Atomic hardy-type spaces between H1 and L1 on metric spaces with non-doubling measures
Atomic hardy-type spaces between H1 and L1 on metric spaces with non-doubling measures
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DOI：
10.1007/s10114-011-9118-7
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发表时间：
2011-11
期刊：
影响因子：
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通讯作者：
Liguang Liu;Dachun Yang;Dongyong Yang
中科院分区：
文献类型：
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作者：
Liguang Liu;Dachun Yang;Dongyong Yang
Letbe（莱贝，|·|，μ），其中|·|是欧几里得距离，μ是满足多项式增长条件的Gauss测度度量空间（Gauss测度，|·|，dλ），或空间（S，d，ρ），其中S ∈ n ∈ n+是（ax+B）-群，di是左不变黎曼度量，ρ是S上指数增长的右Haar测度.本文引入并建立了原子Hardy型空间和BMO型空间的一些性质。设H 1是已知的原子哈代空间，L 0 1是L 1以外的整数为0的子空间.本文证明了当X ∈（0，∞）时，X s是对偶空间，当X ∈（0，1]时，X s = H 1，且X ∞= L 0 1（或L 1）.作为应用，证明了如果T是从H1到L1和从L1到L1，∞有界的线性算子，则对所有的r ∈（1，∞）和∈（r，∞]，T是从Xr到Lorentz空间L1，s有界的，这适用于（n，n）上的Calderón-Zygmund算子，|·|，μ），Ornstein-Uhlenbeck算子在（n，n，|·|，dγ）和与（S，d，ρ）上的谱乘子相关的谱算子。这些结果推广了Sweezy，Abu-Shammala和Torchinsky关于欧氏空间的相应结果.
Letbe (ℝn, |·|, µ), where |·| is the Euclidean distance, µ is a nonnegative Radon measure on ℝnsatisfying the polynomial growth condition, or the Gauss measure metric space (ℝn, |·|,dλ), or the space (S, d, ρ), whereS≡ ℝn⋉ ℝ+is the (ax+b)-group,dis the left-invariant Riemannian metric andρis the right Haar measure onSwith exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spacesand the BMO-type spaces. LetH1be the known atomic Hardy space andL01the subspace off∈L1with integral 0. The authors prove that the dual space ofXsiswhens∈ (0,∞),Xs=H1whens∈ (0, 1], andX∞=L01(orL1). As applications, the authors show that ifTis a linear operator bounded fromH1toL1and fromL1toL1,∞, then for allr∈ (1,∞) ands∈ (r,∞],Tis bounded fromXrto the Lorentz spaceL1,s, which applies to the Calderón-Zygmund operator on (ℝn, |·|, µ), the imaginary powers of the Ornstein-Uhlenbeck operator on (ℝn, |·|,dγ) and the spectral operator associated with the spectral multiplier on (S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.