Anisotropic fractional Gagliardo-Nirenberg, weighted Caffarelli-Kohn-Nirenberg and Lyapunov-type inequalities, and applications to Riesz potentials and $p$-sub-Laplacian systems

In this paper we prove the fractional Gagliardo-Nirenberg inequality on homogeneous Lie groups. Also, we establish weighted fractional Caffarelli-Kohn-Nirenberg inequality and Lyapunov-type inequality for the Riesz potential on homogeneous Lie groups. The obtained Lyapunov inequality for the Riesz potential is new already in the classical setting of $\mathbb{R}^{N}$. As an application, we give two-sided estimate for the first eigenvalue of the Riesz potential. Also, we obtain Lyapunov inequality for the system of the fractional $p$-sub-Laplacian equations and give an application to estimate its eigenvalues

In the works of E. Gagliardo [9] and L. Nirenberg [14] (independently), they obtained the following (interpolation) inequality where 2 ≤ p ≤ ∞ for N = 2, 2 ≤ p ≤ 2N N −2 for N > 2. The Gagliardo-Nirenberg inequality on the Heisenberg group H n has the following form where ∇ H is a horizontal gradient and Q is a homogeneous dimension of H n .In [3], the authors established the best constant for the sub-elliptic Gagliardo-Nirenberg inequality (1.2).Consequently, in [20] the best constants in Gagliardo-Nirenberg and Sobolev inequalities were also found for general hypoelliptic (Rockland operators) on general graded Lie groups.
In [15] the authors obtained a fractional version of the Gagliardo-Nirenberg inequality in the following form: where [u] s,p is Gagliardo's seminorm defined by for N ≥ 1, s ∈ (0, 1), p > 1, α ≥ 1, τ > 0, and a ∈ (0, 1] is such that In this paper we formulate the fractional Gagliardo-Nirenberg inequality on the homogeneous Lie groups.To the best of our knowledge, in this direction systematic studies on the homogeneous Lie groups started by the paper [18] in which homogeneous group versions of Hardy and Rellich inequalities were proved as consequences of universal identities.
In [15] the authors proved the fractional analogues of the Caffarelli-Kohn-Nirenberg inequality in weighted fractional Sobolev spaces.Also, in [1] a fractional Caffarelli-Kohn-Nirenberg inequality for an admissible weight in R N was obtained.
Recently many different versions of Caffarelli-Kohn-Nirenberg inequalities have been obtained, namely, in [24] on the Heisenberg groups, in [22] and [23] on stratified groups, in [19] and [21] on (general) homogeneous Lie groups.One of the aims of this paper is to prove the fractional weighted Caffarelli-Kohn-Nirenberg inequality on the homogeneous Lie groups.
1.3.Fractional Lyapunov-type inequality.Historically, in Lyapunov's work [13] for the following one-dimensional homogeneous Dirichlet boundary value problem (for the second order ODE) it was proved that if u is a non-trivial solution of (1.
Theorem 1.2.Let Ω ⊂ R N be an open set, and let ω ∈ L θ (Ω) with 1 < N sp < θ < ∞, be a non-negative weight.Suppose that problem (1.11) has a non-trivial weak solution u ∈ W s,p 0 (Ω). Then where C > 0 is a universal constant and r Ω is the inner radius of Ω.
In [4], the authors considered a system of ODE for p and q-Laplacian on the interval (a, b) with the homogeneous Dirichlet condition in the following form: on the interval (a, b), with where f, g ∈ L 1 (a, b), f, g ≥ 0, p, q > 1, α, β ≥ 0 and Then we have Lyapunov-type inequality for system (1.13) with homogeneous Dirichlet condition (1.14): where p ′ = p p−1 and q ′ = q q−1 .In [11], the authors obtained the Lyapunov-type inequality for a fractional p-Laplacian system in an open bounded subset Ω ⊂ R N with homogeneous Dirichlet conditions.One of our goals in this paper is to extend the Lyapunov-type inequality for the Riesz potential and for the fractional p-sub-Laplacian system on the homogeneous Lie groups.These results are given in Theorem 5.1 and 5.7.Also, we give applications of the Lyapunov-type inequality for the Riesz potential and for fractional p-sub-Laplacian system on the homogeneous Lie groups.To demonstrate our techniques we consider the Riesz potential in the Abelian case (R N , +) and give two side estimates of the first eigenvalue of the Riesz potential in the Abelian case (R N , +).
Summarising our main results of the present paper, we prove the following facts: • An analogue of the fractional Gagliardo-Nirenberg inequality on the homogeneous group G; • An analogue of the fractional weighted Caffarelli-Kohn-Nirenberg inequality on G; • An analogue of the Lyapunov-type inequality for the Riesz potential on G; • An analogue of the Lyapunov-type inequality for the fractional p-sub-Laplacian system on G.
The paper is organised as follows.First we give some basic discussions on fractional Sobolev spaces and related facts on homogeneous Lie groups, then in Section 3 we present the fractional Gagliardo-Nirenberg inequality on G.The fractional weighted Caffarelli-Kohn-Nirenberg inequality on G is proved in Section 4. In Section 5 we discuss analogues of the Lyapunov-type inequalities for the Riesz potential and fractional p-sub-Laplacian system on G.

Preliminaries
We recall that a Lie group (on R n ) G with the dilation which is an automorphism of the group G for each λ > 0, is called a homogeneous (Lie) group.In this paper, for simplicity, we use the notation λx instead of the dilation D λ (x).The homogeneous dimension of the homogeneous group G is denoted by with the properties i) q(x) = q(x −1 ) for all x ∈ G, ii) q(λx) = λq(x) for all x ∈ G and λ > 0, iii) q(x) = 0 iff x = 0.Moreover, the following polarisation formula on homogeneous Lie groups will be used in our proofs: there is a (unique) positive Borel measure σ on the unit quasi-sphere We refer to [7] for the original appearance of such groups, and to [6] for a recent comprehensive treatment.Let p > 1, s ∈ (0, 1), and let G be a homogeneous Lie group of homogeneous dimension Q.For a measurable function u : G → R we define the Gagliardo quasi-seminorm by Now we recall the definition of the fractional Sobolev spaces on homogeneous Lie groups denoted by W s,p,q (G).For p ≥ 1 and s ∈ (0, 1), the functional space is called the fractional Sobolev space on G. Similarly, if Ω ⊂ G is a Haar measurable set, we define the Sobolev space Now we recall the definition of the weighted fractional Sobolev space on the homogeneous Lie groups denoted by where β 1 , β 2 ∈ R with β = β 1 + β 2 and it depends on β 1 and β 2 .
The mean of a function u is defined by where |Ω| is the Haar measure of Ω ⊂ G.
We will also use the decomposition of G into quasi-annuli A k,q defined by where q(x) is a quasi-norm on G.

Fractional Gagliargo-Nirenberg inequality on G
In this section we prove an analogue of the fractional Gagliardo-Nirenberg inequality on the homogeneous Lie groups.To prove Gagliardo-Nirenberg's inequality we need some preliminary results from [12], a version of a fractional Sobolev inequality on the homogeneous Lie groups.
From now on, unless specified otherwise, G will be a homogeneous group of homogeneous dimension Q. Theorem 3.1 ( [12], Fractional Sobolev inequality).Let p > 1, s ∈ (0, 1), Q > sp, and let q(•) be a quasi-norm on G.For any measurable and compactly supported function u : G → R there exists a positive constant C = C(Q, p, s, q) > 0 such that where Then, where C = C(s, p, Q, a, α) > 0.
Proof of Theorem 3.2.By using the Hölder inequality, for every where p * = Qp Q−sp .From (3.3), by using the fractional Sobolev inequality (Theorem 3.1), we obtain where C is a positive constant independent of u.Theorem 3.2 is proved.
Remark 3.3.In the Abelian case (R N , +) with the standard Euclidean distance instead of the quasi-norm, from Theorem 3.2 we get the fractional Gagliardo-Nirenberg inequality which was proved in [15].

Weighted fractional Caffarelli-Kohn-Nirenberg inequality on G
In this section we prove the weighted fractional Caffarelli-Kohn-Nirenberg inequality on the homogeneous Lie groups.
To prove the fractional weighted Caffarelli-Kohn-Nirenberg inequality on G we will use Theorem 3.2 in the proof of the following lemma.
Let λ > 0 and 0 < r < R and set Then, for every u ∈ C 1 (Ω), we have where C r,R is a positive constant independent of u and λ.
Proof of Lemma 4.3.Without loss of generality, we assume that 0 < s ′ ≤ s and τ ′ ≥ τ are such that and λ = 1, then let Ω 1 be By using Theorem 3.2, Jensen's inequality and [u] s ′ ,p,q,Ω ≤ C[u] s,p,q,Ω , we get where C r,R > 0. Let us set u(λx) instead of u(x), then Thus, we compute The proof of Lemma 4.3 is complete.
Proof of Theorem 4.1.First let us consider the case (4.2), that is, β − σ ≤ s and where A k,q is defined in (2.9) and k ∈ Z. Now by using (4.9) we obtain Then, from (4.10) we get ) .(4.11)Here by (4.1), we have Thus, we obtain s,p,β,q,A k,q q µ (x)u and by summing over k from m to n, we get [u] aτ s,p,β,q,A k,q q µ (x)u where k, m, n ∈ Z and m ≤ n − 2.
To prove (4.3) let us choose n such that where B 2 n is a quasi-ball of G with the radius 2 n .The following known inequality will be used in the proof.
Let us prove the case of β − σ > s.Without loss of generality, we assume that where We also assume that a 1 > 0, 1 > a 2 and τ 1 , τ 2 > 0 with 1 and From (4.32) in the case a p + 1−a α − as Q > 0 with a > 0, β − σ > s and (4.35), we get From (4.32), (4.39) and (4.40), we have Thus, using this, (4.35) and Hölder's inequality, we obtain and .42) where B 1 is the unit quasi-ball.By using the previous case, we establish and The proof of Theorem 4.1 is complete.

Lyapunov-type inequalities for the fractional operators on G
In this section we prove the Lyapunov-type inequality for the Riesz potential and for the fractional p-sub-Laplacian system on homogeneous Lie groups.Note that the Lyapunov-type inequality for the Riesz operator is new even in the Abelian case (R N , +).Also, we give applications of the Lyapunov-type inequality, more precisely, we give two side estimates for the first eigenvalue of the Riesz potential of the fractional p-sub-Laplacian system.
Let us consider the Riesz potential on a Haar measurable set Ω ⊂ G that can be defined by the formula The (weighted) Riesz potential can be also defined by Then ω Proof of Theorem 5.1.In (5.3), by using Hölder's inequality for p, θ > 1 with Let p ′ be such that p ′ = pθ ′ and then θ = 1 2−p .Thus, we get From (5.6) we calculate Finally, since u = 0, this implies Theorem 5.1 is proved.
Let us consider the following spectral problem for the Riesz potential: We recall the Rayleigh quotient for the Riesz potential: where λ 1 (Ω) is the first eigenvalue of the Riesz potential.So, a direct consequence of Theorem 5.1 is Let Ω ⊂ G be a Haar measurable set and Q ≥ 2 > 2s > 0 and let 1 < p < 2. Assume that Then for the spectral problem (5.9), we have where . Proof of Theorem 5.2.By using (5.10),Theorem 5.1 and ω = 1 λ 1 (Ω) , we obtain (5.12) Theorem 5.2 is proved.
In the Abelian group (R N , +) we have the following consequences.To the best of our knowledge, these results seem new (even in this Euclidean case).
Let us consider the Riesz potential on Ω ⊂ R N : and the weighted Riesz potential (5.14) Then we have following theorem: . Assume that u ∈ L .Then for the spectral problem (5.16) we have, is the same as the proof of Theorem 5.2.From [17] we have The proof of Theorem 5.4 is complete.
In [12] the authors proved a Lyapunov-type inequality for the fractional p-sub-Laplacian with the homogeneous Dirichlet condition.Here we establish Lyapunovtype inequality for the fractional p-sub-Laplacian system for the homogeneous Dirichlet problem.Namely, let us consider the fractional p-sub-Laplacian system: (5.18) with homogeneous Dirichlet conditions where Ω ⊂ G is a Haar measurable set, ω i ∈ L 1 (Ω), ω i ≥ 0, s i ∈ (0, 1), p i ∈ (1, ∞) and (−∆ p,q ) s is the fractional p-sub-Laplacian on G defined by Here B q (x, δ) is a quasi-ball with respect to q, with radius δ, centred at x ∈ G, and α i are positive parameters such that n i=1 To prove a Lyapunov-type inequality for the system we need some preliminary results from [12], the so-called fractional Hardy inequality on the homogeneous Lie groups.
Now we present the following analogue of the Lyapunov-type inequality for the fractional p-sub-Laplacian system on G.
Proof of Theorem 5.7.For all i = 1, . . ., n, let us define and where p * i = Q Q−s i p i is the Sobolev conjugate exponent as in Theorem 3.1.Notice that for all i = 1, . . ., n we have γ i ∈ (0, 1) and ξ i = p i θ ′ , where θ ′ = θ θ−1 .Then for every i ∈ {1, . . ., n} we get and by using Hölder's inequality with the following exponents ν i = 1 γ i and 1 On the other hand, from Theorem 3.1, we obtain and from Theorem 5.5, we have Thus, from (5.28) and by taking u i (x) = v i (x) in (5.24), we get , for every i = 1, . . ., n.Therefore, by using Hölder's inequality with exponents θ and θ ′ , we obtain By using Hölder's inequality and (5.21), we get (5.31)Theorem 5.7 is proved.Now, let us discuss an application of the Lyapunov-type inequality for the fractional p-sub-Laplacian system on G.In order to do it we consider the spectral problem for the fractional p-sub-Laplacian system in the following form: where Ω ⊂ G is a Haar measurable set, ϕ ∈ L 1 (Ω), ϕ ≥ 0 and s i ∈ (0, 1), p i ∈ (1, ∞), i = 1, . . ., n.
7) and ω(x) is a real-valued and continuous function on [a, b], then necessarily b a |ω(x)|dx > 4 b − a .

≤ C ω i e i θ θ− 1 L
every e i > 0 we haveΩ |u i (x)| ξ i r γ i s i p i Ω,q dx e i = 1 r e i γ i s i p i Ω,q Ω |u i (x)| ξ i dx e iwhere C is a positive constant.Then, we choose e i , i = 1, . . ., n, such thatα i n j=1 e j p i − e i = 0, i = 1, . . ., n.Consequently, from (5.21) we have the solution of this system e i = α i p i , i = 1, . . ., n.