A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces

By Yongsheng Han, D. Müller, Dachun Yang. 2008

262 citations recorded by the source corpus.

We work on RD-spaces 𝒳 , namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝒳 . An important example is the Carnot-Caratheodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spaces H p ( 𝒳 ) and local Hardy spaces h p ( 𝒳 ) on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces when p ≥ 1 . The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces, inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, and BMO are also presented. Moreover, we prove boundedness results on these Besov and Triebel-Lizorkin spaces for classes of singular integral operators, which include non-isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn-Laplacian on certain classes of model domains in ℂ N . Our theory applies in a wide range of settings.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c9e4fde286506ff8547e81aa7deac75a02a7850c

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In Applied Mathematics

1. Dachun Yang
2. Yuan Zhou
3. Wen Yuan
4. Xuan Thinh Duong
5. Ji Li
6. Liguang Liu
7. Dongyong Yang
8. Pekka Koskela

In Mathematical Physics

1. Dachun Yang
2. Yuan Zhou
3. Wen Yuan
4. Dongyong Yang
5. Xuan Thinh Duong
6. Liguang Liu
7. P. Auscher
8. Ji Li

Citations in the corpus

• A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces (2009, methodology): “In this paper, we introduce spaces of Lipschitz type on RD-spaces, and discuss their relations with Besov and Triebel-Lizorkin spaces in [17] and various Sobolev spaces in [29, 13].”
• Orthonormal bases of regular wavelets in spaces of homogeneous type (2011, background): “Although we do not develop the theory of function spaces in the detail of [15] here, it is clear that our analysis involving the quantity d(x,Y ) will extend to further related questions that we do not explicitly deal with; we believe that it could be used to extend large parts of the recent theory ”
• A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type (2018, methodology): “The already existing works (see [26, 27, 20, 54, 55]) regarding Hardy spaces on RD-spaces show the feasibility of establishing various real-variable characterizations of the atomic Hardy spaces on spaces of homogeneous type via the Calderón reproducing formulae.”
• Maximal function characterizations of Hardy spaces on RD-spaces and their applications (2008, methodology): “(If a0 = 2, this is the classical reverse doubling condition), and equivalently, for all x ∈X and 0 23 , 36].”
• Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces (2011, background): “For the theory of Besov spaces on Rn see, for example, [25, 22, 26], and, on metric measure spaces, see [9, 21, 7].”
• Measure Density and Extension of Besov and Triebel–Lizorkin Functions (2014, background): “For related interpolation results in the metric setting, see [46], [16] and [9].”
• Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type (2012, background): “3) were developed in [33] and [23] (the same kind of identity was established in [31] and [32] on the one-parameter Carnot-Carathéodory-type space).”
• WEIGHTED BESOV AND TRIEBEL–LIZORKIN SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS (2020, methodology): “(i) Using the existence of the approximation of identity, the authors developed the theory of Besov spaces Ḃp,q and Triebel–Lizorkin spaces Ḟ s p,q for a range 1 6 p, q 6 ∞ and s ∈ (−θ, θ) for some θ ∈ (0, 1) on metric measure spaces with polynomial volume growths [39] and on spaces with doubling an”
• Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators (2009, unknown): “§”