Weighted $L^p$-Hardy and $L^p$-Rellich inequalities with boundary terms on stratified Lie groups

In this paper, generalised weighted $L^p$-Hardy,$ L^p$-Caffarelli-Kohn-Nirenberg, and $L^p$-Rellich inequalities with boundary terms are obtained on stratified Lie groups. As consequences, most of the Hardy type inequalities and Heisenberg- Pauli-Weyl type uncertainty principles on stratified groups are recovered. Moreover, a weighted $L^2$-Rellich type inequality with the boundary term is obtained.


Introduction
Let G be a stratified Lie group (or a homogeneous Carnot group), with dilation structure δ λ and Jacobian generators X 1 , . . . , X N , so that N is the dimension of the first stratum of G. We refer to [10], or to the recent books [4] or [9] for extensive discussions of stratified Lie groups and their properties. Let Q be the homogeneous dimension of G. The sub-Laplacian on G is given by It was shown by Folland [10] that the sub-Laplacian has a unique fundamental solution ε, where δ denotes the Dirac distribution with singularity at the neutral element 0 of G. The fundamental solution ε(x, y) = ε(y −1 x) is homogeneous of degree −Q + 2 and can be written in the form for some homogeneous d which is called the L-gauge. Thus, the L-gauge is a symmetric homogeneous (quasi-) norm on the stratified group G = (R n , •, δ λ ), that is, • d(x) > 0 if and only if x = 0, • d(δ λ (x)) = λd(x) for all λ > 0 and x ∈ G, We also recall that the standard Lebesque measure dx on R n is the Haar measure for G (see, e.g. [9, Proposition 1.6.6]). The left invariant vector field X j has an explicit form and satisfies the divergence theorem, see e.g. [9] for the derivation of the exact formula: more precisely, we can write with x = (x ′ , x (2) , . . . , x (r) ), where r is the step of G and x (l) = (x (l) 1 , . . . , x (l) N l ) are the variables in the l th stratum, see also [9, Section 3.1.5] for a general presentation. The horizontal gradient is given by ∇ G := (X 1 , . . . , X N ), and the horizontal divergence is defined by The horizontal p-sub-Laplacian is defined by and we will write N for the Euclidean norm on R N . Throughout this paper Ω ⊂ G will be an admissible domain, that is, an open set Ω ⊂ G is called an admissible domain if it is bounded and if its boundary ∂Ω is piecewise smooth and simple i.e., it has no self-intersections. The condition for the boundary to be simple amounts to ∂Ω being orientable.
We now recall the divergence formula in the form of [19,Proposition 3.1]. Let f k ∈ C 1 (Ω) C(Ω), k = 1, . . . , N. Then for each k = 1, . . . , N, we have (1.5) Consequently, we also have Using the divergence formula analogues of Green's formulae were obtained in [19] for general Carnot groups and in [20] for more abstract settings (without the group structure), for another formulation see also [11]. The analogue of Green's first formula for the sub-Laplacian was given in [19] in the following form: if v ∈ C 1 (Ω) ∩ C(Ω) and u ∈ C 2 (Ω) ∩ C 1 (Ω), then Rewriting (1.7) we have By using ( ∇u)v = ( ∇v)u and subtracting one identity for the other we get Green's second formula for the sub-Laplacian: It is important to note that the above Green's formulae also hold for the fundamental solution of the sub-Laplacian as in the case of the fundamental solution of the (Euclidean) Laplacian since both have the same behaviour near the singularity z = 0 (see [1,Proposition 4.3]).
Weighted Hardy and Rellich inequalities in different related contexts have been recently considered in [15] and [13]. For the general importance of such inequalities we can refer to [2]. Some boundary terms have appeared in [23].
The main aim of this paper is to give the generalised weighted L p -Hardy and L p -Rellich type inequalities on stratified groups. In Section 2, we present a weighted L p -Caffarelli-Kohn-Nirenberg type inequality with boundary term on stratified group G, which implies, in particular, the weighted L p -Hardy type inequality. As consequences of those inequalities, we recover most of the known Hardy type inequalities and Heisenberg-Pauli-Weyl type uncertainty principles on stratified group G (see [21] for discussions in this direction). In Section 3, a weighted L p -Rellich type inequality is investigated. Moreover, a weighted L 2 -Rellich type inequality with the boundary term is obtained together with its consequences.
Usually, unless we state explicitly otherwise, the functions u entering all the inequalities are complex-valued.

Weighted L p -Hardy type inequalities with boundary terms and their consequences
In this section we derive several versions of the L p weighted Hardy inequalities.
2.1. Weighted L p -Cafferelli-Kohn-Nirenberg type inequalities with boundary terms. We first present the following weighted L p -Cafferelli-Kohn-Nirenberg type inequalities with boundary terms on the stratified Lie group G and then discuss their consequences. The proof of Theorem 2.1 is analogous to the proof of Davies and Hinz [8], but is now carried out in the case of the stratified Lie group G. The boundary terms also give new addition to the Euclidean results in [8]. The classical Caffarelli-Kohn-Nirenberg inequalities in the Euclidean setting were obtained in [6].
Let G be a stratified group with N being the dimension of the first stratum, and let V be a real-valued function in L 1 loc (Ω) with partial derivatives of order up to 2 in L 1 loc (Ω), and such that LV is of one sign. Then we have: Let Ω be an admissible domain in the stratified group G, and let V be a real-valued function such that LV < 0 holds a.e. in Ω. Then for any complex-valued u ∈ C 2 (Ω) ∩ C 1 (Ω), and all 1 < p < ∞, we have the inequality Note that if u vanishes on the boundary ∂Ω, then (2.1) extends the Davies and Hinz result [8] to the weighted L p -Hardy type inequality on stratified groups: (Ω) and using Green's first formula (1.7) and the fact that LV < 0 we get On the other hand, let u(x) = R(x) + iI(x), where R(x) and I(x) denote the real and imaginary parts of u. We can restrict to the set where u = 0. Then we have we get that |∇ G |u|| ≤ |∇ G u| a.e. in Ω. Therefore, where we have used Hölder's inequality in the last line. Thus, when ǫ → 0, we obtain (2.1).

2.2.
Consequences of Theorem 2.1. As consequences of Theorem 2.1, we can derive the horizontal L p -Caffarelli-Kohn-Nirenberg type inequality with the boundary term on the stratified group G which also gives another proof of L p -Hardy type inequality, and also yet another proof of the Badiale-Tarantello conjecture [3] (for another proof see e.g. [18] and references therein).

2.2.2.
Badiale-Tarantello conjecture. Theorem 2.1 also gives a new proof of the generalised Badiale-Tarantello conjecture [3] (see, also [18]) on the optimal constant in Hardy inequalities in R n with weights taken with respect to a subspace.

7)
where γ = α + β + 1 and |x ′ | is the Euclidean norm R N . If γ = N then the constant The proof of Proposition 2.3 is similar to Corollary 2.2, so we sketch it only very briefly.

2.3.
Uncertainty type principles. The inequality (2.12) implies the following Heisenberg-Pauli-Weyl type uncertainty principle on stratified groups.
Let Ω ⊂ G be admissible domain in a stratified group G and let V ∈ C 2 (Ω) be real-valued. Then for any complex-valued function u ∈ C 2 (Ω) ∩ C 1 (Ω) we have In particular, if u vanishes on the boundary ∂Ω, then we have (2.14) Proof of Corollary 2.5. By using the extended Hölder inequality and (2.12) we have proving (2.13).
By setting V = |x ′ | α in the inequality (2.14), we recover the Heisenberg-Pauli-Weyl type uncertainty principle on stratified groups as in [17] and [20]: In the abelian case G = (R n , +), taking N = n ≥ 3, for α = 0 and p = 2 this implies the classical Heisenberg-Pauli-Weyl uncertainty principle for all u ∈ C ∞ 0 (R n \{0}): By setting V = d α in the inequality (2.14), we obtain another uncertainty type principle: taking p = 2 and α = 0 this yields

Weighted L p -Rellich type inequalities
In this section we establish weighted Rellich inequalities with boundary terms. We consider first the L 2 and then the L p cases. The analogous L 2 -Rellich inequality on R n was proved by Schmincke [22] (and generalised by Bennett [5]).
Let Ω be an admissible domain in a stratified group G with N ≥ 2 being the dimension of the first stratum. If a real-valued function V ∈ C 2 (Ω) satisfies LV (x) < 0 for all x ∈ Ω, then for every ǫ > 0 we have for all complex-valued functions u ∈ C 2 (Ω) ∩ C 1 (Ω). In particular, if u vanishes on the boundary ∂Ω, we have . Proof of Theorem 3.1. Using Green's second identity (1.8) and that LV (x) < 0 in Ω, we obtain Using the Cauchy-Schwartz inequality we get yielding (3.1).
Corollary 3.2. Let G be a stratified group with N being the dimension of the first stratum. If α > −2 and N > α + 4 then for all u ∈ C ∞ 0 (G\{x ′ = 0}) we have Proof of Corollary 3.2. Let us take V (x) = |x ′ | −(α+2) in Theorem 3.1, which can be applied since x ′ = 0 is not in the support of u. Then we have Let us set C N,α := (α + 2)(N − α − 4). Observing that To obtain (3.2), let us apply the L p -Hardy type inequality (2.2) by taking and then choosing ǫ = (N + α)/4(α + 2) for (3.3), which is the choice of ǫ that gives the maximum right-hand side.
We can now formulate the L p -version of weighted L p -Rellich type inequalities.

Theorem 3.3.
Let Ω be an admissible domain in a stratified group G. If 0 < V ∈ C(Ω), LV < 0, and L(V σ ) ≤ 0 on Ω for some σ > 1, then for all u ∈ C ∞ 0 (Ω) we have Let Ω an admissible domain in a stratified group G. If V ≥ 0, LV < 0, and there exists a constant C ≥ 0 such that