Sharp Hardy Identities and Inequalities on Carnot Groups

1 Introduction

Let G denote a Carnot group with sub-Laplacian ℒ=∑j=1mXj2, horizontal gradient ∇H=(X1,…,Xm), homogeneous dimension Q, and ℒ-gauge d (see preliminaries for precise definitions). In this paper, we employ factorization of differential operators and Bessel pairs to obtain weighted Hardy identities on Carnot groups G of the form

for u∈C0∞⁢(BR⁢(0)∖{0}), where (V,W) is a pair of positive C1-functions on (0,R) which satisfy the condition that there exists a positive C1-solution φ:(0,R)→ℝ to the differential equation

Such a pair of functions is called a Bessel pair. Here, dx is a Haar measure on G, and BR⁢(0) is the gauge ball determined by d with radius R and center being the origin 0. This result is contained in Theorem 2 below. As a particular important application, we obtain weighted Hardy identities on the Heisenberg group HN=ℝ2⁢N+1, the simplest and most important nonabelian Carnot group. It is important to note that, for an appropriately chosen Bessel pair, our results recover the Hardy identity corresponding to known Hardy inequalities on the Heisenberg group and more general Carnot groups, thereby strongly improving these known inequalities by finding exact remainder terms.

In addition to these identities, we also obtain Hardy identities of the form

(1.2)

where ∇H is the horizontal gradient with respect to certain smooth vector fields X1,…,Xm, and dK is the Carnot–Carathéodory distance (also known CC distance) to a closed subset K with respect to this system of vector fields. A key point here is that, by [24], the CC distance function dK satisfies the horizontal eikonal equation |∇H⁡dK|=1, unlike a general homogeneous gauge d for which |∇H⁡d|=1 is typically false. This is the content of Theorem 16.

We recall that, on the Heisenberg group HN=ℝ2⁢N+1, the sharp Hardy inequality

was established by Garofalo and Lanconelli in [14]. The Lp-Hardy inequality on the Heisenberg group for 1<p<Q=2⁢N+2 was established by Niu, Zhang and Wang [26] by using the Picone’s identity for the p-subLaplacian (see also Zhang and Niu [28])

When d is replaced by the Carnot–Carathéodory metric dcc, then there holds

for some constant c>0. See for example [13, 22, 27]. Using the fact that dcc and d are equivalent metrics, one may also obtain a Hardy inequality of the form (1.4) with dcc replaced by d, though the best constant appears to be unknown in either case. Nevertheless, while (1.3) is sharp, there are no extremizers, and so it is natural to try to improve this inequality by adding nontrivial nonnegative terms to the right-hand side of (1.3). As mentioned above, we in fact determine exact remainder terms so that (1.3) becomes

Therefore, we obtain a strict improvement of the known Hardy inequality (1.3), and the counterparts for Carnot groups. This is the content of Theorem 6.

The approach used here is inspired by the recent work of Lam, Lu and Zhang in [20] where they used factorization of differential operators and Bessel pairs to prove, among other things, various Hardy identities on the half-space

Their main identity theorem is stated as follows.

In addition to this theorem, Lam, Lu and Zhang also proved the following Hardy identities on strips of the form A2⁢R=ℝN-1×(0,2⁢R) which imply in particular Hardy identities on the half-space ℝ+N with the Euclidean norm |x| replaced by the distance function d∞⁢(x)=xN, the height from the boundary. Their result is stated as follows.

Actually, similar half-space Hardy inequalities have been established for the Heisenberg group. We recall the following result of Luan and Yang given in [23].

See [18] for similar Hardy inequalities on different Heisenberg half spaces.

In the spirit of Theorems A, B and 1.5, we establish the Hardy identities on strips

and half-gauge balls

Here dR⁢(z,t)=min⁡{t,2⁢R-t} and d∞⁢(z,t)=t is the vertical height function. The identity on strips implies in particular improvements of (1.5). These identities are the contents of Theorems 8 and 12.

We note that, in [23], (1.5) has an interesting improved version by adding to the right side a nonnegative nontrivial term with weight given in terms of the homogeneous gauge d; however, they do not obtain exact remainder terms. Moreover, this inequality is not coordinate independent, and so the best constant appearing in front of ∫|x|2⁢t-2⁢|u|2⁢𝑑x depends on the coordinates chosen for the left invariant vector fields on HN. In particular, our choice of coordinates are different and the best constant turns out to be 116 for our choice of coordinates.

We also mention that, our Hardy identities of the form (1.2) for the general distance function dK are in the spirit of the following Hardy identities for general distance functions (for example, the distance to a subset) on Euclidean space obtained by Lam, Lu and Zhang in [21].

Among the particularly interesting applications of the Hardy identity (1.1) are Hardy identities and inequalities which improve Hardy inequalities of the form

by adding positive nontrivial L2-terms to the right-hand side. In fact, Brezis and Vázquez established the remarkable inequality

where Ω⊂ℝN, N>2, is any bounded domain, u∈H01⁢(Ω), and z0 is the first zero of the Bessel function J0. (See also [4] for the distance function the boundary.) We introduce similar inequalities on Carnot groups (see Theorem 6) by establishing identities of the form

where J0;R⁢(r)=J0⁢(z0⁢rR). This is done by choosing an appropriate Bessel pair (V,W) and applying (1.1). We note that such Brezis–Vázquez-type inequalities [5] on Carnot groups were observed first by Kombe in [19].

We take the opportunity to mention some forthcoming works which continue the program of establishing Hardy identities by use of Bessel pairs, including the work in this paper and the work of Lam, Lu and Zhang already mentioned. First, we mention that Duy, Lam and Lu establish recently in [7] the following Lp-Hardy identities by use of p-Bessel pairs.

We also point out here that we have established in [10] the Lp-Hardy identities for general distance functions in the Euclidean setting in which we generalize Theorem D to the Lp-setting. We have also established in [12] the Lp-Hardy identities for domains in Euclidean space. Our Lp-Hardy identities from [10] for general distance functions is stated as follows.

Another important geometric inequality is the Hardy uncertainty principle given by

The uncertainty principle for the Heisenberg group may be found in [14]. Using factorization, we also obtain a new family of inequalities which generalize (1.6) to a family of uncertainty principles whose weights are given in terms of Bessel pairs. These inequalities are of the form

and are the content of Theorem 13. These inequalities appear to be entirely new even for Euclidean space, and they give a new interpretation of the classical Hardy uncertainty principle.

We recall that Ghoussoub and Moradifam introduced in [15] (see also their book [16]) the notion of Bessel pairs to enhance and unify several improved Hardy-type inequalities in the literature for the Euclidean setting. They established the following important characterization theorem.

It is worth mentioning that these results extend straightforwardly to more general settings. We describe two such generalizations now. The first is the case of simply connected nilpotent Lie groups G where we specify a system of left invariant vector fields X={X1,…,Xm} which bracket generate the Lie algebra of G. Then, as was shown by Nagel, Ricci and Stein in [25], the sub-Laplacian ℒ=∑j=1mXj2 has a global fundamental solution ΓX⁢(x,y). Then (1.1) holds if we take ∇H=(X1,…,Xm), d⁢(x)=dX⁢(x)=ΓX⁢(x,0)2-q for some q>2, and BR(0)={dX(x)<R}.

The second is the case that X={X1,…,Xm} is a system of Hörmander vector fields homogeneous of degree -1 (see the preliminaries for precise details). In fact, recently it was shown in [1] that, for these kinds of vector fields, the Hörmander operator ℒ=∑j=1mXj2 admits a global fundamental solution ΓX⁢(x,y), and one can, similar to the nilpotent case, take d⁢(x)=ΓX⁢(x,0)2-q, where q is the homogeneous dimension arising from the family of dilations used to define X. Analogous Hardy identities may be formulated just as in the nilpotent case just mentioned. These identities are the contents of Theorem 20.

Finally, we mention that the Lp-weighted Hardy’s identities and inequalities (for p≠2) with the p-Bessel pairs on Carnot groups and for Hörmander vector fields have been established in [11]. This is inspired by the earlier works of the authors on Lp-weighted Hardy’s identities and inequalities (for p≠2) with the p-Bessel pairs in Euclidean spaces [7, 10, 12].

2 Preliminaries

We begin by recalling facts about general vector fields, and then proceed to recalling facts and properties about the Heisenberg group and more general Carnot groups. Thus, let X={X1,…,Xm} denote a collection of smooth vector fields on ℝN, i.e.,

with ai⁢j∈C∞⁢(ℝN). A subunit curve for X is a Lipschitz continuous curve γ:[0,T]→ℝN, T>0, such that there exist measurable functions h1,…,hm:[0,T]→ℝ satisfying

for almost every t∈[0,T]. The Carnot–Carathéodory distance between two points x,y∈ℝN is then defined to be

In case the vector fields X1,…,Xm are such that the Lie algebra generated by X is N-dimensional at every point in ℝN, then X is said to satisfy Hörmander’s condition. The Chow–Rashevsky theorem asserts that, if X satisfies Hörmander’s condition, then dcc⁢(x,y)<∞ for all x,y∈ℝN. In particular, dcc becomes a metric on ℝN.

Associated with X is the Hörmander sum of squares operator ℒX=∑j=1mXj2, and, in case X satisfies the Hörmander condition, it is a well-known result of Hörmander that ℒX is hypoelliptic. We also define the horizontal gradient ∇H and divergence as follows:

where a=(a1,…,am):ℝN→ℝm is any sufficiently regular function, and Xj* is the formal adjoint of Xj.

Now, suppose for convenience that ℒX has a global fundamental solution ΓX⁢(x,y). Then motivated by the fact that, for N≥3, the function Γeu⁢(x,y)=1N⁢(N-2)⁢ωN⁢|x-y|2-N, where ωN is the volume of the Euclidean unit ball, is the fundamental solution for the Euclidean Laplacian, and |x|=Γeu⁢(x,0)12-Q defines the Euclidean norm, we may define the function dX⁢(x)=ΓX⁢(x,0)12-N and use it to define the corresponding ball

An important family of Hörmander vector fields are the left-invariant vector fields of the Heisenberg group HN=ℝ2⁢N+1. Recall that H may be defined as ℝ2⁢N+1 equipped with the group law

where a,b,a′,b′∈ℝN are horizontal coordinates, and t,t′∈ℝ are central coordinates. At a given point (z,t)=(x1,…,xN,y1,…,yN,t)∈HN, the corresponding left invariant vector fields are given by

for i=1,…,N. These vector fields satisfy the Heisenberg relation

and so the Lie algebra 𝔥 of H is a graded nilpotent Lie algebra of step 2. The corresponding sub-Laplacian is given by

and the horizontal gradient is given as above.

The Heisenberg group is a homogeneous group in the sense that it has a family of anisotropic dilations δλ:H→H given by δλ⁢(z,t)=(λ⁢z,λ2⁢t) for λ>0. The corresponding homogeneous dimension of H is given by Q=2⁢N+2. Moreover, H has a homogeneous gauge given by

and this norm is such that d2-Q is a scalar multiple of the fundamental solution of ℒ. In particular, for an appropriate c>0, the balls BX,c⁢R⁢(0) are exactly the balls gauge balls BR⁢(0)={x∈HN:d⁢(x)<R}. The Haar measure, which will be denoted by dx or d⁢z⁢d⁢t, on H is given by the Lebesgue measure d⁢x=d⁢z⁢d⁢t on ℝ2⁢N+1. The Haar measure and homogeneous gauge enjoy the following homogeneity properties:

The Heisenberg group belongs to a large family of nilpotent groups, called the Carnot groups, which are important in many fields of analysis and have a rich geometric structure. We recall Carnot groups now. First, recall that a stratified Lie algebra 𝔤 of step r is a Lie algebra with subspaces V1,…,Vr satisfying

If G is a simply connected Lie group with a stratified Lie algebra 𝔤, then G is called a Carnot group. In fact, if Nj=dim⁡Vj, and using the exponential map, we may identify G with ℝN1×⋯×ℝNr so that each point g∈G is identified with a point x=(x(1),…,x(r)) in ℝN1×⋯×ℝNr so that x(j)∈ℝNj; such an identification will always be made. Just as for the Heisenberg group, the Carnot groups have a natural family of anisotropic dilations δλ:G→G given by

The homogeneous dimension Q of the group is defined to be Q=∑j=1mj⁢dim⁡Vj. Let d⁢x denote a Haar measure on G, which can be taken to be the Lebesgue measure on ℝN1×⋯⁢ℝNr. The left-invariant vector fields of the first layer on G may be written in the following way:

where aj,k are degree h-1 polynomials which are homogeneous with respect to δλ. These vector fields generate the Lie algebra of G and so every left invariant vector on G may be obtained from taking linear combinations of brackets of the Xj. The horizontal gradient, horizontal divergence, and sub-Laplacian of G are defined as before.

We now recall the notions of symmetric homogeneous norms and gauges in the following definitions.