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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 15, Issue 1

Issues

Commutators of Littlewood-Paley gκ-functions on non-homogeneous metric measure spaces

Guanghui Lu / Shuangping Tao
Published Online: 2017-11-13 | DOI: https://doi.org/10.1515/math-2017-0110

Abstract

The main purpose of this paper is to prove that the boundedness of the commutator Mκ,b generated by the Littlewood-Paley operator Mκ 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 satisfies a certain Hörmander-type condition, the authors prove that Mκ,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).

Keywords: Non-homogeneous metric measure space; Commutators; gκ*-functions; RBMO (μ); Hardy space

MSC 2010: 42B25; 42B35; 30L99

1 Introduction

In 1958, Stein [1] firstly introduced and studied Littlewood-Paley gκ -functions on ℝn. After that, many authors paid much attention to study the properties of the Littlewood-Paley gκ -functions on various function spaces, for example, see [2, 3, 4, 5, 6, 7]. With deeper research, the boundedness of Littlewood-Paley operators and their commutators under the cases of non-doubling measures is also widely discussed (see [8, 9, 10, 11, 12, 13, 14]).

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, μ) 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 gκ -functions on (𝓧, d, μ).

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].

Definition 1.1

([25]). A metric space (𝓧, d) is said to be geometrically doubling, if there exists some N0∈ ℕ such that, for any ball B(x, r)⊂ 𝓧, there is a finite ball covering {B(xi,r2)}i of B(x, r) such that the cardinality of this covering is at most N0.

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 ϵ ∈ (0, 1) and ball B(x, r)⊂ 𝓧, there exists a finite ball covering {B(xi, ϵr)}i of B(x, r) such that the cardinality of this covering is at most N0ϵn. Here and in what follows, N0 is as Definition 1.1 and n := log2N0.

  3. For every ϵ ∈ (0, 1), any ball B(x, r)⊂ 𝓧 can contain at most N0ϵn centers {xi}i of disjoint balls with radius ϵr.

  4. There exists M ∈ ℕ such that any ball B(x, r)⊂ 𝓧 can contain at most M centers {xi}i of disjoint balls {B(xi,r4)}i=1M.

Now, we recall the definition of upper doubling conditions given in [15].

Definition 1.3

([15]). A metric measure space (𝓧, d, μ) is said to be upper doubling, if μ is Borel measure on 𝓧 and there exist a dominating function λ : 𝓧×(0, ∞)→(0, ∞) and a positive constant Cλ such that, for each x ∈ 𝓧, rλ(x, r) is non-decreasing and, for all x ∈ 𝓧 and r ∈ (0, ∞), μ(B(x,r))λ(x,r)Cλλ(x,r2).(1)

Hytönen et al. proved in [16] that there exists another dominating function λ̃ such that λ̃λ, Cλ̃Cλ and, for all x, y ∈ 𝓧 with d(x, y) ≤ r, λ~(x,r)Cλ~λ~(y,r).(2)

Based on this, from now on, we always assume that the dominating function λ as in (1) satisfies (2).

The following coefficient KB,S which introduced [15] by Hytönen is analogous to Tolsa’s number in [8, 9].

Given any two balls BS, set KB,S:=1+2SB1λ(cB,d(x,cB))dμ(y),(3)

where cB represents the center of the ball B.

Hytönen [15] gave the definition of (α, β)-doubling, that is, a ball B⊂ 𝓧 is called (α, β)-doubling if μ(αB)≤ βμ(B) for α, β > 1. At the same time, Hytönen proved that if a metric measure space (𝓧, d, μ) is upper doubling and β > Cλlog2α =: αν, then for every ball B ⊂ 𝓧, there exists some j ∈ ℤ+ such that αj B is (α, β)-doubling. In addition, let (𝓧, d) be geometrically doubling, β > αn with n = log2N0 and μ Borel measure on 𝓧 which is finite on bounded sets. Hytönen also showed that for μ-a.e x ∈ 𝓧, there exist arbitrarily small (α, β)-doubling balls centered at x. Furthermore, the radius of there balls may be chosen to be form αjr for j ∈ ℕ and any preassigned number r ∈ (0, ∞). Throughout this paper, for any α ∈ (1, ∞) and ball B, the smallest (α, βα)-doubling ball of the form αj B with j ∈ ℕ is denoted by α, where βα:=α3(max{n,ν})+30n+30ν.(4)

For convenience, we always assume α = 6 in this paper and denote 6 simply by .

Now we recall the notion of RBMO (μ) from [15].

Definition 1.4

([15]). Let ν > 1. A function fLloc1(μ) is claimed to be in the space RBMO (μ) if there exist a positive constant C and, for any ball B ⊂ 𝓧, a number fB such that 1μ(νB)B|f(x)fB|dμ(x)C(5)

and, for any two balls B and R such that BR, |fBfR|CKB,R.(6)

The infimum of the constants C satisfying (5) and (6) is defined to be the RBMO (μ) norm of f and denoted byfRBMO(μ).

Next, we recall the definition of the Littlewood-Paley gκ -function given in [17].

Definition 1.5

([17]). Let K(x, y) be a locally integrable function on (𝓧× 𝓧)\ {(x, x) : x ∈ 𝓧}. Assume that there exists a non-negative constant C such that, for all x, y ∈ 𝓧 with xy, |K(x,y)|Cd,(x,y)λ(xd(x,y))(7)

and, for all y, y′ ∈ 𝓧, d(x,y)2d(y,y),[|K(x,y)K(x,y)|+|K(y,x)K(y,x)|]dμ(x)d(x,y)C.(8)

The Littlewood-Paley gκ -function Mκ is defined by Mκ(f)(x):=[X×(0,)(tt+d(x,y))κ|1td(y,z)tK(y,z)f(z)dμ(z)|2dμ(y)dtλ(y,t)t]12,(9)

where x ∈ 𝓧, 𝓧 ×(0, ∞) = {(y, t) : y ∈ 𝓧, t > 0} and κ > 1.

Let b ∈ RBMO(μ) and K(x, y) satisfy (7) and (8). The commutator of Littlewood-Paley gκ -function Mκ,b is formally defined by Mκ,b(f)(x):=[X×(0,)(tt+d(x,y))κ×|1td(y,z)tK(y,z)[b(x)b(z)]f(z)dμ(z)|2dμ(y)dtλ(y,t)t]12.(10)

The following notion of the atomic Hardy space is from [16].

Definition 1.6

(16). Let ρ ∈ (1, ∞) and p ∈ (1, ∞]. A function bLloc1(μ) is called a(p, 1)τ-atomic block, if

  1. there exists a ball B such that supp(b) ⊂ B;

  2. 𝓧b(x)dμ(x) = 0;

  3. for any i ∈ {1, 2}, there exists a function ai supported on a ball BiB and τi ∈ ℂ such that b=τ1a1+τ2a2

    and aiLp(μ)[μ(ρBi)]1p1KBi,B1.(11)

    Moreover, let |b|Hatb1,p(μ):=|τ1|+|τ2|.

Definition 1.7

([16]). Let p ∈ (1, ∞]. A function fL1(μ) is said to belong to the atomic Hardy space Hatb1,p (μ), if there exist (p, 1)τ-atomic blocks {bi}i=1 such that f=i=1bi in L1(μ) and i=1|bi|Hatb1,p(μ) < ∞. The norm of f in Hatb1,p(μ) is defined by fHatb1p(μ):=inf{i=1|bi|Hatb1,p(μ)},

where the infimum is taken overall the possible decompositions of f as above.

According to [14], the definition of the Hörmander-type condition on (𝓧, d, μ) is defined by: supr>0d(y,y)ri=16ir<d(x,y)6i+1ri[|K(x,y)K(x,y)|+|K(y,x)K(y,x)|]dμ(x)d(x,y)C,(12)

which is slightly stronger (8).

Our main results in this paper are formulated as follows.

Theorem 1.8

Let b ∈ RBMO(μ), K(x, y) satisfy (7) and (12) and Mκ,b (f) be as in (10). Suppose that Mκ is bounded on L2(μ). Then Mκ,b (f) is bounded on Lp(μ) for 1 < p < ∞, that is, there exists a constant C > 0, such that for all functions f with bounded support, one has Mκ,b(f)Lp(μ)CbRBMO(μ)fLp(μ).

Theorem 1.9

Let b ∈ RBMO(μ), K(x, y) satisfy (7) and (12) and Mκ,b (f) be as in (10). Suppose that Mκ is bounded on L2(μ). Then there is a positive constant C, such that for all functions f with bounded support, μ({xX:Mκ,b(f)(x)>t})CΦ(bRBMO(μ))XΦ(|f(x)|t)dμ(y),

where Φα(t) = t logα(2 + t) for α ≥ 1.

Theorem 1.10

Let b ∈ RBMO(μ), K(x, y) satisfy (7) and (12) and Mκ,b(f) be as in (10). Suppose that Mκ is bounded on L2(μ). Then Mκ,b(f) is bounded from H1(μ) into L1,∞ (μ), namely, there is a positive constant C, such that for all fH1(μ) and t > 0, one has μ({xX:Mκ,b(f)(x)>t})CbRBMO(μ)fH1(μ)t.

2 Preliminaries

In this section, we shall recall some lemmas used in the proofs of our main theorems. Firstly, we recall some useful properties of KB,S as in (3) (see [15]).

Lemma 2.1

([15]).

  1. For all balls BRS, it holds true that KB,RKB,S.

  2. For any ξ ∈[1, ∞), there exists a positive constant Cξ, such that, for all balls BS with rSξ rB, KB,SCξ.

  3. For any ϱ ∈ (1, ∞), there exists a positive constant Cϱ, depending on ρ, such that, for all balls B, KB,ϱCϱ.

  4. There exists a positive constant c such that, for all balls BRS, KB,SKB,R + cKR,S. In particular if B and R are concentric, then c = 1.

  5. There exists a positive constant such that, for all balls BRS, KB,Rc͠KB,S moreover, if B and R are concentric, then KR,SKB,S.

Now, we recall the following conclusion, which is just [18].

Corollary 2.2

([18]). If f ∈ RBMO(μ), then there exists a positive constant C such that, for any ball B, ϖ ∈ (1, ∞) and r ∈[1, ∞), (1μ(ϖB)B|f(x)mB~(f)|rdμ(x))1rCfRBMO(),(13)

where above and in what follows, m(f) denotes the mean of f over , namely, mB~(f):=1μ(B~)B~f(y)dμ(y).

Moreover, the infimum of the positive constants C satisfying |mB(f) − mS(f)|≤ CKB,S and (13) is an equivalent RBMO (μ) norm of f.

Next, we recall some results from [15, 19].

Lemma 2.3

([15]). (1) Let p ∈ (1, ∞), r ∈ (1, p) and ϕ ∈ (0, ∞). The following maximal operators defined, respectively, be setting, for all fLloc1(μ) and x ∈ 𝓧, Mr,ϕf(x):=supQx(1μ(ϕQ)Q|f(y)|rdμ(y))1r,Nf(x):=supQx,Qdoubling1μ(Q)Q|f(y)|dμ(y)

and Mϕf(x):=supQx1μ(ϕQ)Q|f(y)|dμ(y)

are bounded on Lp(μ) and also boundedfrom L1(μ) into L1,∞ (μ).

(2) For all fLloc1(μ) , it holds true that |f(x)|≤ Nf(x) for μ-almost every x ∈ 𝓧.

The following result is given in [19].

Lemma 2.4

([19]). Let fLloc1(μ) with the extra condition𝓧f(x)dμ(x) = 0 ifμ∥ := μ(𝓧)<∞. Assume that for some p, 1 < p < ∞, inf{1, Nf} ∈ Lp(μ). Then we have NfLp(μ)CMfLp(μ),

where Mf(x):=supBx1μ(6B)B|f(x)mB~(f)|dμ(x)+sup(Q,R)Δx|mQ(f)mR(f)|KQ,RforallfL1oc1(μ) and x ∈ 𝓧, and Δx := {(Q, R) : xQR and Q, R are doubling balls}.

Also, we recall the following Calderón-Zygmund decomposition theorem given in [19]. Suppose γ0 is a fixed positive constant satisfying that γ0 > max{Cλ3log26,63n}, where Cλ is as in (1) and n as in Remark 1.2.

Lemma 2.5

([19]). Let p ∈[1, ∞), fLp(μ) and t ∈ (0, ∞)(t > γ0||f||Lp(μ)μ(X) when μ(𝓧)<∞). Then

  1. there exists a family of finite overlapping balls {6Bi}i such that {Bi}i is pairwise disjoint, 1μ(62Bi)Bi|f(x)|pdμ(x)>tpγ0foralli,1μ(62τBi)τBi|f(x)|pdμ(x)tpγ0foralliandallτ(2,),(14)

    and |f(x)|tforμ-almosteveryxX(i6Bi);(15)

  2. for each i, let Si be a (3×62,Cλlog2(3×62)+1)-doubling ball of the family {(3 × 62)kBi}k∈ℕ, and ωi = χ6Bi/(kχ6Bk). Then there exists a family {φi}i of functions that for each i, supp(φi) ⊂ Si, φi has a constant sign on Si and Xφi(x)dμ(x)=6Bif(x)ωi(x)dμ(x),(16) i|φj(x)|γtforμ-almosteveryxX,(17)

    where γ is some positive constant depending only on (𝓧, μ), and there exists a positive constant C, independent of f, t and i, such that, if p = 1, then φiL(μ)μ(Si)CX|f(x)ωi(x)|dμ(x),(18)

    and if p ∈ (1, ∞), (si|φi(x)|pdμ(x))1p[μ(Si)]1pCtp1X|f(x)ωi(x)|pdμ(x).(19)

Finally, we recall the following John-Nirenberg inequality from [15].

Lemma 2.6

([15]). For every ς > 1, there exists a positive constant C such that, for every b ∈ RBMO(μ) and every ball B, μ({xB:|b(x)mB~(b)|>t})Cμ(ςB)exp(CtbRBMO(μ)).

From Lemma 2.6, it is easy to prove that there are two positive constants B1 and B2 such that, for any ball B and b ∈ RBMO(μ), 1μ(ςB)Bexp(|b(x)mB~(b)|B1bRBMO(μ))dμ(x)B2.

3 Proofs of Theorems 1.81.10

Proof of Theorem 1.8

Let 0 < r < 1, we firstly claim that, for any p ∈ (1, ∞), bL(μ) and all bounded functions f with compact support, μ({xX:Mr[Mκ,b(f)](x)>t})CtpbRBMO(μ)pfLp(μ)p.(20)

Once (20) is established, by the Marcinkiewicz interpolation theorem, it is easy to obtain that Mr[Mκ,b(f)]Lp(μ)CbRBMO(μ)fLp(μ).

By Lemma 2.4, for any p ∈ (1, ∞), bL(μ) and all bounded function fwith compact support and integral zero, Mκ,b(f)Lp(μ)CbRBMO(μ)fLp(μ),

together with the fact that the bounded function f with compact support and integral zero is dense in Lp(μ) (see [19, Theorem 6.4]), we finish the proof of Theorem 1.8.

Now, we turn to estimate (20). Without loss of generality, let ϕ = 6 as in Lemma 2.3 and ∥bRBMO(μ) = 1. For each fixed t > 0 and bounded function f with compact support and integral zero, applying Lemma 2.5 to f, we see that f = g + h, where g:=fχXj6Bj+jφj and h:=j(ωjfφj)=:jhj. By (15) and (17), we easily get gL(μ)Ct.(21)

On the other hand, applying (17), (19) and Hölder inequality, we have gLp(μ)CfLp(μ)+Ct11pjφjL1(μ)1pCfLp(μ)+Ct11p[j(sj|φj(x)|pdμ(x))1p[μ(Sj)]1p]1pCfLp(μ)+Ct11pt1p1(X|f(x)|pdμ(x))1pCfLp(μ).

Further, for 1 < s < p, by applying the result of the Claim and (21), we deduce μ({xX:Mr[Mκ,b(g)](x)>2Ct})μ({xX:Ms,6[Mκ(g)](x)>Ct})+μ({xX:gL(μ)>Ct})CtpMs,6[Mκ(g)]Lp(μ)pCtpgLp(μ)pCtpfLp(μ)p.(22)

From this, we can write μ({xX:Mr[Mκ,b(h)](x)>t})μ({xX:CMr,6[Mκ,b(h)](x)>t})μ({xX:CMr,6[Mκ(j(bm6Bj~(b))hj)](x)>t2})+μ({xX:CMr,6[j|b(x)m6Bj~(b)|Mκ(hj)](x)>t2})=:H1+H2,

where we have used the fact that Mrf(x) ≤ CMr,6 f(x) (see [20]).

By applying the (L1(μ), L1,∞(μ))-boundedness of Mρ, for any σ > 0, we get σμ({xX:Mr,6(u)(x)>σ})Csupτ>Cστμ({xX:|u(x)|>τ}).(23)

Choosing 1 < p1 < p, by hj := j(jφj) and (23), we have H1Ct1supτ>Ctτ×1τp1Mκ[j(bm6Bj~(b))hj]Lp1(μ)p1Ctp1j(bm6Bj~(b))fωjLp1(μ)p1+Ctp1j(bm6Bj~(b))φjLp1(μ)p1=:H11+H12.

For H11. By Hölder inequality, (13) and (14), we have H11Ctp1j(6Bj|f(x)|pdμ(x))p1p(6Bj|b(x)m6Bj~(b)|pp1pp1dμ(x))pp1pCtp1j(6Bj|f(x)|pdμ(x))p1p[μ(62Bj)]1p1pCtp1j(6Bj|f(x)|pdμ(x))p1p(tpBj|f(x)|pdμ(x))1p1pCtpfLp(μ)p.

With a way similar to that used in the proof of D12 in [20], it is easy to obtain H12CtpfLp(μ)p.

Now, we turn to estimate H2. By (23) and hj := j (jφj), write H2Ct1supτ>Ct2τμ({xX:j|bm6Bj~(b)|Mκ(hj)(x)>Cτ})Ct1jX|b(x)m6Bj~(b)|Mκ(hj)(x)dμ(x)Ct1jX6Sj|b(x)m6Bj~(b)|Mκ(hj)(x)dμ(x)+Ct1j6Sj|b(x)m6Bj~(b)|Mκ(φj)(x)dμ(x)+Ct1j656Bj|b(x)m6Bj~(b)|Mκ(fωj)(x)dμ(x)+Ct1j6Sj656Bj|b(x)m6Bj~(b)|Mκ(fωj)(x)dμ(x)=:H21+H22+H23+H24.

An argument similar to that used in the proof of E1 in [17, Theorem 1.10] shows that H21Ct1hjL1(μ),H24Ct1hjL1(μ),

which, together with the fact that ∥hjL1(μ)C t1−p fLp(μ)p, thus, H21 + H24C tp fLp(μ)p.

For H22. By Hölder inequality, the Lp(μ)-boundedness of Mκ, (13) and (19), we have H22Ct1j6Sj|b(x)m6Sj~(b)|Mκ(φj)(x)dμ(x)+Ct1j|m6Sj~(b)m6Bj~(b)|6SjMκ(φj)(x)dμ(x)Ct1jMκ(φj)Lp(μ)[(6Si|b(x)m6Sj~(b)|pdμ(x))1p+μ(6Sj)1p]Ct1jφjLp(μ)μ(6Sj)1pCt1t1pj6Bj|f(x)|pdμ(x)CtpfLp(μ)p.

Similar to the estimate of the H22, we conclude H23Ct1j(656Bj|b(x)m6Bj~(b)|pdμ(x))1pMκ(fωj)Lp(μ)Ct1jμ(62Bj)1pfωjLp(μ)Ctpj6Bj|f(x)|pdμ(x)CtpfLp(μ)p.

Combining the estimates for H21, H22, H23 and H24, we get H2CtpfLp(μ)p,

which, together with H1 and (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 ∈ (1, ∞), p0 ∈ (1, ∞) and bL(μ) . If Mκ is bounded on L2(μ), then there exists a positive constant C such that, for all fL(μ) ∩ Lp0(μ) and for all x ∈ 𝓧, M[Mb,κ(f)](x)C{bRBMO(μ)Ms,6[Mκ(f)](x)+bRBMO(μ)fL(μ)}.(24)

Proof

Without loss generality, we may assume ∥bRBMO(μ) = 1. Let B be an arbitrary ball and S be a doubling ball with BS, denote hB:=mB[Mκ([bmB~(b)]fχX65B)]

and hS:=mS[Mκ([bmS(b)]fχX65B)]

To prove (24), it only needs to prove 1μ(6B)B|Mκ,b(f)(y)hB|dμ(y)CMs,6[Mκ(f)](x)+fL(μ)(25)

and |hBhS|CKB,S2{Ms,6[Mκ(f)](x)+fL(μ)}.(26)

To prove (25), for a fixed ball, xB and fL(μ), we decompose f as f(y)=fχ65B(y)+fχX65B(y)=:f1(y)+f2(y).

Thus, we write 1μ(6B)B|Mκ,b(f)(y)hB|dμ(y)1μ(6B)B|b(y)mB~(b)|Mκ(f)(y)+Mκ([bmB~(b)]f1)(y)+Mκ([bmB~(b)]f2)(y)hB|dμ(y)1μ(6B)B|b(y)mB~(b)|Mκ(f)(y)dμ(y)+1μ(6B)BMκ([bmB~(b)]f1)(y)dμ(y)+1μ(6B)B|Mκ([bmB~(b)]f2)(y)hB|dμ(y)=:E1+E2+E3.

For E1, by Hölder inequality and (13), we have E11μ(6B)B|b(y)mB~(b)|sdμ(y)1sB[Mκ(f)(y)]sdμ(y)1s1μ(6B)B|b(y)mB~(b)|sdμ(y)1s1μ(6B)B[Mκ(f)(y)]sdμ(y)1sCMs,6[Mκ(f)](x).

Applying Hölder inequality, the L2(μ)-boundedness of Mκ and (13), we can conclude E2[μ(B)]12μ(6B)B[Mκ([bmB~(b)]f1)(y)]2dμ(y)12C[μ(B)]12μ(6B)(65B|b(y)mB~(b)|2|f(y)|2dμ(y))12CfL(μ)[μ(B)]12[μ(2×65B)]12μ(6B)(1μ(2×65B)65B|b(y)m65B~(b)|2dμ(y))12+CfL(μ)[μ(B)]12[μ(65B)]12μ(6B)|m65B~(b)mB~(b)|CfL(μ).

Where we use the fact that |mB~(b)m65B~(b)|CKB,65BC.

For E3, it follows that E3=1μ(6B)B|Mκ([bmB~(b)]f2)(y)hB|dμ(y)=1μ(6B)B|Mκ([bmB~(b)]f2)(y)1μ(B)BMκ([bmB~(b)]f2)(x)dμ(x)|dμ(y)1μ(6B)1μ(B)BB|Mκ([bmB~(b)]f2)(y)Mκ([bmB~(b)]f2)(x)|dμ(x)dμ(y),

For any x, yB, by (7) and Minkowski inequality, we have |Mκ([bmB~(b)]f2)(x)Mκ([bmB~(b)]f2)(y)|[X×(0,)|(tt+d(x,x~))κ2(tt+d(y,x~))κ2|2×|d(x~,z)tK(x~,z)[b(z)mB~(b)]f2(z)dμ(z)|2dμ(x~)dtλ(x~,t)t3]12CX|b(z)mB~(b)||f2(z)|[d(x~,z)t[d(x,y)d(x,x~)]2[d(x~,z)]2[λ(x~,d(x~,z))]2dμ(x~)dtλ(x~,t)t3]12dμ(z)CfL(μ)X65B|b(z)mB~(b)|[d(x~,z)t2d(x~,z)>d(x,z)[d(x,y)d(x,x~)]2[d(x~,z)]2[λ(x~,d(x~,z))]2×dμ(x~)dtλ(x~,t)t3]12dμ(z)+CfL(μ)X65B|b(z)mB~(b)|[d(x~,z)t,d(x,x~)<t2d(x~,z)d(x,z)[d(x,y)d(x,x~)]2[d(y,z)]2[λ(y,d(y,z))]2×dμ(x~)dtλ(x~,t)t3]12dμ(z)+CfL(μ)X65B|b(z)mB~(b)|[d(x~,z+t,d(x,x~)>t2d(x~,z)d(x,z)[d(x,y)d(x,x~)]2[d(y,z)]2[λ(y,d(y,z))]2×dμ(x~)dtλ(x~,t)t3]12dμ(z)=:F1+F2+F3.

For F1, we have F1CrBfL(μ)X65B|b(z)mB~(b)|[d(x~,z)2d(x~,z)>d(x,z)[d(x~,z)]2[d(x,z)]2[λ(x~,d(x~,z))]2×dμ(x~)dtλ(x~,t)t3]12dμ(z)CrBfL(μ)X65B|b(z)mB~(b)|d(x,z)[2d(x~,z)>d(x,z)[d(x~,z)]2[λ(x~,d(x~,z))]2×1λ(x~,d(x~,z))(d(x~,z)dtt3)dμ(x~)]12dμ(z)CrBfL(μ)X65B|b(z)mB~(b)|d(x,z)[2d(x~,z)>d(x,z)1λ(x~,d(x~,z))]2×1λ(x~,d(x~,z))dμ(x~)]12dμ(z)CrBfL(μ)X65B|b(z)mB~(b)|d(x,z)[2d(x~,z)>d(x,z)1λ(x~,12d(x,z))]2×1λ(x~,d(x~,z))dμ(x~)]12dμ(z)CrBfL(μ)X65B|b(z)mB~(b)|d(x,z)[k=02kB(z,d(cB,z))2k1B(z,d(cB,z))×1λ(x~,d(x~,z))dμ(x~)]121λ(z,d(x,z))dμ(z)CfL(μ)k=16(k1)6k65B6k165B|b(z)mB~(b)|λ(cB,d(cB,z))dμ(z)CfL(μ)k=16(k1)λ(cB,6k165rB)[6k65B|b(z)m6k65B~(b)|dμ(z)+m6k65B~(b)mB~(b)|]CfL(μ).

Next we estimate F2. For any ∈ 𝓧 and xX65B satisfying d(, x) < t, 2d(, z) ≤ d(x, z) and 12d(x,z)<t, we can conclude F2CrBfL(μ)X65B|b(z)mB~(b)|d(x,z)[2d(x~,z)d(x,z)12d(x,z)[d(x~,z)]2[λ(x~,d(x~,z))]2×dμ(x~)dtλ(x~,t)t3]12dμ(z)CrBfL(μ)X65B|b(z)mB~(b)|d(x,z)[2d(x~,z)d(x,z)[d(x~,z)]2[λ(x~,d(x~,z))]21λ(x~,d(x~,z))×(12d(x,z)dtt3)dμ(y)]12dμ(z)CrBfL(μ)X65B|b(z)mB~(b)|λ(cB,d(cB,z))1d(x,z)dμ(z)CfL(μ)k=16(k1)6k65B6k165B|b(z),mB~(b)|λ(cB,d(cB,z))dμ(z)CfL(μ).

With a way similar to that used in the proof of F2, it follows that F3CfL(μ),

which, together with the estimates of F1 and F2, it is easy to see that E3CfL(μ),

thus, the proof of (25) is finished.

Now, we estimate (26). For any two balls BS with xB and assume N : = NB, S+1, where S is a doubling ball. Write |hBhS|=|mB[Mκ([bmB~(b)]fχX65B)]mS[Mκ([bmS(b)]fχX65B)]||mB[Mκ([bmB~(b)]fχX6NB)]mS[Mκ([bmS(b)]fχX6NB)]|+|ms[Mκ([bms(b)]fχX6NB)]ms[Mκ([bmB~(b)]fχX6NB)]|+|mB[Mκ([bmB~(b)]fχ6NB65B)]|+|mS[Mκ([bmS(b)]fχ6NB65B)]|=:I1+I2+I3+I4.

Following the proof of E3, it is not difficult to see that I1+I4CfL(μ).

Now, we estimate I2, for any y ∈ 𝓧, applying Hölder inequality, we deduce I21μ(S)S|ms(b)mB~(b)|Mκ(fχX6NB)(y)dμ(y)CKB,Sμ(S)(S[Mκ(fχX6NB)(y)]sdμ(y))1sμ(S)11sCKB,SMs,6[Mκ(f)](x),

where we have used the fact that |mS(b)−m(b)| ≤ CKB,S.

Finally, we estimate for I3. For xB, we have I3=|mB[Mκ([bmB~(b)]fχ6NB65B)]||mB[Mκ([bmB~(b)]fχ6NB6B)]|+|mB[Mκ([bmB~(b)]fχ6B65B)]|=:I31+I32.

With a way similar to that used in the proof of E2, it follows that I32CfL(μ).

Meanwhile, following the proof of E3, we have I31CKB,S2fL(μ).

Combining the estimates for I31, I32, I1, I2 and I4, we obtain (26). Thus, we complete the proof of (24). □

Proof of Theorem 1.9

For convenience, we assume ∥bRBMO(μ) = 1. For each fixed t > 0 and functions f with bounded support, applying Lemma 2.5 to |f| with p = 1, and letting Bj, Sj, φj and ωj as the same as Lemma 2.5. We see that f = g+h, where g(x):=fχXj(6Bj)+jφj;h(x):=j[f(x)ωj(x)φj(x)]:=jhj(x).(27)

Noticing that ∥gL1(μ)CfL1(μ). Applying the L2(μ)-boundedness of Mb,κ in Theorem 1.8 and the fact that |g(x)| ≤ Ct, it is not difficult to obtain that μ({xX:Mκ,b(g)(x)>t})Ct1X|f(y)|dμ(y).

From (14), we have μ(j62Bj)jCtfL1(μ), so the proof of Theorem 1.9 is reduced to prove μ({xXj(62Bj):Mκ,b(h)(x)>t})CX|f(y)|tlog(2+|f(y)|t)dμ(y).(28)

For each fixed j and xXjj(62Bj), let bj(x) := b(x)−mj(b) and write Mb,κ(h)(x)j|bj(x)|Mκ(hj)(x)+Mκ(jbjhj)(x):=I(x)+II(x).

With a way similar to that used in the estimate of H21, H22 and H23 in Theorem 1.8, we have μ({xXj(62Bj):I(x)>t})CtfL1(μ).(29)

By hj := jφj, write μ({xX(62Bj):|II(x)|>t})μ({X(62Bj)j:Mκ(jbjfωj)(x)>t2})+μ({X(62Bj)j:Mκ(jbjφj)(x)>t2})CjBjΦ(|b(y)mB~j||f(y)|t)dμ(y)+CXΦ(j|b(y)mB~j||φ(y)|t)dμ(y)=:II1+II2.

For all α ≥ 1, let Φα(t)=tlogα(2+t),Ψ1α(t)=expt1α. For any s, t > 0, we have the following facts Φ(st)C[Φ(s)+Ψ(t)],

and for any s > 0 and t1, t2 > 0, we have Φs(t1t2)CΦs(t1)Φs(t2),

For II1, by (14) and Lemma 2.6, we have II1CjBj[Ψ(|b(y)mBj(b)|B1)+Φ(|f(y)|tB1)]dμ(y)CjBj[exp(|b(y)mBj(b)|B1)+Φ(|f(y)|tB1)]dμ(y)CjBj[μ(62Bj)+Φ(|f(y)|tB1)]dμ(y)CtjBj|f(y)|d(y)+XΦ(|f(y)|t)dμ(y)CXΦ(|f(y)|t)dμ(y).

In order to estimate II2, we assume that Λ ⊂ N is a set of finite index, rj(y):=1t|φj(y)|. By applying the convex property of Φ, we get Φ(jΛrj(y)|b(y)mB~j(b)|)CjΛrj(y)Φ(|b(y)mB~j(b)|).

On the other hand, if we take Λ = N, the above inequality also holds by the property of Φ. With a way similar to H12 in the proof of Theorem 1.8, we have II2C1tjφjL(μ)SjΦ(|b(y)mB~j(b)|)dμ(y)C1tjφjL(μ)Sj|b(y)mB~j(b)|[1+|b(y)mB~j(b)|]dμ(y)C1tjφjL(μ)μ(Sj)CX|f(y)|tdμ(y),

which, together with II1 and (29), imply (28), and hence the proof of Theorem 1.9 is finished. □

Proof of Theorem 1.10

Without loss generality, we may assume that ∥bRBMO(μ) = 1 and ρ = 2 as in Definition 1.6. It suffices to prove that, for any (∞, 1)τ-atomic block h, μ({xX:Mκ,b(h)(x)>t})C|h|Hatb1,(μ)t.(30)

Assume that supp(h) ⊂ R and h=j=12τjaj, where aj is a function supported in BjR such that ∥ajL(μ)[μ(4Bj)]1KBj,R1 and |τ1|+|τ2||h|Hatb1,(μ). Write Mκ,b(h)(x)|b(x)mR~(b)|Mκ(h)(x)+Mκ([mR~(b)b]h)(x)=:J1(x)+J2(x).

By the (L1(μ), L1, ∞(μ))-boundedness of Mκ (see [17]), we have μ({xX:J2(x)>t2})C1tR|b(x)mR~(b)||h(x)|dμ(x)C|τ1|tR|b(x)mR~(b)||a1(x)|dμ(x)+C|τ2|tR|b(x)mR~(b)||a2(x)|dμ(x)=:J21+J22.

Now, we estimate J21, applying Hölder inequality and (13), we have J21C|τ1|ta1L(μ)[Bj|b(x)mBj~(b)|dμ(x)+|mBj~(b)mR~(b)|μ(Bj)]C|τ1|tμ(4Bj)1KBj˙,R1KBj,Rμ(Bj)C|τ1|t.

Similar to the estimate of J21, it is easy to obtain that J22C|τ2|t.

Thus, we can conclude that μ({xX:J2(x)>t2})Ct1(|τ1|+|τ2|)Ct1|h|Hatb1,(μ).(31)

Now, we turn to estimate J1. Write μ({xX:J1(x)>t2})Ct1X2R|b(x)mR~(b)|Mκ(h)(x)dμ(x)+Ct12R|b(x)mR~(b)|Mκ(h)(x)dμ(x)=:J11+J12

An argument similar to H21 that used Theorem 1.8 and ajL(μ)[μ(4Bj)]1KBj,R1, we have J11Ct1|h|Hatb1,(μ).

It remains to estimate J12. Write J12Ct1j=12|τj|2R|b(x)mR~(b)|Mκ(aj)(x)dμ(x)Ct1j=12|τj|2R62Bj|b(x)mR~(b)|Mκ(aj)(x)dμ(x)+Ct1j=12|τj|62Bj|b(x)m62Bj~(b)|Mκ(aj)(x)dμ(x)+Ct1j=12|τj||mR~(b)m62Bj~(b)|62BjMκ(aj)(x)dμ(x)=:U1+U2+U3.

With an argument similar to that used in the proof of V11 in [17, Theorem 1.11], it is easy to get U3Ct1|b|Hatb1(μ).

For U2, applying Hölder inequality, the L2(μ)-boundedness of Mκ, (13) and ∥ ajL(μ)[μ(4Bj)]1KBj,R1, we have U2Ct1j=12|τj|KBj,RMκ(aj)L2(μ)μ(63Bj)12Ct1j=12|τjKBj,RajL(μ)μ(Bj)12μ(63Bj)12Ct1|b|Hatb1(μ).

By the L2(μ)-boundedness of Mκ and an argument similar to H24 in Theorem 1.8, U1Ct1|b|Hatb1(μ).

Combining the whole estimates as above, we finish the proof of Theorem 1.10. □

4 Conclusions

In this work we proved that the commutators Mκ,b generated by the Littlewood-Paley operators Mκ and RBMO (μ) functions were bounded on Lp(μ) with 1 < p < ∞, and bounded from the spaces L log L(μ) to the weak Lebesgue spaces over non-homogeneous metric measure spaces in the sense of Hytönen. Also, we obtained the boundedness of the commutators Mκ,b on Hardy spaces.

With the results of the commutators given herein, we shall consider the boundedness of the Mκ,b on Morrey spaces and generalized Morrey spaces over non-homogeneous metric measure spaces in the follow-up work.

Acknowledgement

The authors would like to thank referees for several valuable comments and suggestions. Thanks also go to the editor for the timely communications. This work was supported by the National Natural Science Foundation of China (No. 11561062).

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About the article

Received: 2016-07-10

Accepted: 2017-09-29

Published Online: 2017-11-13


Conflict of interest The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1283–1299, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0110.

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