New characterizations and applications of inhomogeneous Besov and Triebel–Lizorkin spaces on homogeneous type spaces and fractals

Han, Yongsheng; Yang, Dachun

doi:10.4064/dm403-0-1

Cited by 50 publications

(72 citation statements)

References 23 publications

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“…This class includes R n , some Riemannian manifolds, some self-similar fractals and, in particular, the post critically finite self-similar fractals of [33,43]. Thus, as well as indicating how to define Lipschitz-type spaces of order no less than 1 related to Besov and Triebel-Lizorkin spaces on spaces of homogeneous type (particularly on metric-measure spaces and fractals), our results also give some new characterizations of Besov and Triebel-Lizorkin spaces of order less than 1 using discrete square fractional derivatives (see [13,14,23]). Moreover, Theorem 3.12 below contains both a discrete and an inhomogeneous version of Theorem 2 of [13].…”

Section: Introductionmentioning

confidence: 89%

“…To state the definition of the inhomogeneous Besov spaces B s pq (X) and the inhomogeneous Triebel-Lizorkin spaces F s pq (X) studied in [23][24][25][26], we need the following approximations to the identity, first introduced in [21]. …”

Section: Relations With Besov and Triebel-lizorkin Spacesmentioning

confidence: 99%

“…In [25] and [24] (see also [23]), Han, Lu and Yang introduced Besov spaces on spaces of homogeneous type. It was observed by Triebel in [48, pp.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, motivated by [18,[30][31][32], we introduce some new spaces of Lipschitz type on metric-measure spaces and show that these new Lipschitz-type spaces are just the well-known Besov spaces or Triebel-Lizorkin spaces introduced in [23][24][25][26] when the smooth index is less than 1. Our results also indicate that some of the Hölder-Zygmund spaces of Strichartz [43] are the same as some of our function spaces for small orders of smoothness (see also [31,48,52]).…”

Section: Introductionmentioning

confidence: 99%

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Spaces of Lipschitz Type on Metric Spaces and Their Applications

Yang¹,

Lin²

2004

Proceedings of the Edinburgh Mathematical Society

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New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel-Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Haj lasz-Sobolev spaces, Korevaar-Schoen-Sobolev spaces, Newtonian Sobolev space and Cheeger-Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy-Littlewood-Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

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Section: Introductionmentioning

confidence: 89%

Section: Relations With Besov and Triebel-lizorkin Spacesmentioning

confidence: 99%

“…In [25] and [24] (see also [23]), Han, Lu and Yang introduced Besov spaces on spaces of homogeneous type. It was observed by Triebel in [48, pp.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spaces of Lipschitz Type on Metric Spaces and Their Applications

Yang¹,

Lin²

2004

Proceedings of the Edinburgh Mathematical Society

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“…Again, using formulae (1.5), one showed that Besov spaces are independent of the choice of approximations to the identity {S k } ∞ k=−∞ , and, moreover, all other properties such as embedding, interpolation, duality, atomic decomposition and the T 1 theorem were obtained; see [12,10,11,13,14].…”

Section: Introductionmentioning

confidence: 99%

Besov spaces with non-doubling measures

Deng¹,

Han

Yang

2005

Trans. Amer. Math. Soc.

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Abstract. Suppose that µ is a Radon measure on R d , which may be nondoubling. The only condition on µ is the growth condition, namely, there is a constant C 0 > 0 such that for all x ∈ supp (µ) and r > 0,where 0 < n ≤ d. In this paper, the authors establish a theory of Besov spaceṡ B s pq (µ) for 1 ≤ p, q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure µ, C 0 , n and d. The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.

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