A unified approach to weighted Hardy type inequalities on Carnot groups

We find a simple sufficient criterion on a pair of nonnegative weight functions \begin{document}$V(x)$\end{document} and \begin{document}$W(x) $\end{document} on a Carnot group \begin{document}$\mathbb{G},$\end{document} so that the general weighted \begin{document}$L^{p}$\end{document} Hardy type inequality \begin{document}$\begin{equation*}\int_{\mathbb{G}}V\left( x\right) \left\vert \nabla _{\mathbb{G}}\phi \left(x\right) \right\vert ^{p}dx\geq \int_{\mathbb{G}}W\left( x\right) \left\vert\phi \left( x\right) \right\vert ^{p}dx\end{equation*}$ \end{document} is valid for any \begin{document}$φ ∈ C_{0}^{∞ }(\mathbb{G})$\end{document} and \begin{document}$p>1.$\end{document} It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on \begin{document}$\mathbb{G}.$\end{document} We also present some new results on two-weight \begin{document}$L^{p}$\end{document} Hardy type inequalities with remainder terms on a bounded domain \begin{document}$Ω$\end{document} in \begin{document}$\mathbb{G}$\end{document} via a differential inequality.

On the Euclidean space R n , the classical Hardy inequality asserts that and holds for every φ ∈ C ∞ 0 (R n ) if 1 ≤ p < n, and for every φ ∈ C ∞ 0 (R n \{0}) if n < p < ∞. Here the subscript zero signifies compact support. It is also known that the positive constant on the right-hand side of (1) is sharp but, for p > 1, that equality is only possible if φ = 0 a.e. In the critical case n = p an inequality of type (1) fails for every positive constant on the right-hand side, while the following sharp inequality is valid for all φ ∈ C ∞ 0 (B 1 (0)), where B 1 (0) is the unit ball in R n centered at the origin; see Edmunds and Triebel [12]. The stronger version of (2) was then presented by Adimurthi and Sandeep in [2].
On the other hand in the case of sub-Riemannian spaces, especially on Carnot groups G, Hardy type inequalities have been also intensively investigated, see [11], [22], [34], [10], [27], [26], [29], [36]. For instance, D'Ambrosio in [11] and Goldstein and Kombe in [22] established, among the other things, the following L p Hardy type inequality on polarizable Carnot groups G, for all φ ∈ C ∞ 0 (G\ {0}) , provided that Q ≥ 3 and 1 < p < Q. Here, Q is the homogeneous dimension of G, N : G −→ [0, ∞) is the homogeneous norm associated with the fundamental solution for the sub- . . , X m ) is the horizontal gradient on G and X 1 , . . . , X m are the generators of G (see Section 2 for definitions and preliminaries).
Later in [27] Kombe discovered the sharp weighted Hardy inequality for the p = 2 case on general Carnot groups G having the form where φ ∈ C ∞ 0 (G\ {0}) and α ∈ R, Q ≥ 3, 2 < Q + α. Niu and Wang [34] then extended the inequality (4) to the L p case on polarizable Carnot groups G and showed that for any φ ∈ C ∞ 0 (G\ {0}) one has whenever α ∈ R, 1 < p < Q + α, γ > −1. We note that all the constants appearing in (3) , (4) and (5) are sharp but are never achieved. We also mention that Jin and Shen [26], recently have proved a weighted L p Hardy inequality on general Carnot groups G by using a special class of weighted p-sub-Laplacian and the corresponding fundemantal solution. More recently, Lian also has got a similar result on the same groups with a sharp constant, see [29].
All of these works motivate us to investigate a constructive method to derive Hardy type inequalities with different weights on G. In this direction, we provide an approach that recovers and improves most of the Hardy type inequalities that have appeared to date. More precisely, we verify that if V ∈ C 1 (G) and W ∈ L 1 loc (G) are nonnegative functions and Φ ∈ C ∞ (G) is a positive function such that almost everywhere in a general Carnot group G, then the inequality is valid for every φ ∈ C ∞ 0 (G), where p ≥ 2 and c p > 0. It is worth emphasizing here that one can readily obtain as many weighted Hardy type inequalities as one can construct a weight function V and a function Φ satisfying the above hypotheses (see Applications of Theorem 3.1). We remark that a similar inequality with a different nonnegative remainder term also exists for the case 1 < p < 2 (see Theorem 3.1).
We also give new results on two-weight L p Hardy type inequalities with remainders on a bounded domain Ω in polarizable Carnot groups G. The primary tool which we employ in constructing these type of inequalities is a differential inequality involving a nonnegative general weight function V, the homogeneous norm N and a positive smooth function δ (see Theorem 4.1). We show some concrete examples by specializing the functions V and δ (see Applications of Theorem 4.1).

Preliminaries and notations.
We first give an account of some of the basic definitions, terminology and background results of analysis on Carnot groups G that will be used throughout the article. For further details on this topic we refer the interested readers to [3], [4], [7], [15], [17], [32], and the references therein.
A Carnot group is a connected, simply connected, nilpotent Lie group G ≡ (R n , ·) whose Lie algebra G admits a stratification. That is, there exist linear subspaces This defines an s-step Carnot group and the integer s ≥ 1 is called the step of G. Via the exponential map, it is possible to induce on G a family of automorphisms of the group, called dilations, δ λ : R n −→ R n (λ > 0), such that where 1 = α 1 = · · · = α m < α m+1 ≤ · · · ≤ α n are integers and m = dim(V 1 ). The group law can be written in the following form where P : R n × R n −→ R n has polynomial components and P 1 = · · · = P m = 0 (see [32], Chapter 12, Section 5). Note that the inverse x −1 of an element x ∈ G has the form Let X 1 , . . . , X m be a family of left invariant vector fields that form an orthonormal basis of V 1 ≡ R m at the origin, that is, The vector fields X j have polynomial coefficients and can be assumed to be of the form where each polynomial a ij is homogeneous with respect to the dilations of the group, that is, Then, the Carnot-Caratheodory distance d cc (x, y) between two points x, y ∈ G is defined to be the infimum of the lengths b a γ (t), γ (t) 1/2 dt of all horizontal curves γ : [a, b] −→ G such that γ(a) = x and γ(b) = y. Notice that d cc is a homogeneous norm and satisfies the invariance property for all x, y, z ∈ G and is homogeneous of degree one with respect to the dilation δ λ . The Carnot-Caratheodory balls are defined by B(x, R) = {y ∈ G|d cc (x, y) < R}.
The n-dimensional Lebesgue measure, L n , is the Haar measure of group G. This is the homogeneous dimension of G.
The nonlinear operator is the p-sub-Laplacian on Carnot group G. If p = 2 then we have the linear sub- is the horizontal gradient on G and X 1 , . . . , X m are the generators of G. The fundamental solution u for ∆ G is defined to be a weak solution to the equation −∆ G u = δ 0 , where δ 0 denotes the Dirac distribution with singularity at the neutral element 0 of G. In [14], Folland proved that in any Carnot group G, there exists a homogeneous norm N such that u = N 2−Q is harmonic in G\ {0} and is a positive multiple of the fundamental solution for ∆ G . We now start with u and set and recall that a homogeneous norm on G is a continuous function N : G −→ [0, ∞), smooth away from the origin, which satisfies the conditions N (δ λ (x)) = λN (x) , N x −1 = N (x) and N (x) = 0 iff x = 0. Using the homogeneous norm N, we define the N -ball B N in G with center zero and radius R by A Carnot group G is called polarizable if the homogeneous norm N = u 1/(2−Q) , associated to Folland's solution u for the sub-Laplacian ∆ G , satisfies the following ∞-sub-Laplace equation This class of groups was introduced by Balogh and Tyson [3] and admits the analogue of polar coordinates. It is known that the Euclidean space, the Heisenberg group H n and Kaplan's H-type group are polarizable Carnot groups. In [3], the same authors also proved that for every 1 < p < ∞ the function Moreover, for each 1 < p < ∞ there exists a constant l p > 0 such that −∆ G,p u p = l p δ 0 in the sense of distributions.

Weighted Hardy type inequalities.
Here is the main result of this section.
Theorem 3.1. Let V ∈ C 1 (G) and W ∈ L 1 loc (G) be nonnegative functions and Φ ∈ C ∞ (G) be a positive function such that almost everywhere in a general Carnot group G. There exists a positive constant c p depending only on p such that, if p ≥ 2, then and if 1 < p < 2, then for all φ ∈ C ∞ 0 (G) . Proof. We now recall the following inequalities that will be used in this article (see, for example, [30]). For any 1 < p < ∞ there exists a positive constant c p depending only on p such that for all a, b ∈ R n we have and |a + b| p ≥ |a| p + p |a| p−2 a · b + c p |b| 2 (|a| + |b|) 2−p , for 1 < p < 2.
Let ϕ be a new variable ϕ := φ Φ , where 0 < Φ ∈ C ∞ (G) and φ ∈ C ∞ 0 (G) . Applying the inequality (9) with a = ϕ∇ G Φ and b = Φ∇ G ϕ, we get Multiplying the inequality (11) by V (x) on both sides and integrating by parts over G yield As a next step, by using the weighted p−Laplacian inequality (6) , we conclude that Making the change of variable ϕ = φ Φ in the above integrals, we obtain the desired result (7) . Note that the Theorem 3.1 holds also for 1 < p < 2 and in this case we use the inequality (10) with the same choices of a and b as in the above derivation. This finishes the proof of Theorem 3.1.
Applications of Theorem 3.1. Let > 0 be given. To make following arguments rigorous we should replace the function N with its regularization N := (u + ) 1 2−Q and after the computation take the limit as −→ 0. However, for the sake of simplicity we will proceed formally.
As we have already mentioned most of the known Hardy type inequalities on polarizable Carnot groups G such as (3) , (4) and (5) , and as well as other new results can be obtained, via the above approach, by making suitable choices for V and Φ. As a first example, note that the choice easily yields the following important result due to J. Wang and P. Niu [34]. Corollary 1. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q and let α ∈ R, 1 < p < Q + α, γ > −1. Then the inequality on B N , we recover the weighted L p Hardy type inequality (3.40) presented in [11].
Corollary 2. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q and let Q = p > 1, α < −1. Then the inequality is valid for all φ ∈ C ∞ 0 (B N ). One can however apply the Theorem 3.1 to obtain other new inequalities on G. For instance, let us take then we readily get the following result.
Corollary 3. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q and let α ∈ R, Q + α > p > 1. Then the inequality On the other hand, by considering the functions we obtain Carnot version of the inequality (5.1) proved in [31] for the Euclidean context. Corollary 4. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q and let 1 < p < Q, α > 1. Then for every φ ∈ C ∞ 0 (G), one has

Another application of Theorem 3.1 with the special functions
leads us to the subsequent improved Carnot analogue of the inequality (42) established in [19] for the Euclidean setting.
We now take the units on B N , and we have the following Hardy type inequality (12) that was first proved in [24] for the Heisenberg group H n and then in [11] for polarizable Carnot groups G by slightly different methods. Corollary 6. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q and let Q = p > 1. Then for every φ ∈ C ∞ 0 (B N ), one has Uncertainty Principle Inequalities. The first and most famous uncertainty principle goes back to Heisenberg's seminal work, which was developed in the context of quantum mechanics [25]. The mathematical details of this principle were provided by Pauli and Weyl [35] and hence it is sometimes referred to as the Heisenberg-Pauli-Weyl inequality. In the Euclidean setting, the uncertainty principle inequality with sharp constant can be stated as where φ ∈ C ∞ 0 (R n ) . There exists much literature devoted to deriving various uncertainty principle type inequalities in the Euclidean and other settings (see [16], [27]). For instance, in polarizable Carnot groups G, Kombe [27] showed that for any φ ∈ C ∞ 0 (G) , the following inequality is valid We should mention that Theorem 3.1 does not only give us weighted Hardy inequalities but also gives the Heisenberg-Pauli-Weyl type inequalities with the best constant. For instance, we now consider the pair where α > 0, and we immediately obtain Then the above inequality takes the form Aα 2 + Bα + C ≤ 0 for every α ∈ R which implies that B 2 − 4AC ≤ 0. In other words, we have the inequality (14). Now we make the following special choices of functions V and Φ in Theorem 3.1 where α > 0. We get Arguing as above, we have the following version of the Heisenberg uncertainty principle inequality.
Corollary 7. Let G be a polarizable Carnot group with homogeneous norm N = u Finally, let us consider the pair V ≡ 1 and Φ = e −αN , α > 0 then we get following inequality.
Corollary 8. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q . Then for every φ ∈ C ∞ 0 (G), one has

WEIGHTED HARDY TYPE INEQUALITIES ON CARNOT GROUPS 2017
4. Two-Weight Hardy type inequalities with remainders. We now prove an improved two-weight L p Hardy type inequality via a differential inequality involving a general nonnegative weight function V , the homogeneous norm N and a positive smooth function δ.
Theorem 4.1. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q and let Ω be a bounded domain with smooth boundary ∂Ω in G. Assume V is a nonnegative C 1 −function and δ is a positive C ∞ −function such that Proof. For any φ ∈ C ∞ 0 (Ω) we set ϕ := N −γ φ with γ < 0, a constant that will be chosen later. By direct computation we have Applying the inequality (9) with a = γN γ−1 ϕ∇ G N and b = N γ ∇ G ϕ yields Multiplying both sides of (16) by V (x) N α and then using integration by parts gives Taking into account that ∆ G N = (Q − 1) |∇ G N | 2 N and ∆ G,∞ N = 0 we obtain on a bounded domain Ω with smooth boundary in G, where R > sup x∈Ω N (x) . It is obvious that they fulfill all hypotheses in the Theorem 4.1, hence we have the weighted L p Hardy type inequality containing a logarithmic remainder.
Corollary 9. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q and let Ω be a bounded domain with smooth boundary ∂Ω in G. Then for all φ ∈ C ∞ 0 (Ω), we have where Q + α > p ≥ 2, α ∈ R, c p > 0 and R > sup x∈Ω N (x) .

Remark 3.
In the Abelian case, when G = R n , with the ordinary dilations, one has G = V 1 = R n so that Q = n. Now it is clear that the above inequality with the homogeneous norm N (x) = |x| and α = 0 recovers the inequality (1.4) proved by Adimurthi et al. in [1].
We now apply Theorem 4.1 with the pair V ≡ 1 and δ = log(log R N ), R > e sup x∈Ω N (x) , and we obtain the following result including a different logarithmic remainder.
Corollary 10. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q and let Ω be a bounded domain with smooth boundary ∂Ω in G. Then for all φ ∈ C ∞ 0 (Ω), we have where Q + α > p ≥ 2, α ∈ R, c p > 0 and R > e sup x∈Ω N (x) .
On the other hand, by making the choices V = e N and δ = e −N , we derive the subsequent two-weight L p Hardy type inequality involving two nonnegative remainders.
Corollary 11. Let G be a polarizable Carnot group with homogeneous norm N = u 1 2−Q and let Ω be a bounded domain with smooth boundary ∂Ω in G. Then for all φ ∈ C ∞ 0 (Ω), we have where Q + α > p ≥ 2, α ∈ R and c p > 0.
Another consequence of the Theorem 4.1 with the special functions V ≡ 1 and δ = R − N on the N -ball B N in G is the following inequality.
Remark 4. The lack of regularity on the above choices can be readily handled by replacing the function N with a suitable N and then passing to the limit as −→ 0.