A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces.
New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals
Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, , which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces and Triebel-Lizorkin spaces have the lifting properties for |s| < θ. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional integrals...
Triebel-Lizorkin spaces with non-doubling measures
Suppose that μ is a Radon measure on , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function inequality...
Some new spaces of Besov and Triebel-Lizorkin type on homogeneous spaces
New norms for some distributions on spaces of homogeneous type which include some fractals are introduced. Using inhomogeneous discrete Calderón reproducing formulae and the Plancherel-Pólya 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...
Factorization theorem for product Hardy spaces
We extend the well known factorization theorems on the unit disk to product Hardy spaces, which generalizes the previous results obtained by Coifman, Rochberg and Weiss. The basic tools are the boundedness of a certain bilinear form on ℝ²₊ × ℝ²₊ and the characterization of BMO(ℝ²₊ × ℝ²₊) recently obtained by Ferguson, Lacey and Sadosky.
Para-accretive functions, the weak boundedness property and the Tb theorem.
G. David, J.-L. Journé and S. Semmes have shown that if b and b are para-accretive functions on R, then the Tb theorem holds: A linear operator T with Calderón-Zygmund kernel is bounded on L if and only if Tb ∈ BMO, T*b ∈ BMO and MTM has the weak boundedness property. Conversely they showed that when b = b = b, para-accretivity of b is necessary for Tb Theorem to hold. In this paper we show that para-accretivity of both b and b is necessary for the Tb Theorem to hold in general. In addition, we...
Littlewood-Paley theory and ϵ-families of operators
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