Commutators associated with Schrödinger operators on the nilpotent Lie group

Assume that G is a nilpotent Lie group. Denote by L=−Δ+W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L=-\Delta +W $\end{document} the Schrödinger operator on G, where Δ is the sub-Laplacian, the nonnegative potential W belongs to the reverse Hölder class Bq1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B_{q_{1}}$\end{document} for some q1≥D2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q_{1} \geq \frac{D}{2}$\end{document} and D is the dimension at infinity of G. Let R=∇(−Δ+W)−12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{R}=\nabla (-\Delta +W)^{-\frac{1}{2}}$\end{document} be the Riesz transform associated with L. In this paper we obtain some estimates for the commutator [h,R]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[h,\mathcal{R}]$\end{document} for h∈Lipνθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h\in \operatorname{Lip}^{\theta }_{\nu }$\end{document}, where Lipνθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{Lip}^{\theta }_{\nu }$\end{document} is a function space which is larger than the classical Lipschitz space.


Introduction
Assume G to be a connected and simply connected nilpotent Lie group and g to be its Lie algebra identified with the space of left invariant vector fields. Given X = {X 1 , . . . , X l } ⊆ g, a Hörmander system of left invariant vector fields on G. Let = l i=1 X 2 i be the sub-Laplacian on G associated with X and the gradient operator ∇ be denoted by ∇ = (X 1 , . . . , X l ). Following [1], one can define a left invariant metric d associated with X which is called the Carnot-Carathéodory metric: let x, y ∈ G, and where γ is a piecewise smooth curve satisfying (1) where d and D denote the local dimension and the dimension at infinity of G, and there is D ≥ d > 0. At this time, the Lie group G is also called a Lie group of polynomial growth. If G is a stratified Lie group, then D = d (cf. [1]). Also, there exist positive constants C 2 , C 3 > 1 such that Throughout this paper, we always assume that d ≥ 2.
Let L = -+ W be the Schrödinger operator, where is the sub-Laplacian on G and the nonnegative potential W belongs to the reverse Hölder class B q 1 for some q 1 ≥ D 2 and D > 3. The Riesz transform R associated with the Schrödinger operator L is defined by Let b be a locally integrable function on G and T be a linear operator. For a suitable function f , the commutator is defined by [b, T]f = bT(f ) -T(bf ). Many researchers have paid attention to the commutator on R n . It is well known that Coifman, Rochberg and Weiss [4] proved that [b, T] is a bounded operator on L p for 1 < p < ∞ if and only if b ∈ BMO(R n ), when T is a Calderón-Zygmund operator. Janson [5] proved that the commutator is bounded from L p (R n ) into L q (R n ) if and only if b ∈ Lip ν (R n ) with ν = ( 1 p -1 q )n, where Lip ν (R n ) is the Lipschitz space. Sheng and Liu [6] proved the boundedness of the commutator [b, R] from the Hardy space H p L (R n ) into L q (R n ) when b belongs to a larger Lipschitz space. Comparatively, there has been much less research on the commutator on nilpotent Lie groups. The goal of this paper will be to obtain some estimates for the commutator related to the Schrödinger operator on nilpotent Lie groups. The complicated structure of nilpotent Lie group will bring some essential difficulties to our estimates in the following sections.
Note that a non-negative locally L q integrable function W on G is said to belong to B q (1 < q < ∞) if there exists C > 0 such that the reverse Hölder inequality holds for every ball B in G.
We first introduce an auxiliary function as follows.
Now we define the space Lip θ ν (G) on the nilpotent Lie group.
Definition 2 Let θ > 0 and 0 < ν < 1, the space Lip θ ν (G) consists of the functions f satisfying holds true for all x, y ∈ G, x = y. The norm on Lip θ ν (G) is defined as follows: It is easy to see that this space is exactly the Lipschitz space when θ = 0 if G is a stratified Lie group (cf. [7] and [8]). We also introduce the following maximal functions.
Definition 3 Let f ∈ L 1 loc (G). For 0 < γ < D, the fractional maximal operator is defined by where the supremum on the right-hand side is taken over all balls B ⊆ G and r is the radius of the ball B.
We are in a position to give the main results in this paper.
where D denotes the dimension at infinity of the nilpotent Lie group G. Let Denote the adjoint operator of R byR = (-+W ) -1 2 ∇. Then, for any h ∈ Lip θ ν (G), 0 < ν < 1, the commutator [h,R] is bounded from L q (G) into L p (G), 1 We immediately deduce Corollary 1 by duality.
where D denotes the dimension at infinity of the nilpotent Lie group G. Let Then, for any h ∈ Lip θ ν (G), Throughout this paper, unless otherwise indicated, C will be used to denote a positive constant that is not necessarily the same case at each occurrence and it depends at most on the constants in (3) and (6). We always denote δ = 2 -D q 1 . By A ∼ B, we mean that there exist constants C > 0 and c > 0 such that c ≤ A B ≤ C.

Estimates for the kernels of R andR
In this section we recall some estimates for the kernels of Riesz transform R and the dual Riesz transformR, which have been proved in [3].

Lemma 4
There exist constants C, l 0 > 0 such that

Lemma 5
There exist constants C > 0 and l 1 > 0 such that Using Lemma 4, we immediately have the following lemma.

Lemma 6
There exist l 0 > 0, C > 0 such that, for any x and y in G, Let Γ (x, y, λ) denote the fundamental solution for the operator -+ W + λ, namely, .
Particularly, if W ∈ B q 1 for some q 1 ≥ D, then there exists C N > 0 such that, for x = y, By the functional calculus, we may write where Similarly, the adjoint operator of R is defined to bẽ whereK We recall estimates of the kernels for R andR (cf. [3]). and and . (16) and for some δ > 0 and 0 < d(x, x ) < d(x,y) 16 . If W ∈ B q 1 for some q 1 ≥ D, then and 3 Some technical lemmas and propositions , then there exists a positive constant C such that, for all B = B(x, r) with x ∈ G and r > 0, where we have used Lemma 6 in the penultimate inequality.
Similar to the proof of Proposition 1, we immediately get the following.
For the proofs of Proposition 2 and Proposition 3, one can refer to [3].

Proposition 4
There exists a sequence of points {x k } ∞ k=1 in G, so that the set of critical balls is a sequence of balls as in Proposition 4, then The above propositions have been proved in [9] and [10] in the case of a homogeneous space, respectively.

Lemma 11
Let W ∈ B q 1 for q 1 ≥ D 2 , and let and h ∈ Lip θ ν (G). Then, for q 2 < m < ∞, there exists a positive constant C such that holds true for all f ∈ L m loc (G) and every ball Q = B(x 0 , ρ(x 0 )), where M mν is a fractional maximal operator.
Proof Throughout the proof of the lemma, we always assume D 2 ≤ q 1 < D. Let f ∈ L p (G) and Q = B(x 0 , ρ(x 0 )). For we need to consider the average on Q for each term. By the Hölder inequality with m > q 2 and Proposition 1, for x ∈ Q, and using (17) in Lemma 9, we splitRf 2 (x) into two parts To deal with I 1 (x), noting that ρ(x) ∼ ρ(x 0 ) and d(z, x) ∼ d(z, x 0 ), we split d(z, x 0 ) > 2ρ(x 0 ) into annuli to obtain Secondly, we consider the term I 2 (x). We have, for x ∈ Q, Let q 2 < m < D. Using the Hölder inequality and the boundedness of the fractional integral I 1 : L m → L q 1 with Since W ∈ B q 1 , we obtain where in the last two inequalities we have used doubling measure and the definition of ρ, respectively. Therefore, Finally, observing that and using that 1 We still split f = f 1 + f 2 . Choose q 2 <m < m and set t =m m m-m . Using the boundedness ofR on Lm(G) and the Hölder inequality, we get where we have applied Proposition 1 to the last but one inequality. Similarly, for x ∈ Q and using (17) in Lemma 9, we have We start by observing that for 1 ≤m < m, t =m m m-m , and by Lemma 6, ForĨ 1 (x), using (24) withm = 1, we havẽ To deal withĨ 2 (x), we discuss as in the estimate for I 2 (x) with (hh Q )f instead of f and m andq 1 instead of m and q 1 , but we cannot avoid to discuss the different cases where 2 k ρ(x 0 ) ≥ 1 and 2 k ρ(x 0 ) < 1. Let Using (24), we similarly havẽ where we choose N large enough to ensure the above series converges.
Lemma 12 LetR = (-+ W ) -1 2 ∇ be the adjoint operator of the Riesz transform R. Then there exists C > 0 such that, for any f ∈ L m loc (G) and h ∈ Lip θ ν (G), Proof Let f ∈ L m loc (G), x ∈ G and a ball B = B(x 0 , r) with x ∈ B and r < ρ(x 0 ), > 0, we need to control J = 1 |B| B |[h,R]f (y) -c| dy by the right-hand side of (25) for some constant c, which will be designated later.