New properties of Besov and Triebel-Lizorkin spaces on RD-spaces

By Dachun Yang, Yuan Zhou. 2009

104 citations recorded by the source corpus.

An RD-space $${\mathcal X}$$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $${\mathcal X}$$. In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on $${\mathcal X}$$ are independent of the choice of the regularity $${\epsilon\in (0,1)}$$; as a result of this, the Besov and Triebel-Lizorkin spaces on $${\mathcal X}$$ are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and Triebel-Lizorkin spaces by the norms of homogeneous Besov and Triebel-Lizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7fe7b65cfdb1201933d480edbbfd6d66f7c0b04d

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In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Yuan Zhou
4. Jun Liu
5. Pekka Koskela
6. Liguang Liu
7. L. Ihnatsyeva
8. T. Heikkinen

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. Yuan Zhou
4. Jun Liu
5. Liguang Liu
6. Tuomas Hytönen
7. Pekka Koskela
8. P. Auscher

Citations in the corpus

• Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces (2021, unknown): “One special case of spaces of homogeneous type is the RD-space, originally introduced in [30] (see also [29, 79]), which is a doublingmetricmeasure space satisfying the following additional reverse doubling condition: There exist constants C(μ) ∈ (0, 1] and κ ∈ (0, ω] such that, for any ball B(x, r)”
• Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications (2018, unknown): “Observe that (R, | · |~a, dx) is an RD-space (see [36, 83]).”
• Multilinear analysis on metric spaces (2014, background): “RD-spaces have become the underlying context in numerous areas of analysis and PDEs, we refer the reader to [22, 23], [60, 61, 86], [71, 72, 40], [107, 109, 108] and references therein.”
• Anisotropic Variable Hardy-Lorentz Spaces and Their Real Interpolation (2017, background): “Observe that (R, ρ, dx) is an RD-space (see [39, 83]).”
• A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions☆ (2009, methodology): “As shown in [40], the definition of Ḟ s p, q(X ) is independent of the choices of ǫ, β, γ and the approximation of the identity as in Definition 4.”
• Measure Density and Extension of Besov and Triebel–Lizorkin Functions (2014, background): “For various definitions in a metric measure setting, see [9], [11], [16], [27], [32], [37], [47] and the references therein.”
• Orthonormal bases of regular wavelets in spaces of homogeneous type (2011, background): “In several more recent papers [14, 15, 42, 43] on function spaces on (quasi-)metric spaces, this issue is circumvented by assuming the ‘reverse doubling’ property μ(B(x,Cr)) ≥ (1 + ε)μ(B(x, r)), 0 < r < diam(X)/C, which is equivalent to the non-emptyness of annuli: B(x,R) \ B(x, r) 6= ∅ whenever R/r”