Commutators of Square Functions Related to Fractional Differentiation for Second-Order Elliptic Operators

Abstract

Let be a second-order divergence form elliptic operator, where is an accretive matrix with bounded measurable complex coefficients in In this paper, we mainly establish the boundedness for the commutators generated by and the square function related to fractional differentiation for second-order elliptic operators.

1. Introduction

Let be a linear operator in a measurable function space; then, the commutator formed by and is defined by For , set where is a cube in and . Then, the space is defined as

Let , and consider the fractional differentiation operators of even and odd parities, defined for tempered distributions , by Let be the Riesz potential operator of order and be defined in the space of tempered distribution modulo polynomials by setting The Sobolev space is the image of under Equivalently, if and only if . Let , subsequently, if satisfies , in which the supremum is taken over all and . For , is a space of functions modulo constants that is properly contained in (see [1, 2]).

Before presenting our main theorem, we introduce the second-order elliptic operator as follows: For , denote its complex conjugate by Let be an matrix of complex coefficients defined on that satisfy the ellipticity condition: for and for some such that Here, the inner product notation Therefore, Associated with such a matrix , we define a second-order divergence form operator:which we interpret in the usual weak sense via a sesquilinear form. The operator generates a semigroup and the gradient of the semigroup .

In this paper, we first present a general criterion for weak-type boundedness of commutators with square functions and .

Theorem 1. Let , , , and be an integer greater than 1. Let and be two closed subsets of with a Euclidean distance between each other, and let be a family of sublinear operators acting on Assume that, for and ,where . Furthermore, ifThen, we have

We recall a square function, which is representative of larger classes of square functions associated with , given as follows [3]: In 2007, Aushcer [3] proved that for . The interval is the maximal open interval required for the semigroup to be bounded. We recall that is the maximal open interval required for the semigroup to be bounded. In [3], the author has shown in general that , if ; , if and .

Many researchers have contributed to the commutators associated with the second-order elliptic operator, and among the numerous studies, some related to development and applications have been cited herein [4–9]. In particular, commutators with fractional differentiations associated with play an important role in the theory of linear partial differential equations and harmonic analysis [10–14]. Naturally, the case of the commutators of square functions being related to fractional differentials associated with is worth studying.

In this paper, we define a square function related to the fractional differential operator associated with as follows: Moreover, for and , the commutator of can be defined by In this paper, we also establish the boundedness for .

Theorem 2. Let be a second-order elliptic operator in divergence form defined by (5), , and . Then, for , we have

The remainder of this paper is organised as follows: in Section 2, we present some lemmas that play an important role in the proof of the main results; in Section 3, we prove Theorem 1; in Section 4, we prove Theorem 2. For , denotes the dual exponent of , i.e., Throughout this paper, the letter “” will stand for a positive constant that is independent of the essential variables but will not necessarily have the same value for each occurrence.

2. Preliminary Lemmas

The second-order elliptic operator in divergence form is defined by (5) and has the following off-diagonal estimates (see [3, 4, 9] and references therein).

Lemma 3 (see [3, 4, 9]). Let be a second-order elliptic operator defined by (5), let and be two closed sets of , and let set denote the distance between and . Then, for , the complex-valued function , and vector-valued functions , the following statements hold:
(i) Let and supported in , (ii) Let , supported in and , (iii) Let and supported in , In particular, if we choose , the abovementioned off-diagonal estimates become estimates.

Another very useful and well-known lemma for off-diagonal estimates is introduced here, which could be proved by using a similar argument for the proof of a previous lemma [9, lemma 2.3].

Lemma 4 ([3, 9]). Let and be arbitrary closed subsets of . Assume that the two families of the operators and satisfy the following off-diagonal estimates: Then, for and supported in E, we have

Next, let us introduce a criterion that deals with the boundedness of the commutators of the operators satisfying off-diagonal estimates, which can be proved in [15].

Lemma 5 (see [15]). Let and be two closed subsets of with a Euclidean distance , and let be a family of sublinear operators acting on Assume that, for and with ,If , then for and , we havewhere is independent of , and .

The following two lemmas are about the off-diagonal estimates related to some commutators of the Lipschitz function and semigroups for second-order elliptic operators.

Lemma 6 (see [16]). Let and be two closed subsets of with a Euclidean distance . Assume that and and . If , then for and supported in , we obtain the following for some :

Lemma 7 (see [16]). Let be the second-order elliptic operator in divergence form defined by (5), and be two closed sets of , and express the distance between and . Assume that , , and . Then, for , supported in , we obtain where is independent of .

3. Proof of Theorem 1

For any fixed , without loss of generality, we may assume that is nonnegative. Let us write for the Hardy-Littlewood maximal function. We use the Calderón-Zygmund decomposition for at height . Then, there exists a collection of pairwise disjoint cubes such that and they satisfy the following property:Then, we write , where After estimating (24), and the standard arguments yield for almost every . Then, We estimate every term separately. For , we use (8) and the properties of to obtain Now, we proceed with . Let us fix an integer . We write , where stands for the side length of the cube . We use the notation , where, in general, we write for the -dilated , i.e., for the cube with the same centre as and with the side length . Let . Because we obtain The first term can be estimated as follows: Now, we complete the estimate of By Chebychev’s inequality, we obtain where the supremum is taken over all the functions with . We set Let us recall that . Because supp , we have Where, in the last inequality, we used (6). Because , for , we obtain . Recall that . Subsequently, we obtain Then, using , we obtain Then, because the Hardy-Littlewood maximal function is of weak-type , we use and Kolmogorov’s lemma to obtain Then, we plug the estimate into (31) to obtain Applying of the weak-type , we obtain We now examine . Recall that Then, Thus, from Chebychev’s inequality, We fix . Then, for , by (8), we obtain Hence, by (see [9]) we obtain For , by (7) and (see [3]), we obtain Then, where we use the fact that is bounded on with the bound (see [16]). Next, we estimate the abovementioned two norms, and . Now, taking with , Note that, for all , satisfies the off-diagonal estimates (see Lemma 3(ii)); let , and we obtain (47), which is controlled byThus, from (47), (48), and the fact that is of weak-type , we obtainFor the second term of , , by Lemma 6 and the same procedures performed previously (48): Then, we haveCombining (49) and (51), we obtainCombining the estimates of and , we obtain The proof of Theorem 1 is now completed by combining the estimates of , , and with .

4. Proof of Theorem 2

First, we introduce a lemma that will be used to prove Theorem 2.

Lemma 8 (see [16]). For , let Suppose that is a family of operators satisfyingfor an and all . Assume thatin the sense of for all Then, we haveIf, in addition,then

Recall that The proof of Theorem 2 is presented in two steps.

Step 1 (the boundedness). Let . By Lemma 8, we only need to prove that satisfies (54)-(56) and (58). Because , it is easy to verify that satisfies (54), (56), and (58). Moreover, it is easy to verify that satisfies (55) by the off-diagonal estimate of the operator (see Lemma 3(i, iii)). We obtain

Step 2 (weak-type boundedness for ). We first prove that the commutator is of the weak-type for . We apply Theorem 1 with to prove this result. Because and are bounded on , verification of satisfies (6). Taking , by the Minkowski inequality, we obtain Now, we study each operator separately. For the first operator, we have and, then, For , by Lemma 7(i), we have where . Now, with , by the commutative property of the semigroup and Lemma 7(i), we obtain Collecting this estimate and the one proved for , we obtain Next, we proceed with the estimate of : Let be two closed sets and be such that supp . For , by a similar previous argument [9], we can prove that uniformly on . Then, by Lemma 4, we haveWe write as follows: First, we consider : Let us observe that, because of (71) and Lemma 7(i) in the composition of the operators, each of them verifies an off-diagonal estimate. This fact allows us to employ Lemma 4 to obtain where . From Lemma 5 and (71), it is easy to obtainFinally, for ,We know that the composition of the operators above (75) or Lemma 3 verifies an off-diagonal estimate. It follows from Lemma 4 that Collecting the estimates for and , we obtain which combined with the estimate of verify that satisfies (6). Thus, we have proven Theorem 2.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Y. Chen is supported by NSFC (11471033) and the Fundamental Research Funds for the Central Universities (FRF-BR-17-001B).