Uniform Boundedness for Approximations of the Identity with Nondoubling Measures

Yang, Dachun; Yang, Dongyong

doi:10.1155/2007/19574

Cited by 5 publications

(1 citation statement)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 3.8. In fact, from Theorem 3.1 in [28], we see that Proposition 3.7 and Corollary 3.2 below also hold for S k with k ≤ 0 when R d is not an initial cube.…”

Section: Proposition 37 Let K ∈ N and S K Be As In Section 2 Ifmentioning

confidence: 85%

$h^1$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures

Hu¹,

Yang²,

Yang³

2009

Rev. Mat. Iberoam.

Self Cite

View full text Add to dashboard Cite

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 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).

show abstract

“…Remark 3.8. In fact, from Theorem 3.1 in [28], we see that Proposition 3.7 and Corollary 3.2 below also hold for S k with k ≤ 0 when R d is not an initial cube.…”

Section: Proposition 37 Let K ∈ N and S K Be As In Section 2 Ifmentioning

confidence: 85%