Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

Abstract: This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq(Sn−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces andMorrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.


Introduction
An active topic of current research is the investigation on the variational inequalities for various operators. The rst work was due to Lépingle [21] in 1976 when he established the variational inequality for general martingales (see [31] for a simple proof). Lépingle's result was later used by Bourgain [2] to establish similar variational estimates for the ergodic averages. Since then, Bourgain's work has inaugurated a new research direction in ergodic theory and harmonic analysis. We can consult [2,16,17] for the ergodic averages, [24,25] for the di erential operators, [3,13] for the Hilbert transform, [8,13] for the Riesz transforms, [4,5,9,18] for the singular integrals with rough kernels, [6,24,25,35,36] for the Calderón-Zygmund singular integrals and their commutators as well as [26,27] for the discrete singular integral operators. Recently, Liu and Cui [22] established the boundedness and compactness for variation operators of Calderón-Zygmund singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces. Based on this, we are interested in two types of results: -Boundedness and compactness properties for variation operators of singular integrals with rough kernels and their commutators on Morrey spaces. -Boundedness and continuity properties for variation operators of Calderón-Zygmund singular integrals and their commutators on Besov spaces.
These contents are the main motivations of this work. It should be pointed out that this is the rst work focusing on the boundedness and compactness for variation operators of singular integrals with rough kernels and their commutators on Morrey spaces as well as the boundedness and continuity of variation operators of Calderón-Zygmund singular integrals and their commutators on Besov spaces.

. Background
Let T = {Tϵ} ϵ> be a family of bounded operators satisfying (1.5) Then T K (resp., T m K,b ) is just the usual (resp., the m-th order commutator of) singular integral operator with rough kernel Ω. We denote T K = T Ω and T m K,b = T m Ω,b for m ≥ . • When T K is bounded on L (R n ) and the kernel K is a standard Calderón-Zygmund kernel, which satis es the size condition |K(x, y)| ≤ A |x − y| n , for x ≠ y; (1.6) and the regularity conditions for some δ > |K(x, y) − K(z, y)| ≤ A|x − z| δ |x − y| n+δ , for |x − y| > |x − z|; (1.7) |K(y, x) − K(y, z)| ≤ A|x − z| δ |x − y| n+δ , for |x − y| > |x − z|. (1.8) Then T K (resp., T m K,b ) is the (resp., the m-th order commutator of) standard Calderón-Zygmund singular integral operator on R n .
Throughout this paper, we always assume that ρ > since the ρ-variation in the case ρ ≤ is often not bounded (see [1,2]). Let us attribute the developments of the variation operators for singular integrals to two stages.
Stage 1 (n = ). The variation operators for singular integrals were rst studied by Campbell et al. [3] who showed that Vρ(H) is of type (p, p) for < p < ∞ and of weak type ( , ). The above result was later extended to weighted version in [8,13]. Same conclusions hold for Vρ(R ± ) (see [8,Theorem A]). A general result was given by Liu and Wu [23] who proved that Vρ(T K ) is bounded on L p (w) for < p < ∞ and w ∈ Ap(R), provided that n = and Vρ(T K ) is of type (p , p ) for some p ∈ ( , ∞) and K satis es the conditions (1.6)-(1.8).
Stage 2 (n ≥ ). In 2002, Campbell et al. [4] rst established the L p (R n ) ( < p < ∞) bounds for Vρ(T Ω ), provided that Ω ∈ L log + L(S n− ). This result was essentially improved by Ding et al. [9] to the case Ω ∈ H (S n− ) since L log + L(S n− ) H (S n− ), which is a proper inclusion. The weighted result for Vρ(T Ω ) was rst considered by Ma et al. [25] who proved that Vρ(T Ω ) is bounded on L p (w) for < p < ∞ and w ∈ Ap(R n ), provided that Ω ∈ Lip α (S n− ) for α > . Later on, the above result was improved by Chen et al. [5] to the case Ω ∈ L q (S n− ) for some q > . In [13], Gillespie and Torrea studied the variation operators for Riesz transforms and showed that Vρ(R j ) is bounded on L p (|x| α ) for < p < ∞ and − < α < p − . Recently, Zhang and Wu [35] extended the above result to general Ap weight. Particularly, Ma et al. [25] proved that Vρ(T K ) is bounded on L p (w) for all < p < ∞ and w ∈ Ap(R n ) if K satis es (1.6)-(1.8) and the following priori estimate: (1.9) for some p ∈ ( , ∞).
The variation operator for the commutators was rst studied by Liu and Wu [23] who showed that Vρ(T m K,b ) is bounded on L p (w) for < p < ∞ and w ∈ Ap(R), provided that m ≥ , b ∈ BMO(R) and K satis es (1.6)-(1.9). As applications, they obtained the L p (w) bounds for Vρ(H m b ) and Vρ(R m ±,b ) for < p < ∞ and w ∈ Ap(R) if b ∈ BMO(R). Recently, Liu and Cui [22] extended the above results to the general case n ≥ . For the commutator of rough singular integral, Chen et al. [6] proved that Vρ(T m Ω,b ) is bounded on L p (w) for < p < ∞ if Ω ∈ L q (S n− ) for some q > satisfying (1.5), m = , b ∈ BMO(R) and one of the following conditions holds: (a) q ′ ≤ p < ∞, p ≠ and w ∈ A p/q ′ (R n ); (b) < p ≤ q, p ≠ ∞ and w − p− ∈ A p ′ /q ′ (R n ). Actually, applying the above result, the method in the proof of [6, Theorem 1.1] and induction arguments as in getting [11,Theorem 1], one can conclude that Vρ(T m Ω,b ) is bounded on L p (w) for < p < ∞ if Ω ∈ L q (S n− ) for some q > satisfying (1.5), m ≥ , b ∈ BMO(R) and one of the following conditions holds: (a) q ′ ≤ p < ∞, p ≠ and w ∈ A p/q ′ (R n ); (b) < p ≤ q, p ≠ ∞ and w − p− ∈ A p ′ /q ′ (R n ). These conclusions together with the main result of [5] imply the following result.
and Ω ∈ L q (S n− ) for some q > satisfying (1.5). Then, for < p < ∞, we have Recently, Guo et al. [14] rst studied the compactness for Vρ(T K,b ) on L p (w). They proved that Vρ(T K,b ) is a compact operator on L p (w) for < p < ∞ and w ∈ Ap(R n ), provided that b ∈ CMO(R n ) and K satis es (1.6)-(1.9). Here CMO(R n ) is the closure of C ∞ c (R n ) in the BMO(R n ) topology, which coincides with the space of functions of vanishing mean oscillation. Very recently, Liu and Cui [22] showed that Vρ(T m K,b ) is a compact operator on L p (w) for < p < ∞ and w ∈ Ap(R n ), provided that m ≥ , b ∈ CMO(R n ) and K satis es (1.6)-(1.9).

. Boundedness and compactness on Morrey spaces
As a natural extension of the classical Lebesgue spaces, the Morrey spaces play key roles in partial di erential equations and harmonic analysis. Let us recall one de nition. ([19]). Let ≤ p < ∞ and ≤ β < . For a weight w de ned on R n , the weighted Morrey space M p,β (w) is de ned by

De nition 1.2. (Weighed Morrey spaces)
where the supremum is taken over all balls in R n . Particularly, the M p,β (w) is just the classical weighted Lebesgue space L p (w) when β = .
When w ≡ , M p,β (w) reduces to the classical Morrey space M p,β (R n ), which was rst introduced by Morrey [28] to study the local behavior of solutions to second order elliptic partial di erential equations. The weighted Morrey spaces M p,β (w) were originally introduced by Komori and Shirai [19] who established the bounds for the Hardy-Littlewood maximal operator, fractional integral operator and the Calderón-Zygmund singular integral operator on M p,β (w). Later on, more and more scholars have devoted to investigating the boundedness of various operators on M p,β (R n ) (cf. e.g. [12], [29], [30]).
The boundedness of variation operators of singular integrals and their commutators on Morrey spaces was rst studied by Zhang and Wu [36] who proved that Vρ(T K ) is bounded on M p,β (w) for < β < , < p < ∞ and w ∈ Ap(R n ), provided that n = and K satis es ( . )-( . ). Very recently, Liu and Cui [22] studied the boundedness and compactness for variation operators of Calderón-Zygmund singular integrals and their commutators on weighted Morrey spaces. To be more precise, they showed that Vρ(T m K,b ) is bounded on M p,β (w) for < β < , < p < ∞ and w ∈ Ap(R n ), provided that m ∈ N, b ∈ BMO(R n ) and K satis es (1.6)-(1.9). They also proved that Vρ(T m K,b ) is a compact operator on M p,β (w) for < β < , < p < ∞ and w ∈ Ap(R n ), provided that m ≥ , b ∈ CMO(R n ) and K satis es (1.6)-(1.9).
Particularly, Liu and Cui [22] obtained the following result.

Theorem B ([22]).
Let Ω ∈ Lip α (S n− ) for some α > and Ω satisfy (1.5). Let ρ > , < p < ∞ and ≤ β < . Then It is well known that This is one of the main motivations of this work. In this paper, we shall establish the following results.
Theorem 1.1. Let m ∈ N, ρ > , ≤ β < and < p < ∞. Assume that b ∈ BMO(R n ) and Ω ∈ L q (S n− ) for some q > satisfying (1.5). Then Let Ω ∈ L q (S n− ) for some q > and satisfy (1.5). For r ≥ , de ne Here wr(δ) denotes the integral modulus of continuity of order r of Ω de ned by and ρ is a rotation in R n and ρ := sup x ′ ∈S n− |ρx ′ − x ′ |. Assume that b ∈ CMO(R n ) and F( ) < ∞, then the operator Vρ(T m Ω,b ) is a compact operator on M p,β (R n ).
(ii) The condition (1.11) is strictly weaker than the condition Ω ∈ Lip α (S n− ) with some α > . Thus, Theorems 1.1 and 1.2 essentially improve the conclusions of Theorem B. (iii) When β = , Theorems 1.1 and 1.2 imply the boundedness and compactness of Vρ(T m Ω,b ) on the Lebesgue spaces L p (R n ). (iv) Theorem 1.1 for the case < β < is new, even in the special case m = .
(v) Theorem 1.2 is new, even in the special case β = and m = .
Based on the above, some driving questions are the following Question 1.5. Do the conclusions in Theorems 1.1 and 1.2 hold under the condition that Ω ∈ L log + L(S n− ) or Ω ∈ H (S n− ) or Ω ∈ Fα(S n− ) for some α > ?

. Boundedness and continuity on Besov spaces
The second motivation of this work is to investigate the boundedness and continuity for variation operators of singular integrals and their commutators. For s ∈ R and < p, q ≤ ∞ (p ≠ ∞), we denote byḂ p,q s (R n ) (resp., B p,q s (R n )) the homogeneous (resp., inhomogeneous) Besov spaces. It is well known that In [23], Liu and Wu established the following criterion on the boundedness and continuity of a class of sublinear operators on Besov spaces.

Proposition 1.3. ([23]). Let T be a sublinear operator. Assume that T
for any x, ζ ∈ R n . Here ∆ ζ (f ) is the di erence of f for an arbitrary function f de ned on R n and ζ ∈ Rn, i.e., . Then T is bounded onḂ p,q s (R n ) for < s < and < q < ∞. Specially, if T also satis es the following for arbitrary functions f , g de ned on R n . Then T is continuous from B p,q s (R n ) toḂ p,q s (R n ) for < s < and < q < ∞.
Note that the operator Vρ(T K ) is sublinearity and commutes with translations, i.e. Vρ( . One can easily check that Vρ(T K ) satis es (1.13) and (1.14). Applying Proposition 1.3, we have the following result.

Proposition 1.4. Let ρ > and Vρ(T K ) be given as in
As applications of Proposition 1.4, the following results are valid. Corollary 1.5. Let ρ > and Vρ(T K ) be given as in (1.3). Assume that K(x, y) = K(x − y) and K satis es the conditions (1.6)-(1.9). Then Vρ(T K ) is bounded and continuous on B p,q s (R n ) for < s < , < p < ∞ and < q < ∞. Corollary 1.6. Let ρ > and one of the following conditions hold: Here Fα(S n− ) for α > denotes the set of all integrable functions over S n− which satisfy Then the operator Vρ(T) is bounded and continuous on B p,q s (R n ) for < s < , < p < ∞ and < q < ∞. Remark 1.6. (i) It should be pointed out that Corollary 1.5 follows from Theorem 1 in [25] and Proposition 1.4. (ii) The corresponding results in Corollary 1.6 for the cases (i)-(iii) follow from the known bounds for the corresponding operators and Proposition 1.4. It was shown in [9] (see Theorem 1.2 and Corollary 1.6 in [9]) that Vρ(T Ω ) is bounded on L p (R n ) for all p ∈ ( , ∞) under the condition that Ω ∈ H (S n− ) or Ω ∈ α> Fα(S n− ). This together with Proposition 1.4 yields the conclusion of Corollary 1.6 for case (iv). (iii) We remark that the space Fα(S n− ) was introduced by Grafakos and Stefanov [15] in the study of L p boundedness of singular integral operator with rough kernels. Clearly, q> L q (S n− ) Fα(S n− ) for any α > . Moreover, the examples in [15] show that It should be pointed out that the operator Vρ(T m K,b ) does not satisfy the condition (1.13), even in the special case m = and K(x, y) = K(x − y), which makes that Proposition 1.3 does not apply for Vρ(T m K,b ). Therefore, it is natural to ask the following Question 1.7. Is the operator Vρ(T m K,b ) bounded and continuous on B p,q s (R n ) for some < s < and < p, q < ∞ when m ≥ ?
In this paper we shall present a positive answer to this question, which is another one of main motivations. Before presenting the rest of main results, let us introduce the following de nition.

De nition 1.8. (Lipschitz space).
The homogeneous Lipschitz spaceΛ(R n ) is given bẏ The rest of main results can be formulated as follows: is bounded and continuous on B p,q s (R n ) for < s < and < q < ∞. Particularly, As some applications of Proposition 1.7, we obtain Corollary 1.8. Let ρ > , m ≥ and Vρ(T m K,b ) be given as in (1.4). Assume that b ∈ Λ(R n ), K(x, y) = K(x − y) and K satis es the conditions (1.6)-(1.9). Then Vρ(T m K,b ) is bounded and continuous on B p,q s (R n ) for < s < , < p < ∞ and < q < ∞. Particularly, Corollary 1.9. Let m ≥ , ρ > , b ∈ Λ(R n ) and one of the following conditions hold: Then the operator Vρ(T) is bounded and continuous on B p,q . This together with Proposition 1.7 yields the conclusions of Corollary 1.9 for case (iv).
Some interesting questions can be formulated as follows: .

Outline of this paper and some notations
The rest of this paper is organized as follows. Section 2 is devoted to presenting the proof of Theorem 1.1. The proof of Theorem 1.2 will be given in Section 3. Finally, we shall prove Proposition 1.7 in Section 4. We would like to remark that the proofs of Theorems 1.1 and 1.2 are motivated by the methods from [7]. The proof of Proposition 1.7 is based on some known arguments from [22,23]. However, some new techniques are needed to be explored. Throughout this paper, for any p ∈ ( , ∞), we let p ′ denote the dual exponent to p de ned as /p + /p ′ = . For x ∈ R n and r > , we denote by B(x, r) the open ball centered at x with radius r. For t > and B := B(x, r) with x ∈ R n and r > , we denote tB = B(x, tr). We end this section by presenting an useful inequality: for all x ∈ R n , any arbitrary functions F and f de ned on R n × R n , where ρ > and {ε i } is an increasing or decreasing sequence of positive numbers.

Proof of Theorem 1.1
In this section we shall prove Theorem 1.1. At rst, let us introduce some notations and lemmas, which are the main ingredients of our proof. For Ω ∈ L (S n− ), the maximal operator with rough kernel Ω is de ned by The following lemma was proved by Chen et al. [7].
). Let < β < and Ω ∈ L q (S n− ) for some q > satisfying (1.5). Then for < p < ∞, there exists an ϵ > such that for any k ∈ N and f ∈ M p,β (R n ), where B := B(t, r) is an arbitrary xed ball and f k = fχ k+ B\ k B .
Motivated by the idea in the proof of Theorem 1.8 in [7], we have the following result.
and Ω ∈ L q (S n− ) for some q > satisfying (1.5). Let T b be a linear or sublinear operator satisfying where C > . If there exist p ∈ ( , ∞) and C > such that T b satis es Proof. Proposition 2.2 for the case m = was proved by Chen et al. in [7] (see Theorem 1.8 in [7]). We shall prove the case m ≥ by adopting the method as in the proof of Theorem 1.8 in [7]. Let B = B(x , r), where x ∈ R n and r > . To prove (2.3), it su ces to show that where C > is independent of x , r and b.
Decompose f as f = fχ B + fχ ( B) c . By Minkowski's inequality and the sublinearity of T b , one gets From the assumption (2.2) we see that (2.6) Next we estimate I . Fix x ∈ B. We get from (2.1) that (2.7) Fix k ≥ . Note that when z ∈ k+ B \ k B. By (2.8), we havê (2.9) Let f k = fχ k+ B\ k B and s ∈ ( , min{p, q}). By Hölder's inequality and the well-known property for b BMO(R n ) , one has On the other hand, by (2.8) and a change of variable, we have (2.11) In light of (2.10) and (2.11) we would have (2.14) It remains to estimate J k, (x). Let us consider two cases: (Case 1: p ≥ q ′ ). In this case we have that Ω ∈ L p ′ (S n− ). By Hölder's inequality, a change of variable and (2.8), we getˆ Then we have (2.16)

Note that f sq/u ′ /(qs)
. By the arguments similar to those used in getting (2.16), we havê

Proof of Theorem 1.2 . Preliminaries
To prove Theorem 1.2, we need the following characterization that a subset in M p,β (R n ) is a strongly precompact set. Proposition 3.1. Let < p < ∞ and ≤ β < . Then a subset F of M p,β (R n ) is strongly pre-compact set in M p,β (w) if F satis es the following conditions: (ii) F uniformly vanishes as in nity, i.e. The following result follows from [10].

Lemma 3.2. ([10]
). Let ≤ β < n and Ω ∈ L (S n− ) satisfying (1.5). Then for R > , there exists a constant C > independent of R such that for x ∈ R n with |x| < R/ , Here w(δ) is given as in Theorem 1.2.

. Proof of Theorem 1.2
We divide the proof of Theorem 1.2 into four steps: Step 1. Reduction via smooth truncated techniques. We shall adopt the truncated techniques followed from [20] to prove Theorem 1.
. For any η > , we de ne the function Ωη by It is clear that Ωη ∈ L q (S n− ) and Ωη satis es (1.5). In what follows, let us x b ∈ CMO(R n ), < p < ∞ and ≤ β < . We shall prove that there exists a constant C > independent of η such that Note thatˆ|

This together with (3.3) leads to
Note that Combining this with the L p (R n ) bounds for M Ω with < p < ∞ and Proposition 2.2 yields that Combining (3.5) with (3.4) leads to (3.2). By (3.2) and [34, p. 278, Theorem (iii)], to prove the compactness for Vρ(T m Ω,b ), it su ces to prove the compactness for Vρ(T m Ωη ,b ) when η > is small enough. For β > , let To prove the compactness for Vρ(T m Ωη ,b ), it is enough to show that the set F satis es the conditions (i)-(iii) of Proposition 3.1 when η > is small enough.
Step 3. A veri cation for condition (ii) of Proposition 3.1. Assume that b ∈ C ∞ (R n ) and is supported in a ball B = B( , r). Fix f ∈ M p,β (R n ) with f M p,β (R n ) ≤ and E N := {x ∈ R n : |x| > N} with N ≥ max{nr, }. Note that b(x) = when x ∈ E N since N ≥ nr. By (1.16) and a change of variable, we have For a xed ballB =B(x , t). We consider two cases: (1) (q < p). By Hölder's inequality and (3.6), we have By Minkowski's inequality, we have (2) (q ≥ p). By Hölder's inequality and (3.6), we get Then by Fubini's theorem, we get Combining (3.8) with (3.7) implies that F satis es condition (ii) of Proposition 3.1.
Step 4. A veri cation for condition (iii) of Proposition 3.1. We want to show that for a xed η ∈ ( , ).

Proof of Proposition 1.7
This section is devoted to proving Proposition 1.7. The proof will be divided into two parts: Step 1: Proof of the boundedness part. It was proved by Yabuta [33] that if < s < , ≤ p < ∞ and ≤ q ≤ ∞, then It was shown in [22] (see [22, (5.16)]) that     On the other hand, it was shown in [22] (see [22, (5.11)]) that Vρ(T m K,b )(f ) L p (R n ) ≤ C b m L ∞ (R n ) f L p (R n ) .