A Note on Commutators of the Fractional Sub-Laplacian on Carnot Groups

In this manuscript, we provide a point-wise estimate for the $3$-commutators involving fractional powers of the sub-Laplacian on Carnot groups of homogeneous dimension $Q$. This can be seen as a fractional Leibniz rule in the sub-elliptic setting. As a corollary of the point-wise estimate, we provide an $(L^{p},L^{q})\to L^{r}$ estimate for the commutator, provided that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-\frac{\alpha}{Q}$.


Introduction and Main Results
In this paper we propose to investigate commutator type estimates for fractional powers of the sub-Laplacian on a Carnot group G. In fact, we will be focusing mainly on a fractional type Leibniz rule. These kind of estimates are very effective in studying the regularity or certain fractional PDEs. The study of these type of PDEs can be of many interests, namely, from an analytical point of view, but also from a geometric point of view, after the paper of [7], where a family of sub-elliptic fractional conformally invariant operators were exhibited. These operators are the CR parallel to the GJMS operators [8] defined in the Riemannian setting.
Because of this geometric motivation, the Riemannian GJMS operators where intensively investigated and in the special case of the euclidean space R n , the fractional powers of the Laplacian became a main research focus. For instance, fractional type Leibniz rules were proved and investigated in [11] using potential analysis and then another alternative proof was provided using harmonic extensions in [9]. As is the case of local operators, the main tool to study regularity of certain PDEs is localizing the functions with a cut-off. For non-local operators this procedure becomes more complicated. That is why a Leibnitz type rule is valuable in this case. In our work, we will follow the potential analysis approach developed in [11] in order to provide a fractional sub-elliptic Leibniz type rule in a general Carnot group. Then, in Section 6 we discuss the case of the Heisenberg group and the two kinds of fractional type operators related to the sub-Laplacian.
We start here by defining the commutators and introducing the main results. Given a Carnot group G we denote by Q its homogeneous dimension. For this purpose, we fix 0 < α < Q, u, v ∈ S(G), the 3-commutator H α (·, ·) is then defined by Also, given 0 < τ < Q and β, δ > 0 such that β + α < min{1, τ } we define the operator T τ,β,δ by Our first result, involves a point-wise estimate, similar to the one in [11], that can be seen as a fractional type Leibniz rule for 0 < α < 2.

Preliminaries and Notations
In what follows, G will be a Carnot group with homogeneous dimension Q and ∆ b will be its sub-Laplacian as defined in Folland [4] and Folland-Stein in [6]. We will start by recalling some definitions and properties of the fractional sub-Laplacian and adapt the following comparaison notations: We write f (x) ≈ g(x), if there exists a constant C > 0 such that Also, we will write f (x) g(x) if there exists a constant C > 0 such that Recall that there exists a gauge | · | on G that we can assume symmetric, that induces a quasidistance (see [1] Chap 5, for more details). Moreover we have that the existence of two positive constants c and C such that We can assume without loss of generality that c < 1 and C > 1. We provide now a very usefull lemma, that we will be using later on in our computations Proof: A version of this Lemma was proved in [4] for 2|y| < |x|. In fact, using the homogeneity of f , we can rescale and assume that |x| = 1 and |y| ≤ 1 2 , therefore |xy| = 0 and hence, the proof follows from the regular intermediate calue theorem, leading to It remains then to prove the case when |y| ≈ |x|. Again, if |xy| > 1 4 , then the proof is done, or else, we have that |xy| < 1 4 , then |y| > 1 2 , hence, we have that We move now to the definition of the fractional sub-Laplacian and its inverse. For this purpose, we consider the heat semi-group defined by H t = e t∆ b , then one can define as in [4,6], the following fractional operators for α > 0: for any integer k > α. We have now the following: converges absolutely and it satisfies the following properties: • For f ∈ L p (G) and 1 < p < ∞, we have that As a corollary of this proposition one has α−Q defines a G-homogeneous norm, smooth away from the origin and it induces a quasi-distance that is equivalent to the left-invariant Carnot-Caratheodory distance. In a similar way, one can define the functionR α , introduced in [3 Again, it is easy to see thatR α is G-homogeneous of degree α − Q and Using this function, one can define another representation for the fractional sub-Laplacian, that will be fundamental in the proof of our results.
For the sake of notation, we will use in this paper the following notation: 3 Proof of Theorem 1.1 We are interested in the study of the commutator We take now Therefore, one needs to study the kernel k defined by: In order to simplify the computations, we introduce the kernel k τ defined by and we split the space into three parts using the following characteristic functions: We first notice that Therefore, using Lemma 2.1, we have, for 0 < δ < min{τ, 1}, that and similarly Now notice that using "polar coordinates", if one takes 0 < δ 1 + δ 2 − α < ε, where ε is positive and small, we have for ǫ > 0 that This yields, Therefore, one can state that proving the desired result.

Proof of Theorem 1.2
From now on, we will let τ > 0 and two non-negative numbers β and δ so that β + δ < min{τ, 1}. We then, define the commutator T τ,β,δ (u, v) by Notice that As in the previous proof, we propose to study the kernel k defined by But, in this situation, we need a better splitting of the space, adapted to the different quantities involved. So we define the sets We can verify that each collection of sets exhaust (G) 4 . In a first step, we will split the kernel k so that k ≤ k 1 + k 2 where Step I: Estimate on C 1 ∪ C 2 .
Notice that on C 1 ∪ C 2 we have that Therefore, Using the same procedure that was done in Section 3, we have the existence of L > 0 and numbers s k ,s k > 0 and t k ,t k ∈ (0, τ ) withs k < ε, and s k + t k =s k +t k = τ − β − δ so that which is the required estimate. On the other hand, the estimate for k 1 is not as straight forward as it is for k 2 . Indeed, we will split the set C 1 ∪ C 2 using the sets B j , j = 1, 2, 3. For instance, in the set (C 1 ∪ C 2 ) ∩ B 1 , we have that Therefore, |z −1 x| and |wy −1 z| are comparable and we can use Lemma 2.1, to get for any ε ∈ [0, 1] Now notice that on C 1 , we have that Thus, So if we take ε ∈ (β, 1) and integrate on w, we get Therefore, we have the desired inequality if one chooses ε ∈ (δ +β, τ ), s = τ −ε and t = ε−β −δ.
The case of C 1 ∪ C 2 ∩ B 2 : Notice here that Therefore, we have that Again, on B 2 we have that |wy −1 x|χ B 2 > 2|z −1 x|χ B 2 . Hence, This yields So if we integrate over w while taking ε > β, we get and therefore, we have the same conclusion as in B 1 .
Step 2: Estimate on C 3 : In this step we will rewrite the kernel k by noticing that and we will be using the splitting induced by the A i , i = 1, 2, 3. So first we will look at k on the set Notice that on C 3 , |y −1 x| and |wy −1 x| are comparable. Moreover, So again |wy −1 z| and |y −1 z| are comparable. Therefore, by Lemma 2.1, we have for ε ∈ [0, 1] that Taking ε = ε 1 + ε 2 , we find that Once again, we choose ε 1 > δ and ε 2 > β, and ε = ε 1 + ε 2 < min{1, τ }, to have after integration over w on C 3 Thus, We focus now on the last part, that is C 3 ∩ A 3 . Notice that since |z −1 x| > |y −1 z| on A 3 , and |y −1 x| ≈ |wy −1 x| on C 3 , we have from (4.10), that (4.13) We will estimate k 3 and k 4 separately. For k 3 we have that which is similar to case C 3 ∩ (A 1 ∪ A 2 ). Hence, the desired inequality holds .
We move now to the term k 4 . We have that So in the set {2|w| < |y −1 z|}, we have by Lemma 2.1 and we are in the same situation as k 3 above. So we treat the estimate in 2|w| ≥ |y −1 z|.
On the other hand, since integrating again with respect to w yields and in the end, Which finishes the proof.

The case of the Heisenberg group
In the case where G = H n , and therefore Q = 2n + 2, there are two kinds of fractional powers of the sub-Laplacian. Indeed, there is the usual fractional sub-Laplacian as defined above in this manuscript, and the conformally invariant one that we will denote by L α , we will refer to this last one as the geometric fractional sub-Laplacian. As in [10], these two operators can be distinguished in terms of their spectral Fourier multipliers (we refer the reader to [12] for the definition of the Fourier transform and Fourier spectral multipliers on the Heisenberg group). Indeed, we have that the operator (−∆ b ) corresponds to the spectral multiplier A(k, λ) := (2k + n)|λ|. On the other hand, the spectral multiplier of L α is A(k, λ, α) := (2|λ|) α 2 Γ( 2k+n 2 + 2+α 4 ) Γ( 2k+n 2 + 2−α 4 ) .
In particular, we have thatÃ (k, λ, 2) = A(k, λ, 2) = A(k, λ), and therefore L 2 = −∆ b . As opposed to the case of a general Carnot group and the potential R α , the fundamental solution G α of L α can be computed explicitly. Indeed, as shown in [10] extending the result in ?? for the case of the sub-Laplacian, there exists a constant c n,α such that G α (x) = c n 1 |x| Q−α .
Also, we have a similar integral representation of L α whereR −α can be computed explicitely. Indeed, there exists a constantc n,α , such that for u ∈ C ∞ 0 (H n ) we have L α u(x) =c n,α H n u(x) − u(y) |y −1 x| Q+α dy.