The boundedness of variation associated with the commutators of approximate identities

In this paper, we establish $L^{p}$
 
 L
 p
 
 -boundedness and endpoint estimates for variation associated with the commutators of approximate identities, which are new for variation operators. As corollaries, we obtain the corresponding boundedness results for variation associated with the commutators of heat semigroups and Poisson semigroups.


Introduction and main results
The intension of this paper is to obtain boundedness of variation associated with the commutators of approximate identities. Before stating our results, we first recall some relevant background.
The boundedness of commutators of singular integrals, especially the endpoint estimates, attracts lots of scholars' interest. Recall that given a locally integral function b and a linear or nonlinear operator T, the commutator [b, T] is defined by (1.1) And we say that b belongs to BMO(R n ) spaces if b BMO(R n ) := sup Coifman, Rochberg, and Weiss [8] first studied the L p -boundedness of commutators of singular integrals with the symbol b ∈ BMO(R n ). While the endpoint case p = 1 was the remarkable work of Pérez [23], it was shown in [23] that the commutators of Calderón-Zygmund operators enjoy an L log L estimate instead of being of weak type (1, 1). Moreover, it was observed in [22] that the commutators do not always map H 1 (R n ) into L 1 (R n ).
In fact, Pérez [23] proved that the commutators of Calderón-Zygmund operators are bounded from H 1 b (R n ) (see its definition in [23]) into L 1 (R n ). Precisely, Pérez proved this result by establishing that where a is a b-atom (see its definition in [23]). However, Ky [15] pointed out that the above inequality does not suffice to illustrate that [b, T] is bounded from H 1 b (R n ) into L 1 (R n ). Ky filled this gap and proved that [b, T] is bounded from In this paper, we study variation associated with the commutators of approximate identities. We first recall some definitions, notations, and background. Let I ⊂ R + and a := {a t } t∈I be a family of complex numbers. For ρ > 2, the ρ-variation of a is defined by where the supremum is taken over all finite decreasing sequences {t k } in I. Let F := {F t } t>0 be a family of operators. We also define the ρ-variation of the family F by is the space of Schwartz functions. We consider the following family of operators: where φ t (x) := t -n φ(x/t). Let b ∈ L 1 loc (R n ). We will also take into account the corresponding family of commutators of operators The ρ-variation of families and In [16], Lépingle improved the well-known Doob maximal inequality by proving a variational inequality for martingales. Relying upon this result, Bourgain [2] obtained corresponding variational estimates for the Birkhoff ergodic averages and pointwise convergence results. This work has set up a new research subject in harmonic analysis and ergodic theory. Afterwards, the study of variational inequalities has been spilled over into harmonic analysis, probability and ergodic theory. Particularly, we refer readers to see [5] for the classical work of ρ-variation operators for singular integrals. In [5], the authors gave the L p -bounds and weak type (1, 1) bounds for ρ-variation operators of truncated Hilbert transform if ρ > 2, and then they extended the results to higher dimension in [6].
For results of rough kernels and weighted cases, see [10,13,20]. In 2009, Crescimbeni et al. [9] established the L p -bounds and weak type (1, 1) bounds for variation operators of heat semigroups and Poisson semigroups. Betancor et al. [1] also considered the L pboundedness of variation operators of heat semigroups associated with Schrödinger operators, which was generalized to metric spaces endowed with nonnegative Borel measure by Bui [4] and to weighted cases by Tang and Zhang [25]. The acting on Hardy spaces for variation operators associated with approximate identities, which include the variation operators of heat semigroups, was given by Liu [19]. For recent works on variation inequalities, we refer readers to [3,11,21,26,27].
The other side of the shield, the variational inequalities for the commutators of singular integrals also have been intensively studied. In 2013, Betancor et al. [1] studied the mapping property of variation operators for the commutators of Riesz transforms in the Euclidean and Schrödinger setting. A few years later, Liu and Wu [18] obtained the weighted L p -boundedness for variation operators of commutators of truncated singular integrals with the Calderón-Zygmund kernels. Recently, variation operators of commutators with rough kernels were also obtained in [7]. For more works on variation operators of commutators, see [17,28,29] From the known results about variation inequalities, none of the boundedness estimates for variation of commutators have been established, can we establish the corresponding boundedness results for variation of commutators? In this paper, we give a firm answer to this question.
Our contribution of this paper is the following.
Specially, we will consider the families of operators , the heat semigroup e t and the Poisson semigroup And we will also take into account the corresponding families of commutators of operators , respectively. It is well known that the kernel of the heat semigroup φ t (x) = (πt) -n/2 e -|x| 2 /t belongs to S(R n ) and satisfies R n φ(x) dx = 1. Following the steps in proving Theorem 1.2 and Theorem 1.3, the endpoint estimates also hold for variation associated with the commutators of Poisson semigroups. Therefore, we have the following corollary.
with {t i } being a fixed sequence decreasing to 0. We remark that from the proofs of the above theorems, the corresponding results also hold for the oscillation associated with the commutators of approximate identities. Remark 1.6 For a family of operators given in Remark 1.5, define the α-jump operator for F at x by We refer readers to [10,14] for more details about jump operators. It is trivial that We organize the rest of the paper as follows. In Sect. 2, we give some preliminaries. Section 3 is devoted to proving Theorems 1.1-1.3.
We end this section by making some conventions, we denote a positive constant by C, which may change at each occurrence. We also denote f g, f ∼ g if f ≤ Cg and f g f , respectively. For any ball B := B(x 0 , r) ⊂ R n , x 0 and r are the center and the radius of B, respectively, and f B means the mean value of f over B, χ B represents the characteristic function of B.

Preliminaries
In this section, we introduce some necessary definitions and lemmas.
For δ > 0, we denote M δ (f ) := M(|f | δ ) 1/δ , where M is the Hardy-Littlewood maximal operator. The sharp maximal function M is defined by In this paper, we use the following famous Fefferman-Stein inequality, see [12]: for all function f such that the left-hand side is finite, where 0 < p < ∞.
Next, let us recall some known facts concerned with H 1 (R n ).

Lemma 2.3 (Cf. [24]) A function f ∈ L 1 (R n ) belongs to H 1 (R n ) if and only if f = i λ i a i in
where the infimum is taken over all the above atomic decomposition of f .
As we stated in the introduction, the space H 1 b (R n ) is introduced to fill the gap left by Pérez [23]. We present its definition as follows.
In [15], Ky gave the following criterion to verify the mapping property of operators on Hardy type spaces.

Lemma 2.5 (Cf. [15]) Let b ∈ BMO(R n ), T be a linear or nonlinear operator, and the commutator be defined as (1.1). If T is bounded from H
for any ψ-atom a of H 1 (R n ) with supp a ⊂ Q, then the commutator [b, T] maps H 1 b (R n ) into L 1 (R n ).
Remark 2. 6 We remark that we do not give the definition of ψ-atom of H 1 (R n ) since we need to state so many backgrounds of it, but only use the properties a L 2 (Q) ≤ |Q| -1/2 and Q a(y) dy = 0, where Q is a cube. We refer readers to [15] for its definition.
In this paper, we also need the following lemma established in [19].

The boundedness of commutators
To prove Theorem 1.1, we first establish the following lemma.
Lemma 3.1 Let φ ∈ S(R n ) satisfy R n φ(x) dx = 1 and b ∈ BMO(R n ), then for ρ > 2 and any 1 < s < ∞, there is a constant C > 0 such that Proof For x ∈ R n , let B := B(x 0 , r) be any ball containing x. We write f = f 1 + f 2 , where f 1 = f χ 4B . For y ∈ B, one can see that We first estimate I. Using Hölder's inequality with exponents s and s , then To deal with II, choose 1 < μ, q < ∞ such that μq = s. Thus, Lemma 2.7 yields that Finally, we consider III. Note that For z ∈ R n \4B, applying the mean value theorem, we get that Observe that And note that 2 -j-1 1 + (j + 1) ∼ r -1/s b BMO(R n ) .
Combining with the above two estimates, we conclude that which implies that This together with the estimate of I and II completes our proof. Now, we are in the position to prove Theorem 1.1.

Proof of Theorem
is finite, then for f ∈ C ∞ c (R n ) and any s satisfying 1 < s < p, we make use of the Fefferman-Stein inequality, Lemma 3.1, and Lemma 2.7 to get that then by density we get our result. Now, we return to proving that If b is a bounded function, then by Lemma 2.7, we have Then it is easy to check that b N BMO(R n ) ≤ c b BMO(R n ) , where the implicit constant is independent of N . Keeping in mind that f ∈ C ∞ c (R n ) and φ ∈ S(R n ), using |b N | ≤ |b| and b ∈ L 1 loc (R n ), by the Lebesgue dominated convergence theorem, we get From this, one can check that Letting N → ∞ and applying the Fatou lemma, we deduce that Next, we give the proof of Theorem 1.2.
Proof of Theorem 1.2 Let f ∈ H 1 (R n ), then by Lemma 2.3, we can write f = i λ i a i , where a i is an (1, ∞) atom. To prove this conclusion, we only need to prove it for f being a finite sum f = j λ j a j with j |λ j | ≤ 2 f H 1 (R n ) , where a j is a (1, ∞) atom and supp a j ⊂ B j := B(x j , r j ). In fact, assume that V ρ (( f ) b ) is bounded from H 1 (R n ) to L 1,∞ (R n ) for such f , then for the general f , one can select a sequence {f k } k with f k being a finite sum as above such that f k converges to f in H 1 norm or almost everywhere when k → ∞. Therefore, by a limit argument, Theorem 1.2 follows from the L 2 -boundedness of V ρ (( f ) b ). Now, in the following, we assume that f = j λ j a j is a finite sum satisfying j |λ j | ≤ 2 f H 1 (R n ) , where a j is a (1, ∞) atom and supp a j ⊂ B j . Note that Firstly, we consider I 1,j (x). By Hölder's inequality and Theorem 1.1, we obtain Thus, by the Chebyshev inequality and j |λ j | ≤ 2 f H 1 (R n ) , x ∈ R n : I 1,j (x) > α/3 Next, we pay our attention to I 2,j (x). By Lemma 2.7 and we deduce that Finally, we deal with I 3,j (x). We make use of (3.1) and the vanishing property of a j and Minkowski's inequality to obtain Observe that B j a j (y) dy ≤ 1.