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Localized Hardy spaces H 1 related to admissible functions on RD-spaces and applications to Schrödinger operators. (English) Zbl 1217.42044

Let 𝒳 be an RD-space, which means that 𝒳 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝒳. A positive function ρ on 𝒳 is called admissible if there exist positive constants C 3 and k 0 such that for all x,y𝒳,

ρ(y)C 3 [ρ(x)] 1 1+k 0 [ρ(x)+d(x,y)] k 0 1+k 0 ,

where d is the metric on 𝒳. A nontrivial class of admissible function is the well-known reverse Hölder class. Let 𝒢 0 ε (β,γ) be the completion of the set which is composed of test functions with the additional property. The Hardy space H ρ 1 (𝒳) associated to ρ is defined as follows:

H ρ 1 (𝒳)={f𝒢 0 ε (β,γ) ' :f H ρ 1 (𝒳) =G ρ (f) L 1 (𝒳) <},

where ε(0,1), β,γ(0,ε) and G ρ (f) is the grand maximal function associated to ρ.

At first, the authors obtain an atomic decomposition characterization of H ρ 1 (𝒳). They show that H ρ 1 (𝒳)=H ρ 1,q (𝒳) with equivalent norms, where H ρ 1,q (𝒳) is the atomic Hardy space associated to ρ. Secondly, they establish a radial maximal function characterization of H ρ 1 (𝒳) and obtain another characterization of H ρ 1 (𝒳) via a variant of the radial maximal function, where the radial maximal function is associated to the admissible function ρ. Moreover, they prove the boundedness of certain localized singular integrals on H ρ 1 (𝒳) via a finite atomic decomposition characterization of some dense subspace of H ρ 1 (𝒳). The theory in this paper can be applied, respectively, to the Schrödinger operator or degenerate Schrödinger operator on n , or to the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups and some new results are also obtained.

Reviewer: Liu Yu (Beijing)
MSC:
42B30H p -spaces (Fourier analysis)
42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory
42B35Function spaces arising in harmonic analysis
42B37Harmonic analysis and PDE