Let Γ be a compact d-set in Rn with 0 < d ≤n, which includes various kinds of fractals. The author shows that the Besov spaces Bspq(Γ) defined by two different and equivalent methods, namely, via traces and quarkonial decompositions in the sense of Triebel are the same spaces as those obtained by regarding Γ as a space of homogeneous type when 0 < s < 1, 1 < p <∞　 and 1≤q≤∞.

New norms for some distributions on spaces of homogeneous type which include some fractals are introduced. Using inhomogeneous discrete Calderon reproducing formulae and the Plancherel-Polya inequalities on spaces of homogeneous type, the authors prove that these norms give a new characterization for the Besov and Triebel-Lizorkin spaces with p; q > 1 and can be used to introduce new inhomogeneous Besov and Triebel-Lizorkin spaces with p; q≤ 1 on spaces of homogeneous type. Moreover, atomic decompositions of these new spaces are also obtained. All the results of this paper are new even for Rn.

In this paper, the authors establish the inhomogeneous Plancherel-Polya inequalities on spaces of homogeneous type by use of the inhomogeneous discrete Calderon reproducing formulas. As an application, the authors prove that the Lebesgue norms of the inhomogeneous Littlewood-Paley g-function and S-function on spaces of homogeneous type are equivalent. All results are new even for Rn.

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, Hajlasz–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.

Let μ be a Radon measure on Rd which satisfies the growth condition only namely, there is a constant C > 0 such that for all x ∈ Rd,r > 0 and for some fixed 0 < n ≤d, μ(B(x, r)) ≤Crn, where B(x, r) is the ball centered at x and having radius r. In this paper, we first give a new atomic characterization of the Hardy space H1(μ) introduced by X. Tolsa. As applications of this new characterization, we establish the (H1(μ), L1,∞(μ)) estimate of the commutators generated by RBMO(μ) functions with the Calderon–Zygmund operators whose kernels satisfy only the size condition and a certain minimum regularity condition. Using this endpoint estimate and a new interpolation theorem for operators which is also established in this paper and has independent interest, we further obtain the Lp(μ) (1 < p < ∞) boundedness of these commutators.