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

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(x_i, ϵr)}_i of B(x, r) such that the cardinality of this covering is at most N₀ϵ⁻ⁿ. 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 N₀ϵ⁻ⁿ centers {x_i}_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 {x_i}_i of disjoint balls {B(xi,r4)}i=1M.

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 C_λ such that, for each x ∈ 𝓧, r → λ(x, r) is non-decreasing and, for all x ∈ 𝓧 and r ∈ (0, ∞),

Definition 1.4

([15]). Let ν > 1. A function f ∈ Lloc1(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 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 (μ) norm of f and denoted by ∥f∥_RBMO(μ).

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 x ≠ y,

and, for all y, y′ ∈ 𝓧,

Definition 1.6

(16). Let ρ ∈ (1, ∞) and p ∈ (1, ∞]. A function b ∈ Lloc1(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 a_i supported on a ball B_i ⊂ B and τ_i ∈ ℂ such that

   b=τ1a1+τ2a2

   and

   ∥ai∥Lp(μ)≤[μ(ρBi)]1p−1KBi,B−1. (11)

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

Definition 1.7

([16]). Let p ∈ (1, ∞]. A function f ∈ L¹(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=1∞bi in L¹(μ) and ∑i=1∞|bi|Hatb1,p(μ) < ∞. The norm of f in Hatb1,p (μ) is defined by

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

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 L^p(μ) for 1 < p < ∞, that is, there exists a constant C > 0, such that for all functions f with bounded support, one has

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,

where Φ_α(t) = t log^α(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 L^1,∞ (μ), namely, there is a positive constant C, such that for all f ∈ H¹(μ) and t > 0, one has

Lemma 2.1

([15]).

1. For all balls B ⊂ R ⊂ S, it holds true that K_B,R≤K_B,S.

2. For any ξ ∈[1, ∞), there exists a positive constant C_ξ, such that, for all balls B ⊂ S with r_S ≤ ξ r_B, K_B,S ≤ C_ξ.

3. For any ϱ ∈ (1, ∞), there exists a positive constant C_ϱ, depending on ρ, such that, for all balls B, K_B,B͠^ϱ ≤ C_ϱ.

4. There exists a positive constant c such that, for all balls B ⊂ R ⊂ S, K_B,S≤K_B,R + cK_R,S. In particular if B and R are concentric, then c = 1.

5. There exists a positive constant c͠ such that, for all balls B ⊂ R ⊂ S, K_B,R≤c͠K_B,S moreover, if B and R are concentric, then K_R,S ≤ K_B,S.

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, ∞),

where above and in what follows, m_B͠(f) denotes the mean of f over B͠, namely,

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

Lemma 2.3

([15]). (1) Let p ∈ (1, ∞), r ∈ (1, p) and ϕ ∈ (0, ∞). The following maximal operators defined, respectively, be setting, for all f ∈ Lloc1(μ) and x ∈ 𝓧,

and

are bounded on L^p(μ) and also boundedfrom L¹(μ) into L^1,∞ (μ).

(2) For all f ∈ Lloc1(μ) , it holds true that |f(x)|≤ Nf(x) for μ-almost every x ∈ 𝓧.

Lemma 2.4

([19]). Let f ∈ Lloc1(μ) with the extra condition ∫_𝓧f(x)dμ(x) = 0 if ∥μ∥ := μ(𝓧)<∞. Assume that for some p, 1 < p < ∞, inf{1, Nf} ∈ L^p(μ). Then we have

where M♯f(x):=supB∋x1μ(6B)∫B|f(x)−mB~(f)|dμ(x)+sup(Q,R)∈Δx|mQ(f)−mR(f)|KQ,Rforallf∈L1oc1(μ) and x ∈ 𝓧, and Δ_x := {(Q, R) : x ∈ Q ⊂ R and Q, R are doubling balls}.

Lemma 2.5

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

1. there exists a family of finite overlapping balls {6B_i}_i such that {B_i}_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-almosteveryx∈X∖(∪i⁡6Bi); (15)

2. for each i, let S_i be a (3×62,Cλlog2⁡(3×62)+1) -doubling ball of the family {(3 × 6²)^kB_i}_k∈ℕ, and ω_i = χ6Bi/(∑kχ6Bk). Then there exists a family {φ_i}_i of functions that for each i, supp(φ_i) ⊂ S_i, φ_i has a constant sign on S_i and

   ∫Xφi(x)dμ(x)=∫6Bif(x)ωi(x)dμ(x), (16)

   ∑i|φj(x)|≤γtforμ-almosteveryx∈X, (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

   ∥φi∥L∞(μ)μ(Si)≤C∫X|f(x)ωi(x)|dμ(x), (18)

   and if p ∈ (1, ∞),

   (∫si|φi(x)|pdμ(x))1p[μ(Si)]1p′≤Ctp−1∫X|f(x)ωi(x)|pdμ(x). (19)

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,

Proof of Theorem 1.8

Let 0 < r < 1, we firstly claim that, for any p ∈ (1, ∞), b ∈ L^∞(μ) and all bounded functions f with compact support,

Once (20) is established, by the Marcinkiewicz interpolation theorem, it is easy to obtain that

By Lemma 2.4, for any p ∈ (1, ∞), b ∈ L^∞(μ) and all bounded function fwith 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(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 ∥b∥_RBMO(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:=fχX∖∪j⁡6Bj+∑jφj and h:=∑j(ωjf−φj)=:∑jhj. By (15) and (17), we easily get

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 Mr♯ f(x) ≤ CM_r,6 f(x) (see [20]).

By applying the (L¹(μ), L^1,∞(μ))-boundedness of M_ρ, for any σ > 0, we get

Choosing 1 < p₁ < p, by h_j := ∑j (fω_j − φ_j) and (23), we have

For H₁₁. By Hölder inequality, (13) and (14), we have

With a way similar to that used in the proof of D₁₂ in [20], it is easy to obtain

Now, we turn to estimate H₂. By (23) and h_j := ∑j (fω_j − φ_j), write

An argument similar to that used in the proof of E₁ in [17, Theorem 1.10] shows that

which, together with the fact that ∥h_j∥_L¹(μ) ≤ C t^1−p ∥f∥Lp(μ)p, thus, H₂₁ + H₂₄ ≤ C t^−p ∥f∥Lp(μ)p.

For H₂₂. By Hölder inequality, the L^p(μ)-boundedness of Mκ∗ , (13) and (19), we have

Similar to the estimate of the H₂₂, we conclude

Combining the estimates for H₂₁, H₂₂, H₂₃ and H₂₄, we get

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 b ∈ L^∞(μ) . If Mκ∗ is bounded on L²(μ), then there exists a positive constant C such that, for all f ∈ L^∞(μ) ∩ L^p₀(μ) and for all x ∈ 𝓧,

Proof

Without loss generality, we may assume ∥b∥_RBMO(μ) = 1. Let B be an arbitrary ball and S be a doubling ball with B ⊂ S, denote

and

To prove (24), it only needs to prove

and

To prove (25), for a fixed ball, x ∈ B and f ∈ L^∞(μ), we decompose f as

Thus, we write

For E₁, by Hölder inequality and (13), we have

Applying Hölder inequality, the L²(μ)-boundedness of Mκ∗ and (13), we can conclude