ R, then this with the group property of the dilations U(·) gives that then it follows from (3.7) and the change of variables that (3.9) 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 −tE * E } t>0 on homogeneous groups.
Theorem 3.1. Let G be a homogeneous group of homogeneous dimension Q. Let x = ry and r = |x| with y ∈ S. Then the semigroup e −tE * E is given by (3.10) Proof of Theorem 3.1. Let us first show that 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 (3.14) 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 Here using M = F • F , one has Then, a direct calculation gives that = r −Q/2 ϕ t (ln r, y) where we have used the change of variables z = e s in the last line.
Now let us give the following representation for e −tA 2 , which is useful to obtain Gagliardo-Nirenberg type inequalities (see Section 4): Corollary 3.2. Let F and F −1 be mappings as in (3.1) and (3.2), respectively. Then we have

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 α (R×S) as the space of all tempered distributions f on R × S with the norm (4.1) We will also use the one-dimensional case of the following result: Theorem 4.1 ([Led03, Theorem 1]). Let 1 ≤ p < q < ∞. Then for every function f ∈ L p 1 (R n ) there exists a positive constant C = C(p, q, n) such that Now we state the Gagliardo-Nirenberg type inequalities: Theorem 4.2. Let G be a homogeneous group of homogeneous dimension Q. Let 1 ≤ p < q < ∞ and let f be such that Rf ∈ L p (R × S) and f ∈ B p/(p−q) (R × S). Then there exists a positive constant C = C(p, q) such that (4.3) Proof of Theorem 4.2. Using Theorem 4.1 with n = 1 and Corollary 3.2, we obtain R |f (r, y)| q dr ≤ C q R ∂f (r, y) ∂r for any y ∈ S, in view of (4.1). One obtains (4.3) after integrating the above inequality with respect to y over S.
Corollary 4.11. Let G be a homogeneous group of homogeneous dimension Q. Let | · | be a homogeneous quasi-norm. Let 2 < q < ∞. Let f, Ef ∈ L 2 (G) and Then there exists a positive constant C such that Now we prove (4.12) with sharp constant, which can be viewed as an analogue of Stubbe's inequality [Stu90] on homogeneous groups.
Theorem 4.12. Let G be a homogeneous group of homogeneous dimension Q ≥ 3. Let | · | be a homogeneous quasi-norm. Then we have for all f ∈ C ∞ 0 (G) and 0 ≤ δ < with sharp constant, where and (4.20) Remark 4.13. In the abelian case G = (R n , +) and Q = n, the inequality (4.19) gives that (4.21) 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 [BEHL08, Corollary 4.4] and in [Xia11, Theorem 1.1].
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.
Theorem 5.1. Let φ and ψ be positive functions defined on G. Then there exists a positive constant C such that holds for all positive f if and only if Remark 5.2. In the abelian case G = (R n , +) and Q = n, the inequality (5.1) was studied in [HKK01] for n = 1, and in [DHK97] for n ≥ 1.

Further inequalities
In this section we discuss a number of related inequalities, also interesting on their own. Theorem 6.1. For any quasi-norm | · |, all differentiable | · |-radial functions φ, all p > 1, Q ≥ 2, and all f ∈ C 1 0 (G) we have In (6.1) taking φ = log |x| in the Euclidean (Abelian) case G = (R n , +), n ≥ 2, we have Q = n, and taking p = n ≥ 2, so for any quasi-norm | · | on R n it implies the new inequality: for all f ∈ C 1 0 (R n ). Thus, when n = 2 inequality (6.6) gives the critical case of the Heisenberg-Pauli-Weyl uncertainty principle. Moreover, since 2 n−2 ≥ 2 n , n ≥ 3, inequality (6.6) is an improved version of (6.7). Note that equality case in (6.6) holds for the function f = C exp(−b x ), b > 0.
hand, such analysis is important since, as we mentioned in the introduction, in general there is no homogeneous gradient on homogeneous (Lie) groups. On the other hand, these inequalities give new inequalities even in Euclidean cases as well as cover classical inequalities with gradients. However, in addition to these methods there are other techniques to obtain gradient free Hardy-Sobolev type inequalities. To conclude these discussions let us introduce the following functional where • is the group operation on G.
In the Euclidean (Abelian) case by using this functional, H.-M. Nguyen and M. Squassina (see, e.g. [Ng08] and [NS17]) obtained nonlocal verions of the classical Hardy-Sobolev type inequality. Note that from their inequalities in the singular limit δ ց 0 one recovers the classical Hardy-Sobolev type results since in the Euclidean case the functional I δ 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. Proof of Proposition 6.3. The proof of Proposition 6.3 relies on the same technique as the proof of [NS17, Theorem 1.1] with the difference that now the quasi-norm (instead of the Euclidean distance) is used. For the proof we only need to recall the fact that the Lebesque measure on R n gives the Haar measure for G. The rest of the proof is exactly the same as in the proof of [NS17, Theorem 1.1].