Let(X,ρ,μ)d,θ be a space of homogeneous type,ε∈ (0,θ],|s|＜εand max{d/(d +ε),d/(d+s+ε)}＜q≤∞.The author introduces the new Triebel-Lizorkin spaces Fs∞q(X) and establishes the frame characterizations of these spaces by first establishing a Plancherel-Polya-type inequality related to the norm of the spaces Fs∞q(X).The frame characterizations of the Besov space Bspq(X) with |s|＜ε,max{d/(d+ε),d/(d+s+ε)}＜p≤∞ and 0＜q≤∞ and the Triebel-Lizorkin space Fspq(X) with |s|＜ε,max {d/(d+ε),d/(d+s+ε)}＜p＜∞ and max{d/(d+ε),d/(d+s+ε)}＜q≤∞ are also presented.Moreover,the author introduces the new Triebel-Lizorkin spaces bFs∞q(X) and HFs∞q(X) associated to a given para-accretive function b.The relation between the space bFs∞q(X) and the space 0 and q=2,then resented.The author further proves that if s=HFs∞q(X) is also pHFs∞q(X) = Fs∞q(X),which also gives a new characterization of the space BMO(X),since Fs∞q(X)=BMO(X).
This paper introduces the fractional Sobolev spaces on spaces of homogeneous type, includingmetric spaces and fractals. These Sobolev spaces include the well-known Hajfasz-Sobolev spaces as specialmodels. The author establishes various characterizations of (sharp) maximal functions for these spaces. Asapplications, the author identifies the fractional Sobolev spaces with some Lipscitz-type spaces. Moreover,some embedding theorems are also given.
