Commutators of Littlewood-Paley gκ∗ $g_{\kappa}^{*} $-functions on non-homogeneous metric measure spaces

Abstract The main purpose of this paper is to prove that the boundedness of the commutator Mκ,b∗ $\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator Mκ∗ $\mathcal{M}_{\kappa}^{*} $ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of Mκ∗ $\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that Mκ,b∗ $\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from the space L log L(μ) to the weak Lebesgue space L1,∞(μ), and is bounded from the atomic Hardy spaces H1(μ) to the weak Lebesgue spaces L1,∞(μ).

To solve the unity of the homogeneous type spaces and the metric spaces endowed with measures satisfying the polynomial growth condition, in 2010, Hytönen [15] introduced a new class of metric measure space satisfying the so-called geometrically doubling and the upper doubling conditions (see Definitions 1.1 and 1.3, respectively), which is now called non-homogeneous metric measure space. So, it is interesting to generalize and improve the known results to the non-homogeneous metric measure spaces, see [16][17][18][19][20][21][22][23][24].
In this paper, .X ; d; / stands for a non-homogeneous metric measure space in the sense of Hytönen in [15]. In this setting, we will discuss the boundedness of commutators of Littlewood-Paley g Ä -functions on .X ; d; /.
Before stating the main results, we firstly recall some definitions and remarks. The following notion of the geometrically doubling condition was originally introduced by Coifman and Weiss in [25]. Remark 1.2. Let .X ; d / be a metric space. Hytönen in [15] showed the following statements are mutually equivalent: (1) .X ; d / is geometrically doubling.
(2) For any 2 .0; 1/ and ball B.x; r/ X , there exists a finite ball covering fB.x i ; r/g i of B.x; r/ such that the cardinality of this covering is at most N 0 n . Here and in what follows, N 0 is as Definition 1.1 and n WD log 2 N 0 : (3) For every 2 .0; 1/, any ball B.x; r/ X can contain at most N 0 n centers fx i g i of disjoint balls with radius r: (4) There exists M 2 N such that any ball B.x; r/ X can contain at most M centers fx i g i of disjoint balls fB.x i ; r 4 /g M iD1 .
Now, we recall the definition of upper doubling conditions given in [15].
Hytönen et al. proved in [16] that there exists another dominating function Q such that Q Ä , C Q Ä C and, for all x; y 2 X with d.x; y/ Ä r, Q .x; r/ Ä C Q Q .y; r/: Based on this, from now on, we always assume that the dominating function as in (1) satisfies (2). The following coefficient K B;S which introduced [15] by Hytönen is analogous to Tolsa's number in [8,9]. Given any two balls B S, set where c B represents the center of the ball B.
Hytönen [15] gave the definition of .˛;ˇ/-doubling, that is, a ball B X is called .˛;ˇ/-doubling if .˛B/ Ä .B/ for˛;ˇ> 1. At the same time, Hytönen proved that if a metric measure space .X ; d; / is upper doubling andˇ> C log 2˛ DW˛ , then for every ball B X , there exists some j 2 Z C such that˛j B is .˛;ˇ/-doubling. In addition, let .X ; d / be geometrically doubling,ˇ>˛n with n D log 2 N 0 and Borel measure on X which is finite on bounded sets. Hytönen also showed that for -a.e x 2 X , there exist arbitrarily small .˛;ˇ/-doubling balls centered at x. Furthermore, the radius of there balls may be chosen to be form˛ j r for j 2 N and any preassigned number r 2 .0; 1/. Throughout this paper, for any˛2 .1; 1/ and ball B, the smallest .˛;ˇ˛/-doubling ball of the form˛j B with j 2 N is denoted by Q B˛, wherě˛W For convenience, we always assume˛D 6 in this paper and denote e B 6 simply by e B: Now we recall the notion of RBMO. / from [15].
loc . / is claimed to be in the space RBMO. / if there exist a positive constant C and, for any ball B X , a number f B such that and, for any two balls B and R such that B R, The infimum of the constants C satisfying (5) and (6) is defined to be the RBMO. / norm of f and denoted by kf k RBMO. / .
Next, we recall the definition of the Littlewood-Paley g Ä -function given in [17]. 17]). Let K.x; y/ be a locally integrable function on .X X / n f.x; x/ W x 2 X g. Assume that there exists a non-negative constant C such that, for all x; y 2 X with x ¤ y, and, for all y; y 0 2 X , The Littlewood-Paley g Ä -function M Ä is defined by " " where x 2 X ; X .0; 1/ D f.y; t / W y 2 X ; t > 0g and Ä > 1.
Let b 2 RBMO. / and K.x; y/ satisfy (7) and (8). The commutator of Littlewood-Paley g Ä -function M Ä;b is formally defined by The following notion of the atomic Hardy space is from [16].
where the infimum is taken overall the possible decompositions of f as above.
Our main results in this paper are formulated as follows.
such that for all f 2 H 1 . / and t > 0, one has

Preliminaries
In this section, we shall recall some lemmas used in the proofs of our main theorems. Firstly, we recall some useful properties of K B;S as in (3) (see [15]). Now, we recall the following conclusion, which is just [18]. 18]). If f 2 RBMO. /, then there exists a positive constant C such that, for any ball B, $ 2 .1; 1/ and r 2 OE1; 1/, Next, we recall some results from [15,19].

Lemma 2.3 ([15]
). .1/ Let p 2 .1; 1/, r 2 .1; p/ and 2 .0; 1/. The following maximal operators defined, respectively, be setting, for all f 2 L 1 loc . / and x 2 X , The following result is given in [19]. where Also, we recall the following Calderón-Zygmund decomposition theorem given in [19]. Suppose 0 is a fixed positive constant satisfying that 0 > maxfC 3 log 2 6 ; 6 3n g, where C is as in (1) and n as in Remark 1.2. when .X / < 1/. Then .1/ there exists a family of finite overlapping balls f6B i g i such that fB i g i is pairwise disjoint, 1 for all i and all 2 .2; 1/; where is some positive constant depending only on .X ; /, and there exists a positive constant C , independent of f; t and i , such that, if p D 1, then and if p 2 .1; 1/, Finally, we recall the following John-Nirenberg inequality from [15].
From Lemma 2.6, it is easy to prove that there are two positive constants B 1 and B 2 such that, for any ball B and b 2 RBMO. /, 3 Proofs of Theorems 1.8-1.10 Proof of Theorem 1:8. Let 0 < r < 1, we firstly claim that, for any p 2 .1; 1/, b 2 L 1 . / and all bounded functions f with compact support, Once (20) is established, by the Marcinkiewicz interpolation theorem, it is easy to obtain that kM ] r OEM Ä;b .f /k L p . / Ä C kbk RBMO. / kf k L p . / : By Lemma 2.4, for any p 2 .1; 1/, b 2 L 1 . / and all bounded function f with compact support and integral zero, together with the fact that the bounded function f with compact support and integral zero is dense in L p . / (see [19,Theorem 6.4]), we finish the proof of Theorem 1.8. Now, we turn to estimate (20). Without loss of generality, let D 6 as in Lemma 2.3 and kbk RBMO. / D 1. For each fixed t > 0 and bounded function f with compact support and integral zero, applying Lemma 2.5 to f , we see (15) and (17), we easily get kgk L 1 . / Ä C t: On the other hand, applying (17), (19) and Hölder inequality, we have Further, for 1 < s < p, by applying the result of the Claim and (21), we deduce From this, we can write where we have used the fact that M ] r f .x/ Ä CM r;6 f .x/ (see [20]). By applying the .L 1 . /; L 1;1 . //-boundedness of M , for any > 0, we get Choosing 1 < p 1 < p, by h j WD P j .f ! j ' j / and (23), we have For H 11 . By Hölder inequality, (13) and (14), we have With a way similar to that used in the proof of D 12 in [20], it is easy to obtain H 12 Ä C t p kf k p L p . / : Now, we turn to estimate H 2 . By (23) and h j WD P j .f ! j ' j /, write An argument similar to that used in the proof of E 1 in [17, Theorem 1.10] shows that which, together with the fact that kh j k L 1 . / Ä C t 1 p kf k p L p . / , thus, H 21 C H 24 Ä C t p kf k p L p . / . For H 22 . By Hölder inequality, the L p . /-boundedness of M Ä , (13) and (19), we have Similar to the estimate of the H 22 , we conclude  (22), imply (20) and hence complete the proof of Theorem 1.8.
Next, we come to prove Theorem 1.9. In order to do this, we need the following claim.
Claim. Let K.x; y/ satisfy .7/ and .12/, s 2 .1; 1/, p 0 2 .1; 1/ and b 2 L 1 . /. If M Ä is bounded on L 2 . /, then there exists a positive constant C such that, for all f 2 L 1 . / \ L p 0 . / and for all x 2 X , Proof. Without loss generality, we may assume kbk RBMO. / D 1. Let B be an arbitrary ball and S be a doubling ball with B S, denote (24), it only needs to prove and To prove (25), for a fixed ball, x 2 B and f 2 L 1 . /, we decompose f as f .y/ D f 6 5 B .y/ C f X n 6 5 B .y/ DW f 1 .y/ C f 2 .y/: Thus, we write .6B/ jm f Where we use the fact that jm Q B .b/ m f For any x; y 2 B, by (7) and Minkowski inequality, we have For F 1 , we have Next we estimate F 2 . For any Q x 2 X and x 2 X n 6 5 B satisfying d. Q x; x/ < t , 2d. Q x; z/ Ä d.x; z/ and 1 2 d.x; z/ < t , we can conclude With a way similar to that used in the proof of F 2 , it follows that which, together with the estimates of F 1 and F 2 , it is easy to see that thus, the proof of (25) is finished. Now, we estimate (26). For any two balls B S with x 2 B and assume N WD N B;S C 1, where S is a doubling ball. Write Following the proof of E 3 , it is not difficult to see that Now, we estimate I 2 , for any y 2 X , applying Hölder inequality, we deduce where we have used the fact that jm S .b/ m Q B .b/j Ä CK B;S . Finally, we estimate for I 3 . For x 2 B, we have With a way similar to that used in the proof of E 2 , it follows that I 32 Ä C kf k L 1 . / : Meanwhile, following the proof of E 3 , we have Combining the estimates for I 31 , I 32 , I 1 , I 2 and I 4 , we obtain (26). Thus, we complete the proof of (24).
Proof of Theorem 1:9. For convenience, we assume kbk RBMO. / D 1. For each fixed t > 0 and functions f with bounded support, applying Lemma 2.5 to jf j with p D 1, and letting B j , S j , ' j and ! j as the same as Lemma 2.5.
We see that f D g C h, where Noticing that kgk L 1 . / Ä C kf k L 1 . / . Applying the L 2 . /-boundedness of M b;Ä in Theorem 1.8 and the fact that jg.x/j Ä C t , it is not difficult to obtain that jf .y/jd .y/: From (14), we have .[ j 6 2 B j / Ä C t kf k L 1 . / , so the proof of Theorem 1.9 is reduced to prove For each fixed j and x 2 X n [ With a way similar to that used in the estimate of H 21 ; H 22 and H 23 in Theorem 1.8, we have By h j WD f ! j ' j , write and for any s > 0 and t 1 ; t 2 > 0, we haveˆs For II 1 , by (14) and Lemma 2.6, we have In order to estimate II 2 , we assume that ƒ N is a set of finite index, r j .y/ WD 1 t j' j .y/j. By applying the convex property ofˆ, we getˆ On the other hand, if we take ƒ D N , the above inequality also holds by the property ofˆ. With a way similar to H 12 in the proof of Theorem 1.8, we have which, together with II 1 and (29), imply (28), and hence the proof of Theorem 1.9 is finished. Now, we estimate J 21 , applying Hölder inequality and (13), we have Similar to the estimate of J 21 , it is easy to obtain that J 22 Ä C j 2 j t : Thus, we can conclude that .fx 2 X W J 2 .x/ > t 2 g/ Ä C t 1 .j 1 j C j 2 j/ Ä C t 1 jhj H 1;1 atb . / :