Lipschitz estimates for commutators of singular integral operators associated with the sections

Let H be Monge-Ampère singular integral operator, b∈LipFβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b\in Lip_{\mathcal{F}}^{\beta}$\end{document}, and 1/q=1/p−β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1/q=1/p-\beta$\end{document}. It is proved that the commutator [b,H]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[b,H]$\end{document} is bounded from Lp(Rn,dμ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}(\mathbb{R}^{n},d\mu)$\end{document} to Lq(Rn,dμ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{q}(\mathbb{R}^{n},d\mu)$\end{document} for 1<p<1/β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1< p<1/\beta$\end{document} and from HFp(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{p}_{\mathcal{F}}(\mathbb{R}^{n})$\end{document} to Lq(Rn,dμ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{q}(\mathbb{R}^{n},d\mu)$\end{document} for 1/(1+β)<p≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1/(1+\beta)< p\leq1$\end{document}. For the extreme case p=1/(1+β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p=1/(1+\beta)$\end{document}, a weak estimate is given.


Introduction
In , Caffarelli  As is well known, linear commutators are naturally appearing operators in harmonic analysis that have been extensively studied already. In general, the boundedness results of commutators in harmonic analysis can be used to characterize some important function spaces such as BMO spaces, Lipschitz spaces, Besove spaces and so on (see [-]). Coifman et al. [] applied the boundedness to some non-linear PDEs, which perfectly illustrate the intrinsic links between the theory of compensated compactness and the classical tools of harmonic and real analysis. As for some other essential applications to PDEs such as characterizing pseudodifferential operators, studying linear PDEs with measurable coefficients and the integrability theory of the Jacobians, interested researchers can refer to [-]. It is perhaps for this important reason that the boundedness of commutators attracted vast attention among researchers in harmonic analysis and PDEs. Thus, it is meaningful to identify the behaviors of commutator [b, H] associated with the Monge-Ampère equation. In the sense of Euclidean space R n , the boundedness of commutator [b, T] with b ∈ Lip β acting on Lebesgue spaces, is easily obtained by the inequality where T is a Calderón-Zygmund singular integral operator and I β is the Riesz potential of order β. However, we cannot find a suitable operator to control the commutator [b, H] with b ∈ Lip β F -just as controlling of [b, T] with b ∈ Lip β -due to the particularity of the operator H. Therefore, in this paper we investigate [b, H] directly, and obtain some relatively important properties.
This paper is organized as follows. In Section , we recall some elementary properties of sections. In the first part of Section , we demonstrate the (L p , L q ) boundedness of [b, H] when  < p < /β and /q = /pβ. It is worth mentioning that boundedness of [b, H] from L  (R n , dμ) to weak L /(-β) (R n , dμ) is also obtained, which indicates the differences between the commutator [b, H] with b ∈ Lip β F and that commutator with b ∈ BMO F . Based on these differences, in the second part of Section , we further discuss the behavior of the commutator [b, H] acting on Hardy spaces H p F (R n ), and we see that the commutator x ∈ R n and t > } is monotone increasing in t, i.e., S(x, t) ⊂ S(x, t ) for t ≤ t which satisfies the following criteria: (A) There exist constants K  , K  , K  and  ,  such that given two sections S(x  , t  ), S(x, t) with t ≤ t  satisfying S(x  , t  ) ∩ S(x, t) = ∅, and given T, an affine transformation that "normalizes" S(x  , t  ), that is, and Here B(x, t) denotes the Euclidean ball centered at x with radius t. (B) There exists a constant δ >  such that given a section S(x, t) and y / ∈ S(x, t), if T is an affine transformation that "normalizes" S(x, t), then for any  < <  B T(y), δ ∩ T S x, ( -)t = ∅.
(C) t> S(x, t) = {x} and t> S(x, t) = R n . In addition, we also assume that a Borel measure μ which is finite on compact sets is given, μ(R n ) = ∞, and that it satisfies the doubling property with respect to F , that is, there exists a constant A such that for any section S(x, t) ∈ F . Throughout the paper, the letter C will denote a positive constant that may vary from line to line but remains independent of the main variables. We write A B to indicate that A is majorized by B times a constant independent of A and B, while the notation A ≈ B denotes both A B and B A. Finally, we denote L

Elementary properties of section and notions
According to [], the properties of (A) and (B) imply the following properties: (D) There exists a constant θ ≥ , depending only on δ, K  and  , such that for any y ∈ S(x, t), and its triangular constant is just the θ appearing in (D); that is, Also, Combining (.) and (.), one can see that there exists a constant n  >  and  n  > A such that Thus, (R n , d, μ) becomes a space of homogeneous type. Based on this, one can use the standard real analysis tools as the maximal function Mf and the sharp function M f . In this paper, both of them are defined on (R n , d, μ), namely Here and below, B is a d-ball and f B means the average of f on B. If we write BMO(R n ) := {f : Macías and Segovia [] have found that the quasi-metric d can be replaced by another quasi-metric ρ such that (R n , ρ, μ) is a normal space. Moreover, for the quasi-metric ρ there exist constants C >  and ∈ (, ) such that Let ρ satisfy (.) above and f be a continuous function on R n . Lin [] defined Lipschitz spaces Lip β F associated with sections as follows. Let be given in (.) above. Lin [] found that the function spaces β q,F and Lip β F coincide with equivalent norms for  < β < and  ≤ q ≤ ∞, where β q,F denotes the Campanato spaces associated to the family F of the section. Also, he proved that β q,F are the dual spaces of Hardy spaces H p F (R n ) (/ < p ≤ ). For a locally integral function b, the commutator of Cofiman-Rochberg-Weiss [b, H] is defined as follows: where H is defined by the formula with k(x, y) = i k i (x, y), and each kernel k i satisfies the following properties: where S i (x) = S(x,  i ) for any x ∈ R n , i ∈ Z. If T is an affine transformation that normalizes the section S i (y) then each k i satisfies the Lipschitz condition, |Tu -Tv|; and, finally, if T is an affine transformation that normalizes the section S i (x) then k i satisfies the Lipschitz condition, |Tu -Tv|.
Caffarelli and Gutiérrez [] obtained that H is bounded on L  μ . Subsequently, Incognito [] has given L p μ ( < p < ∞) and the weak-type (, ) estimate of H. Using the property (D) and defining a function σ on R n × R n by σ (x, y) = inf{t >  : y ∈ S(x, t)}, Incognito [] obtained the following conclusions: It is easy to see that and for a given section S(x, t), y ∈ S(x, t) if and only if σ < t.

Boundedness from
In this subsection, we discuss the property of the commutator acting on Lebesgue spaces. In order to prove the theorems above, it is necessary to give the following lemmas.

Lemma . ([])
Let  < p, δ < ∞ and ω ∈ A ∞ . There exists a positive C such that for any smooth function f for which the left-hand side is finite. y). Then there exists a constant C >  such that if σ (y  , x) ≥  k θ  σ (y  , y) and k ≥ .
Lemma . Suppose that b ∈ Lip β F , for  < β < . Let  < δ <  < r < /β. Then there exists a constant C >  such that for any smooth function f and every x ∈ R n , and where Proof Observe that for any constant λ For any fixed ball B = B(x, r).
Let λ and c B be constants to be fixed in the proof. We write For L  , we fix λ = b B . The Hölder inequality and (.) give us From Kolmogorov's inequality and (.), it follows that The estimates for L  , L  , and L  indicate that the proof is completed.

Lemma . ([])
Let local integral function f ∈ L  μ and α > . Then there exists a family of balls {B i } such that: exists an integer N ≥ , independent of f and λ, such that i Now, with the lemmas above, we state the proof of our results.
Proof of Theorem . From Lemma . and Lemma . with  < δ <  < r < p, it follows that Thus, the proof of the theorem is completed.
Proof of Theorem . For f ∈ L  μ and any α > , applying Lemma . with α replaced by α q  with q  =  -β , we obtain, with the same notation as in Lemma ., f = g + h = g + j h j , where . By Lemma ., it is easy to obtain the following properties: By (iii), it is concluded that For K  , we have From (v), Lemma ., (.), and (iii), it follows that The boundedness of [b, H] from L  μ to weak L  μ , (.), (v), and (iii) give us that Combining the estimates for K  , K  and K  , one can finish the proof.

Boundedness from H
In this subsection, we discuss the boundedness of the commutator [b, H] on Hardy spaces H p F (R n ), and obtain the following results in which the symbols  and n  are given in (.) and (.) respectively. Firstly, we recall the definition of the (p, ∞)-atoms and the atomic Hardy spaces H p F (R n ) with respect to a family F of sections and a doubling measure μ.
each a j is a (p, ∞)-atom and j |λ j | p < ∞}, where S(R n ) denotes the space of Schwartz functions and S (R n ) is the dual space of S(R n ). We define the H p F (R n ) norm of f by where the infimum is taken over all decompositions of f = j λ j a j above.
For the extreme case p = /( + β), one gets the following characterization theorem. In general, the (H p , L q ) boundedness of [b, H] fails for p = /( + β), then we give a weak estimate instead.
Next, we show the proofs of the theorems above.
Proof of Theorem . Without loss of generality, we assume that b Lip β F = . By Definition ., we only need to prove that for any (p, ∞)-atom a, Choosing  < p  < /β and /q  = /p β, and noting the (p  , q  ) boundedness of [b, H] and the size condition of a, one can get On the other hand, the cancellation condition of the atom a yields Lemma ., (.), (.), and (.) imply that .
Finally, noting that by the (q, q) boundedness of H, one obtains Combining the estimates for I and II, one can finish the proof.
 holds for any (/( + β), ∞) atom. Thus, we will study the behavior of [b, H] acting on any (/( + β), ∞) atom. Let a be an atom with supp ⊂ The estimation above yields that Equations (.) and (.) imply that the proof is completed.

Conclusions
The authors prove the commutator [b, H] is bounded from L p (R n , dμ) to L q (R n , dμ) for  < p < /β and from H p F (R n ) to L q (R n , dμ) for /( + β) < p ≤  and give the weak estimate at the extreme case p = /( + β) as well, which may give us an essential tool to study the linear or non-linear Monge-Ampère equation. It is a pity that we do not characterize the Lipschitz spaces Lip β F with the boundedness of it due to the particularity of the operator H. But in order to provide more useful ways to study the equation we will continue to perform this work in the future. Moreover, the smoothing effect and the compactness of the commutator [b, H] can be investigated as well.