Spaces of Lipschitz Type on Metric Spaces and Their Applications

Abstract

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.

Heat kernel and Lipschitz–Besov spaces

“…It was proved by Müller and Yang [20] that the space ƒ s p;q coincides with the Besov space B s p;q ; see also Yang and Lin [26] when .M; ; / is an˛-regular metric measure space. It was proved by Müller and Yang [20] that the space ƒ s p;q coincides with the Besov space B s p;q ; see also Yang and Lin [26] when .M; ; / is an˛-regular metric measure space.…”

Difference Characterization of Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

“…When X is a d-set of R n , the Lipschitz-type space L(s, p, q; X), for any given s ∈ R and p, q ∈ (0, ∞], was introduced in [35,37]; see also [15,16]. When X is an Ahlfors n-regular metric measure space, these spaces were introduced in [54]. We also point out that, when X is a metric measure space, these spaces may be non-trivial even when s ∈ (1, ∞) (see, for instance, [54] for more details).…”

“…When X is an Ahlfors n-regular metric measure space, these spaces were introduced in [54]. We also point out that, when X is a metric measure space, these spaces may be non-trivial even when s ∈ (1, ∞) (see, for instance, [54] for more details).…”

“…It is easy to show that the spaces L(s, p, q; X), L(s, p, q; X), Lb (s, p, q; X), and L b (s, p, q; X) are independent of the choice of C (see, for instance, [35] and [54]). These spaces have the following basic properties, whose proofs are similar to those presented in [54], and we omit the details here. Proposition 3.7.…”

New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces

“…However, if X is a totally disconnected subset of R n , for example, X is a fractal of R n , then even for s > 1, f ∈ M s,p (X) is not necessary to be a constant; see [38] for an example.…”