Endpoint estimates for homogeneous Littlewood-Paley g-functions with non-doubling measures

Let μ be a nonnegative Radon measure on R which satisfies the growth condition that there exist constants C0 > 0 and n ∈ (0, d] such that for all x ∈ R and r > 0, μ(B(x, r)) ≤ C0r , where B(x, r) is the open ball centered at x and having radius r . In this paper, when R is not an initial cube which implies μ(R) = ∞ , the authors prove that the homogeneous LittlewoodPaley g -function of Tolsa is bounded from the Hardy space H(μ) to L(μ) , and furthermore, that if f ∈ RBMO (μ) , then [ġ(f)] is either infinite everywhere or finite almost everywhere, and in the latter case, [ġ(f)] belongs to RBLO(μ) with norm no more than C‖f‖RBMO (μ) , where C > 0 is independent of f .


Introduction
Recall that a non-doubling measure μ on R d means that μ is a nonnegative Radon measure which only satisfies the following growth condition, namely, there exist constants C 0 > 0 and n ∈ (0, d] such that for all x ∈ R d and r > 0, (1.1) μ B(x, r) ≤ C 0 r n , where B(x, r) is the open ball centered at x and having radius r .Such a measure μ is not necessary to be doubling, which is a key assumption in the classical theory of harmonic analysis.In recent years, it was shown that many results on the Calderón-Zygmund theory remain valid for nondoubling measures; see, for example, [6,7,8,9,10,11,12,5,3].One of the main motivations for extending the classical theory to the non-doubling context was the solution of several questions related to analytic capacity, like Vitushkin's conjecture or Painlevé's problem; see [13,14,16] or survey papers [15,17,18] for more details.
In particular, Tolsa [11] developed a Littlewood-Paley theory with nondoubling measures for functions in L p (μ) when p ∈ (1, ∞) and used this Littlewood-Paley decomposition to establish some T (1) theorems.The main purpose of this paper is to investigate the behaviors of the homogeneous Littlewood-Paley g -functions of Tolsa in [11] at the extremal cases, namely, in the cases when p = 1 or p = ∞.To be precise, in this paper, when R d is not an initial cube which implies μ(R d ) = ∞ (see [11]), we prove that the homogeneous Littlewood-Paley g -function ġ(f ) of Tolsa is bounded from the Hardy space H 1 (μ) to L 1 (μ), and furthermore, we prove that if f ∈ RBMO (μ), then [ ġ(f )] 2 is either infinite everywhere or finite almost everywhere, and in the latter case, [ ġ(f )] 2 is bounded from RBMO (μ) to RBLO (μ), where RBMO (μ) was introduced by Tolsa in [10] and RBLO (μ) was introduced by Jiang in [3].Notice that L ∞ (μ) ⊂ RBMO (μ) .The last above-mentioned result generalizes the corresponding result of Leckband [4] in replacing L ∞ (R d ) by BMO (R d ), even when μ is the d-dimensional Lebesgue measure and ġ(f ) is the classical homogeneous Littlewood-Paley g -function.When μ(R d ) < ∞, then R d is an initial cube (see [11]) and the homogeneous Littlewood-Paley g -function degenerates into the inhomogeneous Littlewood-Paley g -function g(f ).We also obtain similar results for this inhomogeneous Littlewood-Paley g -function, by first establishing a new theory of local atomic Hardy space h 1, ∞ atb (μ), rbmo (μ) and rblo (μ) in the sense of Goldberg [1].To limit the length of this paper, we will present these results in [2].An interesting open problem is if ġ(f ) and g(f ) can characterize the Hardy space H 1 (μ) and h 1, ∞ atb (μ) , respectively.The organization of this paper is as follows.In Section 2, we recall some necessary definitions and notation, including the definitions of atomic Hardy spaces, RBMO (μ), RBLO (μ) , approximations to the identity and the homogeneous Littlewood-Paley g -function ġ(f ).In Section 3, we establish the boundedness of the homogeneous Littlewood-Paley g -function ġ(f ) from H 1 (μ) to L 1 (μ), and prove that if f belongs to RBMO (μ), then [ ġ(f )] 2 is either infinite everywhere or finite almost everywhere, and in the latter case, [ ġ(f )] 2 belongs to RBLO (μ) with norm no more than C f 2 RBMO (µ) , where C > 0 is independent of f .As a corollary, we also obtain the boundedness of the homogeneous Littlewood-Paley g -function ġ(f ) from RBMO (μ) to RBLO (μ).
Throughout the paper, we always denote by C a positive constant which is independent of the main parameters, but it may vary from line to line.Constant with subscript such as C 1 , does not change in different occurrences.The symbol Y Z means that there exists a constant C > 0 such that Y ≤ CZ .The symbol A ∼ B means that A B A. Moreover, for any D ⊂ R d , we denote by χ D the characteristic function of D .

Preliminaries
In this section, we recall some necessary notions and notation.By a cube Q ⊂ R d , we mean a closed cube whose sides are parallel to the axes and centered at some point of supp (μ), and we denote its side length by l(Q) and its center by where and in what follows, given λ > 0 and any cube Q , λQ denotes the cube concentric with Q and having side length λl(Q).It was pointed out by Tolsa (see [10, pp. 95-96] or [11,Remark 3.1]) that if β > α n , then for any x ∈ supp (μ) and any R > 0, there exists some (α, β)-doubling cube Q centered at x with l(Q) ≥ R , and that if β > α d , then for μ-almost everywhere x ∈ R d , there exists a sequence of (α, β)-doubling cubes {Q k } k∈N centered at x with l(Q k ) → 0 as k → ∞.In what follows, by a doubling cube, we always mean a (2, 2 d+1 )-doubling cube, and for any cube Q , we denote by Q the smallest doubling cube which has the form x Q be the center of Q , and Q R be the smallest cube concentric with Q containing Q and R .The following coefficients were first introduced by Tolsa in [10]; see also [11,12].
We may treat points x ∈ R d as if they were cubes (with side length l(x) = 0).So, for x, y ∈ R d and some cube Q , the notations δ(x, Q) and δ(x, y) make sense; see [11,12] for some useful properties of δ(•, •).We now recall the notion of cubes of generations in [11,12]; see [11,12] for more details.
Throughout this paper, we always assume that R d is not an initial cube.Let A be some big positive constant.In particular, we assume that A is much bigger than the constants 0 , 1 and γ 0 , which appear, respectively, in Lemma 3.1, Lemma 3.2 and Lemma 3.3 of [11].Moreover, the constants A, 0 , 1 and γ 0 depend only on C 0 , n and d.In what follows, for > 0 and a, b ∈ R, the notation a = b ± does not mean any precise equality but the estimate |a − b| ≤ .Definition 2.3.Assume that R d is not an initial cube.We fix some doubling cube R 0 ⊂ R d .This will be our 'reference' cube.For each j ∈ N , let R −j be some doubling cube concentric with R 0 , containing R 0 , and such that δ(R 0 , R −j ) = jA ± 1 (which exists because of Lemma 3.3 of [11]).If Q is a transit cube, we say that Q is a cube of generation k ∈ Z if it is a doubling cube, and for some cube Using Lemma 3.2 in [11], it is easy to verify that for any x ∈ supp (μ) and k ∈ Z, there exists a doubling cube of generation k ; see [11, p. 68].Moreover, the definition of cubes of generations is proved in [11, p. 68] to be independent of the chosen reference R −j in the sense modulo some small errors.Throughout this paper, for any x ∈ supp (μ) and k ∈ Z, we denote by Q x, k a fixed doubling cube centered at x of generation k .On cubes of generations {Q x, k } k∈Z , we have the following simple observation.
Proof.For any given x ∈ supp (μ), we first assume that {x} is not a stopping cube.Then for any N ∈ N, Q x, 0 and Q x, −N are transit cubes (see [11, p. 68 Choosing j ≥ max(j 1 , j 2 ) and using Lemma 3.1 (d) in [11] imply that δ [11] again that [11] shows that there exists a constant C d depending only on d such that On the other hand, since 1 A, then N A ± 6 1 > NA/2 .Therefore, if we take N > 2C d (1 + log M )/A, we then have a contradiction that which implies that the conclusion of Proposition 2.1 is true in the case that {x} is not a stopping cube.
If {x} is a stopping cube, recalling that there exists some k x ∈ Z such that all the cubes of generation k < k x are transit cubes (see [11, p. 68]), we obtain that for ) together with an argument as above, we also have a contradiction, which implies that l(Q x, k ) → ∞ as k → −∞.This finishes the proof of Proposition 2.1.
In [11], Tolsa constructed a class of approximations to the identity , which are integral operators given by kernels S k (x, y) on R d × R d satisfying the following properties: Moreover, Tolsa [11] pointed out that Properties (A-1) through (A-5) also hold if any of Q x, k , Q x , k and Q y, k is a stopping cube.In what follows, without loss of generality, for any x ∈ supp (μ), we always assume that Q x, k is not a stopping cube, since the proofs for stopping cubes are similar.
(3) for j = 1, 2 , there exist functions a j supported on cubes Q j ⊂ R and numbers λ j ∈ R such that b = λ 1 a 1 + λ 2 a 2 , and Then we define |b| , where the infimum is taken over all the possible decompositions of f in p-atomic blocks as above.
Remark 2.2.It was proved by Tolsa [10] that the definition of H 1, p atb (μ) is independent of the chosen constant η > 1 , and for any 1 < p ≤ ∞, all the atomic Hardy spaces H 1, p atb (μ) coincide with equivalent norms.Moreover, a maximal function characterization of H 1, p atb (μ) was also established in [12].Thus, in the rest of this paper, we denote the atomic Hardy space H 1, p atb (μ) simply by H 1 (μ), and when we use the atomic characterization of H 1 (μ), we always assume η = 2 and p = ∞ in Definition 2.4.
) is said to be in the space RBMO (μ) if there exists some constant C 1 ≥ 0 such that for any cube Q centered at some point of supp (μ), and for any two doubling cubes where m Q (f ) denotes the mean of . Moreover, we define the RBMO (μ) norm of f by the minimal constant C 1 as above and denote it by f RBMO (µ) .
Remark 2.3.It was proved by Tolsa [10] that the definition of RBMO (μ) is independent of the choices of η .As a result, throughout this paper, we always assume η = 2 in Definition 2.5.
The following space RBLO (μ) was introduced in [3].It is obvious that Definition 2.6.We say f ∈ L 1 loc (μ) belongs to the space RBLO (μ) if there exists some constant C 2 ≥ 0 such that for any doubling cube Q , and for any two doubling cubes The minimal constant C 2 as above is defined to be the norm of f in the space RBLO (μ) and denoted by f RBLO (µ) .

Main results and their proofs
We begin with the boundedness of the homogeneous Littlewood-Paley gfunction ġ(f ) from H 1 (μ) to L 1 (μ).Recall that R d is assumed not to be an initial cube.

Theorem 3.1. There exists a constant
Proof.Let b be any ∞-atomic block as in Definition 2.4.To be precise, assume that b = λ 1 a 1 + λ 2 a 2 .By the Fatou lemma, to prove Theorem 3.1, it is enough to show that ġ(b) is in L 1 (μ) and Assume that supp (b) ⊂ R and supp (a j ) ⊂ Q j for j = 1, 2 as in Definition 2.4.Since ġ is sublinear, we write Recalling that ġ is bounded on L 2 (μ) (see Theorem 6.1 in [11]), by the Hölder inequality and (2.3), we then see that which is a desired estimate.
For j = 1, 2, let x j be the center of Q j .Notice that for x / ∈ 2Q j and y ∈ Q j , |x − y| ∼ |x − x j |.From this fact, the Hölder inequality, the fact that for any x = y , [11, p. 82]) and (2.3), it follows that .
By Lemma 3.1 (d) in [11], δ(2Q j , 4R) 1 + δ(Q j , R), which in turn implies that We now estimate I 3 .Let x 0 ∈ supp (μ) ∩ R .By the vanishing moment of b , the Minkowski inequality and the Hölder inequality, for x / ∈ 4R , Therefore, Theorem 3.1 is reduced to showing that For any transit cube R and any x ∈ R ∩ supp (μ), let H x R be the largest integer k such that R ⊂ Q x, k .By Proposition 2.1, we know that H x Q exists and is unique.We now claim that for any y ∈ Q j , any integer i ≥ 3 and In fact, by (A-3) and the fact that , which together with Lemma 4.2 (c) in [11] implies that Then the symmetry of S k and (2.2) imply that 2) and (3.3) along with Lemma 3.4 in [11] yield that for any Notice that for any k ∈ Z and x ∈ supp (μ), As a consequence, another application of (2.3) together with shows that On the other hand, since [11, p. 69]), it follows, from (2.3), (3.1) and the fact that for any x / ∈ 4R and y Therefore, I 3 2 j=1 |λ j |, which completes the proof of Theorem 3.1.
To establish the boundedness of the homogeneous Littlewood-Paley g -function ġ(f ) from RBMO(μ) to RBLO(μ) , we need the following estimate.

Lemma 3.1. There exists a constant C > 0 such that for any two cubes
Proof.Without loss of generality, we may assume that f RBMO (µ) = 1.For any Q ⊂ R , set where N Q, R is the smallest integer k such that l 2 k Q ≥ l(R) (see [10]).
It is trivial to check that Notice that from (1.1) and Definition 2.5, it follows that Therefore, to show Lemma 3.1, it suffices to verify that By (1.1) and Lemma 2.1 in [10] together with Definition 2.5, which completes the proof of Lemma 3.1.
The following conclusion is a slight variant of Lemma 9.3 in [10], which can be proved by a slight modification of the proof of Lemma 9.3 in [10].We omit the details.

Lemma 3.2.
There exists some constant P 0 (big enough) depending on C 0 and n such that if x ∈ R d is some fixed point and {f where C depends on C 0 , n and P 0 .Theorem 3.2.For any f ∈ RBMO (μ) , ġ(f ) is either infinite everywhere or finite almost everywhere, and in the latter case, where C > 0 is independent of f .
Proof.We first claim that for any f ∈ RBMO (μ) , if there exists a point Without loss of generality, we may assume that f RBMO (µ) = 1 .For any [11, p. 69]).This fact together with supp It follows from the doubling property of Q along with Remark 2.3, the L 2 (μ) -boundedness of ġ(f ) (see [11,Theorem 6.1]) and Corollary 3.5 in [10] that Thus taking (3.9) into account, to show (3.8), we only need to verify that for μ-a.e. y ∈ Q , We assert that for each k ∈ Z and z ∈ R d , (3.11) Indeed, (2.1) implies that by the vanishing moment of D k , Lemma 3.1 and (3.12), we have Thus, (3.11) holds.From this assertion we see that for x, y ∈ Q , By the symmetry of D k and (3.2), we see that for any fixed integer i ≥ 3 and k ≥ H x Q − i + 4, and all Therefore, from the vanishing moment of D k , we see that 2) and Lemma 3.4 in [11], we further obtain Moreover, by (3.6), we have Therefore, these facts, together with Definition 2.5, (3.4) and [11,Lemma 3.4] imply that Now we turn our attention to J 2 .The estimate (3.12), Lemma 3.4 in [11], (1.1), Definition 2.5 and (3.4) yield On the other hand, notice that by Lemma 3.4 in [11], for [11].Then it follows from these observations and (3.12) together with Definition 2.5 that Combining these estimates above implies Thus (3.10) holds.
To finish the proof of Theorem 3.2, by Lemma 3.2, it suffices to show that for any doubling cubes For any x ∈ supp (μ) ∩ Q , we first consider the case that H By (3.10) with Q replaced by R , we see that Therefore by (3.9) and (3.11), the estimate (3.13) is reduced to proving that By splitting (3.12) that H x R +2 ⊂ 2R and Lemma 3.1 in [11], Thus, by (3.15), (3.6) and Definition 2.5, we have To estimate L 2 , by Lemma 3.4 in [11], we first see that for any integer there exists a unique integer , and for different k , j k is different.It then follows from Definition 2.5, the decreasing property of Q x, k , (3.15) and (3.5) that Consequently, (3.14) follows by combining the estimates for L 1 and L 2 .If H x R ≤ H x Q ≤ H x R + 9 , then by the estimates (3.9) through (3.11), we also see that (3.13) holds, which completes the proof of Theorem 3.2.
From Theorem 3.2, we can easily deduce the following result.Corollary 3.1.For any f ∈ RBMO (μ) , ġ(f ) is either infinite everywhere or finite almost everywhere, and in the latter case, where C > 0 is independent of f .Proof.First, with the aid of (3.8) and the inequality that for any a, b ≥ 0, (3.16) a − b ≤ a 2 − b 2 1/2 , it is easy to see that if essinf y∈Q ġ(f )(y) < ∞, Moreover, in the argument of (3.13), we see that for any doubling cubes From this fact with (3.9) through (3.11), (3.14) and (3.16), we obtain that for any doubling cubes An application of Lemma 3.2 leads to the conclusion of Corollary 3.1.
and we also use D k to denote the corresponding integral operator with kernel D k .The homogeneous Littlewood-Paley g -function ġ(f ) is then defined by ġ