$h^1$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures
Revista matemática iberoamericana
Let µ be a nonnegative Radon measure on R d which satisfies the growth condition that there exist constants C 0 > 0 and n ∈ (0, d] such that for all x ∈ R d and r > 0, µ(B(x, r)) ≤ C 0 r n , where B(x, r) is the open ball centered at x and having radius r. In this paper, we introduce a local atomic Hardy space h 1, ∞ atb (µ), a local BMO-type space rbmo (µ) and a local BLO-type space rblo (µ) in the spirit of Goldberg and establish some useful characterizations for these spaces. Especially, we
... es. Especially, we prove that the space rbmo (µ) satisfies a John-Nirenberg inequality and its predual is h 1, ∞ atb (µ). We also establish some useful properties of RBLO (µ) and improve the known characterization theorems of RBLO (µ) in terms of the natural maximal function by removing the assumption on the regularity condition. Moreover, the relations of these local spaces with known corresponding function spaces are also presented. As applications, we prove that the inhomogeneous Littlewood-Paley g-function g(f ) of Tolsa is bounded from h 1, ∞ atb (µ) to L 1 (µ), and that [g(f )] 2 is bounded from rbmo (µ) to rblo (µ). 2000 Mathematics Subject Classification: Primary: 42B35; Secondary: 42B25, 42B30, 47A30, 43A99. Keywords: Non-doubling measure, approximation of the identity, maximal operator, John-Nirenberg inequality, duality, cube of generation, g-function, RBMO (µ), rbmo (µ), h 1 , bmo, blo and Littlewood-Paley g-functions 597 We remark that some other variants of local atomic Hardy space and local BMO-type space in the sense of Goldberg were also introduced in . However, it seems that they are not natural for the boundedness of the inhomogeneous Littlewood-Paley g-function. A new idea used in this paper is to classify cubes of R d by using the coefficients δ(·, ·) of Tolsa  (see [17, Definition 3.2] or Definition 2.1 below); while in  , cubes are classified by their side lengths as in the case of Euclidean spaces in  . To be precise, in this paper, using the coefficients δ(·, ·), we introduce a class D of cubes, which have "large" side lengths in the sense that if µ is the d-dimensional Lebesgue measure, then Q ∈ D if and only if the side length of Q is no less than C, where C is a positive constant independent of Q. We then use D to define our local Hardy space, local BMO-type space and local BLO-type space. It is well-known that the coefficients δ(·, ·) of Tolsa describe well the geometric properties of cubes of R d ; see Lemma 3.1 in  (or Lemma 2.1 below). These properties play key roles in the whole theory of analysis associated with non-doubling measures. Using these coefficients, Tolsa in [17, 18] further found suitable variants of dyadic cubes, which are now called cubes of generations. These cubes of generations are the basis of the construction on approximations of the identity of Tolsa in  . Another novelty of this paper is that we introduce a quantity, which further clarifies the geometric relations between general cubes and "dyadic" cubes of Tolsa in [17, 18] ; see Lemma 2.2 below. These properties together with the known properties of "dyadic" cubes (see, for example, Lemma 3.4 and Lemma 4.2 in  ) are key tools used in this paper. bmo, blo and Littlewood-Paley g-functions 599 We may treat points x ∈ R d as if they were cubes (with side length l(x) = 0). So, for any x, y ∈ R d and cube Q ⊂ R d , the symbols δ(x, Q) and δ(x, y) make sense. The following useful properties of δ(·, ·), which were proved by Tolsa in [18, pp. 320-321] (see also [17, Lemma 3.1]), play important roles in the whole paper. Lemma 2.1. There exists a positive constant C, which only depends on C 0 , n, d and ρ, such that the following properties hold: In particular, δ(P, Q) ≤ δ(P, R) + 0 and δ(Q, R) ≤ δ(P, R) + 0 . Moreover, if P and Q are concentric, then 0 = 0. (e) For any P , Q, R ⊂ R d , δ(P, R) ≤ C + δ(P, Q) + δ(Q, R). We now recall the notion of cubes of generations in [17, 18] . Definition 2.2. We say that x ∈ R d is a stopping point (or stopping cube) if δ(x, Q) < ∞ for some cube Q x with 0 < l(Q) < ∞. We say that R d is an initial cube if δ(Q, R d ) < ∞ for some cube Q with 0 < l(Q) < ∞. The cubes Q such that 0 < l(Q) < ∞ are called transit cubes. Remark 2.1. In [17, p. 67], it was pointed out that if δ(x, Q) < ∞ for some transit cube Q containing x, then δ(x, Q ) < ∞ for any other transit cube Q containing x. Also, if δ(Q, R d ) < ∞ for some transit cube Q, then δ(Q , R d ) < ∞ for any transit cube Q . Let A be some big positive constant. In particular, as in [17, 18] , 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  . Moreover, the constants A, 0 , 1 and γ 0 depend only on C 0 , n, d and ρ. In what follows, for > 0 and a, b ∈ R, the notation a = b ± does not mean any precise equality but the estimate |a − b| ≤ .