Boundedness of linear operators via atoms on Hardy spaces with non-doubling measures
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Dachun Yang
und
Dongyong Yang
Abstract
Let μ be a non-negative Radon measure on 
which satisfies only the polynomial growth condition. Let 𝒴 be a Banach space and H1(μ) be the Hardy space of Tolsa. In this paper, the authors prove that a linear operator T is bounded from H1(μ) to 𝒴 if and only if T maps all (p, γ)-atomic blocks into uniformly bounded elements of 𝒴; moreover, the authors prove that for a sublinear operator T bounded from L1(μ) to L1, ∞(μ), if T maps all (p, γ)-atomic blocks with p ∈ (1, ∞) and γ ∈ ℕ into uniformly bounded elements of L1(μ), then T extends to a bounded sublinear operator from H1(μ) to L1(μ). For the localized atomic Hardy space h1(μ), the corresponding results are also presented. Finally, these results are applied to Calderón–Zygmund operators, Riesz potentials and multilinear commutators generated by Calderón–Zygmund operators or fractional integral operators with Lipschitz functions to simplify the existing proofs in the related papers.
© de Gruyter 2011
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Artikel in diesem Heft
- Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth
- On the logarithmic summability of Fourier series
- On a relationship between absolutely nonmeasurable functions and Sierpiński–Zygmund functions
- Boundedness of weighted singular integral operators in grand Lebesgue spaces
- A generalization of the Euler beta function and applications
- Normality and shared values concerning differential polynomial
- Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph
- About some geometric characteristics of the generalized Möbius–Listing surfaces
- Variation formulas of solution for a delay differential equation taking into account delay perturbation and the continuous initial condition
- An admissibility for topological degree of variable Besov and Triebel–Lizorkin spaces
- Boundedness of linear operators via atoms on Hardy spaces with non-doubling measures
