Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type

Abstract: In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type (X, d, μ) in the sense of Coifman andWeiss. More precisely, we show that the commutator [b, T] is bounded on the weighted Morrey space Lp,κ ω (X) with κ ∈ (0, 1) and ω ∈ Ap(X), 1 < p < ∞, if and only if b is in the BMO space. We also prove that the commutator [b, T] is compact on the same weighted Morrey space if and only if b belongs to the VMO space. We note that there is no extra assumptions on the quasimetric d and the doubling measure μ.


Introduction
It is well-known that the boundedness and compactness of the commutator of Calderón-Zygmund operators on certain function spaces and their characterizations play an important role in various areas, such as harmonic analysis, complex analysis, (nonlinear) PDE, etc. See for example [3, 9, 10, 13, 18-20, 22, 24, 25] and the references therein. Recently, equivalent characterizations of the boundedness and the compactness of commutators were further extended to Morrey spaces over the Euclidean space by Di Fazio and Ragusa [16] and Chen et al. [5], and to weighted Morrey spaces by Komori and Shirai [30] for commutators of Calderón-Zygmund operator and by Tao, Da. Yang and Do. Yang [34,35] for the commutator of the Cauchy integral and Beurling-Ahlfors transformation, respectively. For more results on the boundedness of operators on Morrey spaces in di erent settings, we refer the reader to other studies as in [1, 15, 17, 26-28, 33, 37, 38] for instance.
Thus, along this literature, it is natural to study the boundedness and compactness of commutators of Calderón-Zygmund operators on weighted Morrey spaces in a more general setting: spaces of homogeneous type in the sense of Coifman and Weiss [8], as Yves Meyer remarked in his preface to [11], "One is amazed by the dramatic changes that occurred in analysis during the twentieth century. In the 1930s complex methods and Fourier series played a seminal role. After many improvements, mostly achieved by the Calderón-Zygmund school, the action takes place today on spaces of homogeneous type. No group structure is available, the Fourier transform is missing, but a version of harmonic analysis is still present. Indeed the geometry is conducting the analysis. " We say that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss if d is a quasimetric on X and µ is a nonzero measure satisfying the doubling condition. A quasi-metric d on a set X is a function d : X × X −→ [ , ∞) satisfying (i) d(x, y) = d(y, x) ≥ for all x, y ∈ X; (ii) d(x, y) = if and only if x = y; and (iii) the quasi-triangle inequality: there is a constant A ∈ [ , ∞) such that for all x, y, z ∈ X, (1.1) We say that a nonzero measure µ satis es the doubling condition if there is a constant Cµ such that for all x ∈ X and r > , µ(B(x, r)) ≤ Cµ µ(B(x, r)) < ∞, (1.2) where B(x, r) is the quasi-metric ball by B(x, r) := {y ∈ X : d(x, y) < r} for x ∈ X and r > . We point out that the doubling condition (1.2) implies that there exists a positive constant n (the upper dimension of µ) such that for all x ∈ X, λ ≥ and r > , µ(B(x, λr)) ≤ Cµ λ n µ(B(x, r)). (1.3) Throughout this paper we assume that µ(X) = ∞ and that µ({x }) = for every x ∈ X. We now recall the de nition of Calderón-Zygmund operators on spaces of homogeneous type.
Throughout this paper we assume that β(t) = t σ , for some σ > .
Note that by the doubling condition there exist two constants C and C such that C V(y, x) ≤ V(x, y) ≤ C V(y, x). As in [12], we assume that for any Calderón-Zygmund operator T as in De nition 1.1 with β(t) → as t → , the following "non-degenerate" condition holds: there exist positive constants c andĀ such that for every x ∈ X and r > , there exists y ∈ B(x,Ār) \ B(x, r), satisfying

|K(x, y)| ≥ c µ(B(x, r))
. (1.6) This condition gives a lower bound on the kernel and in R n . This "non-degenerate" condition was rst introduced in [22]. This is a natural assumption on the kernel of the singular integrals, since it is obviously true for Hilbert transform and Riesz transforms in the Euclidean setting, and for the Beurling-Ahlfors transformation in the complex setting. Beyond these, we note that, for example, on strati ed Lie groups, a similar condition of the Riesz transform kernel lower bound was shown to be true in [13].
Let T be a Calderón-Zygmund operator on X. Suppose that b ∈ L loc (X) and f ∈ L p (X). Let [b, T] be the commutator de ned by Let p ∈ ( , ∞), κ ∈ ( , ) and ω ∈ Ap(X). The weighted Morrey space L p,κ ω (X) is de ned by Our main results are the following: We mainly combine the ideas in [12] and [35] to prove our main result. We also point out that to obtain the above theorem, we provide an equivalent characterisation of VMO(X), which is stated in Lemma 2.4 below, and is of independent interest.
Throughout the paper, we denote by C and C positive constants which are independent of the main parameters, but they may vary from line to line. For every p ∈ ( , ∞), we denote by p the conjugate of p, i.e., p + p = . If f ≤ Cg or f ≥ Cg, we then write f g or f g; and if f g f , we write f ≈ g.

Preliminaries on Spaces of Homogeneous Type
Let (X, d, µ) be a space of homogeneous type as mentioned in Section 1. We now recall the de nition of the BMO and VMO spaces.
where the sup is taken over all quasi-metric balls B ⊂ X and The following John-Nirenberg inequalities on spaces of homogeneous type come from [29].

Lemma 2.2 ([29]
). If f ∈ BMO(X), then there exist positive constants C and C such that for every ball B ⊂ X and every α > , we have We recall the median value α B (f ) (see [4]): for any real valued function f ∈ L loc (X) and B ⊂ X, α B (f ) is the real number such that Moreover, it is known that α B (f ) satis es that And it is easy to see that for any ball B ⊂ X, where the implicit constants are independent of the function b and the ball B. By Lip(β), < β < ∞, we denote the set of all functions ϕ(x) de ned on X such that there exists a nite constant C satisfying for every x and y in X. ϕ β will stand for the least constant C satisfying the condition above. By Lip c (β), we denote the set of all Lip(β) functions with bounded support on X.
De nition 2.3. We de ne VMO(X) as the closure of the Lip c (β) functions X under the norm of the BMO space.
We will make use of the following characterization of VMO(X) whose proof is given in the Appendix. An equivalent characterization exists for the Euclidean and the strati ed Lie groups case; one can refer to [36] and [4].
if and only if f satis es the following three conditions: where r B is the radius of the ball B and x is a xed point in X.
To this end, we recall the de nition of Ap weights.

De nition 2.5.
Let ω(x) be a nonnegative locally integrable function on X. For < p < ∞, we say ω is an Ap weight, written ω ∈ Ap, if Here the supremum is taken over all balls B ⊂ X. The quantity [ω] Ap is called the Ap constant of ω. For p = , we say ω is an A weight, written ω ∈ A , if M(ω)(x) ≤ ω(x) for µ-almost every x ∈ X, and for p = ∞, let A∞ := ∪ ≤p<∞ Ap and we have [ω] A∞ := sup B − B ω exp − B log ω < ∞.
Note that for ω ∈ Ap the measure ω(x)dµ(x) is a doubling measure on X. To be more precise, we have that for all λ > and all balls B ⊂ X, (2.4) where n is the upper dimension of the measure µ, as in (1.3). We also point out that for ω ∈ A∞, there exist γ > such that for every ball B, This implies that for every ball B and for all δ ∈ ( , ), see also [25]. By the de nition of Ap weight and Hölder's inequality, we can easily obtain the following standard properties. Lemma 2.6. Let ω ∈ Ap(X),p ≥ . Then there exists constantsĈ ,Ĉ > and σ ∈ ( , ) such that for any measurable subset E of a quasi-metric ball B.
According to [2, Theorem 5.5], we have the following result for BMO functions on X.

Characterization of Boundedness for Commutators
In this section, we give the proof of Theorem 1.2.

. Proof of Theorem 1.2(i).
In order to prove Theorem 1.2(i), we need the following lemma.
Proof of Theorem 1.2(i). Let < p < ∞. It is su cient to prove that holds for any ball B. Now x a ball B = B(x , r) and decompose f = fχ A B + fχ X\ A B =: f + f . Without loss of generality we assume thorough out the proof that constant A := for the quasi-triangle inequality. Then For the rst term I, by Lemma 3.1 one has . Now for the second term II, observe that for x ∈ B, by (1.4), we have Next, to estimate III and IV, we need to decompose X \ B into suitable annuli. By Noting that k B → X, as k → ∞, we see that Then we choose a smallest j ≥ such that, We claim that such j exist, since otherwise, for all j ≥ , we have µ( j B) ≤ µ(B). Then it contradicts (3.1). We further point out that, since j is the smallest that satis es the criteria (3.3), we get that Then, from the doubling property, we also have Next we choose a smallest j ≥ j + such that

and that µ( j B) ≤ Cµ µ( j B).
Similarly we see that such j exists. By induction, there exists a sequence {j k } ∞ k= such that For III, by using Hölder's inequality, and using Lemma 2.6 and Lemma 2.7, we have For the term IV, using Hölder's inequality and the decomposition for X\B as above, we get Since b ∈ BMO(X), by Lemma 2.2, there exist some constants C > and C > such that for any ball B and α > Then by Lemma 2.6, we have Similarly, we have Together with Lemma 2.6, we have . This completes the proof.

. Proof of Theorem 1.2(ii).
We rst recall another version of the homogeneous condition (formulated in [12]): there exist positive constants ≤ A ≤ A such that for any ball B := B(x , r) ⊂ X, there exist balls B := B(y , r) such that A r ≤ d(x , y ) ≤ A r, and for all (x, y) ∈ (B × B), K(x, y) does not change sign and Then we rst point out that the homogeneous condition (1.6) implies (3.5). (1.6). Then T satis es (3.5).

Lemma 3.2 ([12]). Let T be the Calderón-Zygmund operator as in De nition 1.1 and satisfy the homogeneous condition as in
Since b is real valued, using Lemma 2.6, Hölder's inequality, boundedness of [b, T] on L p,κ ω (X) and (3.5), we get . This nishes the proof of Theorem 1.2(ii).

Compactness Characterization of the Commutator
Now we will prove Theorem 1.3.

. Proof of Theorem 1.3(i).
We will rst give a su cient condition for subsets of weighted Morrey spaces to be relatively compact. Recall that a subset F of L p,κ ω (X) is said to be totally bounded (relatively compact) if the L p,κ ω (X) closure of F is compact.
is totally bounded if the set F satis es the following three conditions: (ii) F uniformly vanishes at in nity, namely, for any ϵ ∈ ( , ∞), there exists some positive constant M such that, for any f ∈ F, The proof of this lemma follows from [32] using a minor modi cation from Euclidean setting to space of homogeneous type, since it only requires following properties of underlying space: metric and doubling measure.
We will now establish the boundedness of maximal operator T * of a family of smooth truncated operators {Tη} η∈( ,∞) as follows. For η ∈ ( , ∞), let The maximal operator T * is de ned as Recall the Hardy-Littlewood maximal Operator M is de ned by for any f ∈ L loc (X) and x ∈ X, where the supremum is taken over all balls B of X that contain x. Then we have the following lemmas.
Proof. Let f ∈ L loc (X). For any x ∈ X, we have which completes the proof of the Lemma 4.2.

Lemma 4.3.
Let p ∈ ( , ∞), κ ∈ ( , ) and ω ∈ Ap (X) . Then there exists a positive constant C such that, for any f ∈ L p,κ ω (X), Proof. For the boundedness of M on L p,κ ω (X) one can refer to [2]. We only consider the boundedness of T * . For any xed ball B ⊂ X and f ∈ L p,κ ω (X) , we write Again, following the argument in (3.4), there exist j k ∈ N such that Observe f ∈ L p ω (X) . Then, from the boundedness of T * on L p ω (X) (see, for example, [23, Theorem 1.1]), the Hölder inequality, size and smoothness of Kernel, we deduce that where, in the fourth inequality, we used Lemma 2.6 with some σ ∈ ( , ). This nishes the proof of Lemma 4.3.
ε f L p,κ ω (X) . Moreover, by using Lemmas 4.2 and 4.3, we get Now it su ces to show that, for any b ∈Lip c (β), < β < ∞ and η ∈ ( , ∞) small enough, [b, Tη] is a compact operator on L p,κ ω (X) , which is equivalent to show that, for any bounded subset F ⊂ L p,κ ω (X) , [b, Tη] F is relatively compact. That is, we need to verify [b, Tη] satis es the conditions (i) through (iii) of Lemma 4.1.
Next, let x be a xed point in X. Since b ∈Lip c (β), we may further assume b L ∞ = . Observe that there exists a positive constant R such that supp (b) ⊂ B (x , R ). Let M ∈ ( R , ∞) . Thus, for any y ∈ B (x , R ) and x ∈ X with d(x , x) > M, d(x, y) ∼ d(x , x). Then, for x ∈ X with d(x , x) > M, by Hölder inequality and using that V(x, y) ∼ µ (B(x , d(x, x ))) we conclude that Therefore, for any xed ball B := B( x, r) ⊂ X, by Lemma 2.6 , we have Note that As b ∈Lip c (β), it follows that, for any y ∈ B(x, r) To estimate L (x, y), we rst observe that Kη(x, z) = , Kη(y, z) = for any y ∈ B(x, r), d(x, z) ≤ η A and r < η A . Moreover, by the de nition of Kη we know that, for any y ∈ B(x, r), d(x, z) > η A and r < η A , This in turn implies that, for any y ∈ B(x, r) Using the estimates of L (x, y) and L (x, y), we have Then, by Lemma 4.3 and the boundedness of M on L p,κ ω (X) , we obtain Consequently  Then there exist real-valued functions f j j∈N ⊂ L p,κ ω (X) , positive constants K large enough, C , C and C such that, for any j ∈ N and integer k ≥ K , f j L p,κ ω (X) ≤C ,

2)
where B k j := A k− B j is the ball associates with A k− B j in (3.5) and Proof. For each j ∈ N, we de ne function f j as follows: where B j is as in the assumption of Lemma 4.4 and a j ∈ R is a constant such that Then, by the de nition of a j , (2.1) and (2.2) we have a j ≤ / , supp f j ⊂ B j and, for any x ∈ B j , Moreover, since a j ≤ / , we can obtain that, for any x ∈ B j, ∪ B j, , and hence Observe that, for any k ∈ N, we have Using Kernel estimates, (4.4), (4.6) and the fact that d x, x j ∼ d(x, ξ ) for any x ∈ B k j with integer k ≥ and ξ ∈ B j , we have, for any x ∈ B k j , As b BMO(X) = by John-Nirenberg inequality(c.f. [6]), for each k ∈ N and ball B ⊂ X, we have where the last inequality is due to the fact that Since ω ∈ Ap (X) , there exists ϵ ∈ ( , ∞) such that the reverse Hölder inequality holds for any ball B ⊂ X. Then by the Hölder inequality, (4.11), (4.7) and (4.10) we can deduce that there exists a positive constant C such that, for any k ∈ N By Lemma 3.1, (4.5), (4.6), (2.3), (4.1) and (1.6) for any x ∈ B k j , we have Then together with (4.8) we obtain that there exists a positive constant C such that Now we take K ∈ ( , ∞) large enough such that, for any integer k ≥ K From this and (4.9), (4.12) and (4.13), we have This implies (4.2). On the other hand, since supp f j ⊂ B j , by (4.6) and (2.3) and b BMO(X) = , we obtain that, for any Therefore, by (4.12) with B k j replaced by A k+ B j \ A k B j , we can deduce that, for any integer k ≥ K This completes the proof of Lemma 4.4.
We also need the following technical result to handle the weighted estimate for the necessity of the compactness of the commutators. where C := A K > C := A K for some K ∈ N large enough. (ii) r j j∈N is either non-increasing or non-decreasing in j, or there exist positive constants C min and Cmax such that, for any j ∈ N C min ≤ r j ≤ Cmax.
Then there exists a positive constant C such that, for any j, m ∈ N Proof. Without loss of generality, we may assume that b BMO(X) = and r j j∈N is non-increasing. Let f j j∈N , C , C be as in Lemma 4.4 associated with B j j∈N . By (4.2), (4.8), Lemma 2.6 with ω ∈ Ap (X), we nd that, for any j ∈ N, for some positive constant C independent of γ and A . We next prove that, for any j, m ∈ N, Since supp f j+m ⊂ B j+m , from (2.3), (4.6), (4.14) and b BMO(X) = , it follows that, for any And hence we have Moreover, from (4.6) we deduce that, for any (4.18) By using (4.18), the fact {r j } j∈N is non-increasing in j and Hölder's and reverse Hölder's inequalities we have Notice that, for C large enough, by (4.14) we know that d x j , x j+m is also large enough and hence Using (4.17), (4.18) and (4.19), we deduce that Note that lim k→∞ µ(A k B j+m ) = ∞. Then for C large enough, we have This implies that C This nishes the proof of (4.16). By (4.15) and (4.16) we know that, for any j, m ∈ N and C large enough This nishes the proof of Lemma 4.7.
Proof of Theorem 1.3(ii). Without loss of generality, we may assume that b BMO(X) = . To show b ∈ VMO (X), noticing that b ∈ BMO (X) is a real-valued function, we can use a contradiction argument via Lemmas 2.4, 4.4 and 4.5. Now observe that, if b ∉ VMO (X) , then b does not satisfy at least one of (i) through (iii) of Lemma 2.4. We show that [b, T] is not compact on L p,κ ω (X) in any of the following three cases. Case (i) b does not satisfy condition (i) Lemma 2.4. Then there exist γ ∈ ( , ∞) and a sequence of balls in X satisfying (4.1) and that r ( ) j → as j → ∞. Let x be a xed point in X. We will now consider the following two subcases.
Let f j j∈N be associated with B j j∈N ,C C , K and C be as in Lemmas 4.4 and 4.5. Let p ∈ ( , p) be such that ω ∈ Ap (X) and C := A K > C = A K for K ∈ N large enough such that We then have We will rst consider the term F . Assume that E j := J\J ≠ ∅. Then E j ⊂ C B ( ) j +m by (4.21) we have (4.23) Now let be the ball associates with A k− B ( ) j in (3.5). Then using (4.23), we have By this, we further know that there exist nite mutually disjoint B ( ) intersecting E j . By (4.2) and Lemma 2.6, we conclude that If E j l := J \ J = ∅, the inequality is still true. Note that lim k→∞ µ(A k B ( ) j l+m ) = ∞. Then there exist j k ∈ N such that Moreover, from the proof of (4.3), Lemma 4.4, (4.20) and (4.21), we deduce that By (4.21), (4.22), (4.24) and (4.25) we obtain Thus, [b, T]f j j∈N is not relatively compact in L p,κ ω (X) , which implies that [b, T] is not compact on L p,κ ω (X). Therefore, b satis es condition (i) of Lemma 2.4.
Subcase (ii) There exists a subsequence B ( ) Observe that A C B ( ) j ⊂ A C B ( ) j + for any j ∈ N and hence We can use a similar method as that used in Subcase (i) of Case (i) and rede ne our sets in a reversed order. That is, for any xed , k ∈ N, let and for any i ≠ m,

Appendix: Characterisation of VMO(X)
In this section, we provide a characterisation of the VMO space on X by giving the proof of Lemma 2.4.
Proof of Lemma 2.4. In the following, for any integer m, we use B m to denote the ball B(x , m ), where x is a xed point in X.
Necessary condition: Assume that f ∈ VMO(X). If f ∈ Lip c (β), then (i)-(iii) hold. In fact, by the uniform continuity, f satis es (i). Since f ∈ L (X), f satis es (ii). By the fact that f is compactly supported, f satis es (iii). If f ∈ VMO(X)\Lip c (β), by de nition, for any given ε > , there exists fε ∈ Lip c (β) such that f −fε BMO(X) < ε. Since fε satis es (i)-(iii), by the triangle inequality of BMO(X) norm, we can see (i)-(iii) hold for f . Su cient condition: In this proof for j = , , · · · , , the value α j is a positive constant depending only on n and α i for ≤ i < j. Assume that f ∈ BMO(X) and satis es (i)-(iii). To prove that f ∈ VMO(X), it su ces to show that there exist positive constants α , α such that, for any ε > , there exists ϕε ∈ BMO(X) satisfying and ϕε − f BMO(X) < α ε. We rst establish a cover of X. Observe that For each R jε ν,−iε , ν = , , · · · , jε+iε − , letB which is contradict to the fact that Bx ∩ B x ≠ ∅ (Without loss of generality, here we assume that A = in the quasi-triangle inequality. Otherwise, we just need to take r B m = ([ A ] + ) m and make some modi cations). Now we de ne ϕε. By (ii), there exists mε > jε large enough such that when r B > mε−iε−jε , we have De ne We claim that there exist positive constants Observe that supp (hε) ⊂ B mε and there exists a function hε ∈ Cc(X) such that for any x ∈ X, |hε(x) − hε(x)| < ε. Let η(s) be an in nitely di erentiable function de ned on [ , ∞) such that ≤ η(s) ≤ , η(s) = for ≤ s ≤ and η(s) = for s ≥ . And let h t ε (x) = X ρ(x, y, t)hε(y)dµ(y).
we can obtain (5.1) by letting t go to and by taking α = . Now we show (5.2). To this end, we only need to prove that for any ball B ⊂ X, We rst prove that for every Bx with x ∈ B mε , Therefore, Then (5.2) holds by taking α = max{ + α α , + α , C (α α + )}. This nishes the proof of Lemma 2.4.