Maximal function characterizations of Hardy spaces on RD-spaces and their applications
Maximal function characterizations of Hardy spaces on RD-spaces and their applications
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DOI：
10.1007/s11425-008-0057-4
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发表时间：
2008-09
期刊：
影响因子：
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通讯作者：
L. Grafakos;Liguang Liu;Dachun Yang
中科院分区：
文献类型：
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作者：
L. Grafakos;Liguang Liu;Dachun Yang
设一个rd -空间，即Coifman和Weiss意义上的齐次型空间，它具有反向加倍性质。假设x的维数是n。对于α∈(0,∞)，分别用h αp(X),Hdp(X)和h *，p(X)表示对应的Hardy空间onx，分别由非切极大函数，并矢极大函数和极大函数定义。利用一个新的非齐次Calderón再现公式，证明了这些Hardy空间在p∈(1,∞)时与lp (X)重合，在p∈(n/(n+ 1), 1)时彼此重合。建立了h∗，p(X)与p∈(n/(n+ 1)， 1]的原子刻划；并且在∈(n/(n+ 1)，1]范围内，证明了空间eh *，p(X)，由Littlewood-Paley函数定义的Hardy空间hp (X)，与Coifman和weiss的原子Hardy空间重合。进一步证明了子线性算子t从hp （X)唯一地扩展到某个拟巴那赫空间bif的有界子线性算子，且只有当q∈(p,∞)∩[1,∞）或连续(p,∞)-原子映射到b的一致有界元素。
LetXbe an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume thatXhas a “dimension”n. Forα∈ (0, ∞) denote byHαp(X),Hdp(X), andH*,p(X) the corresponding Hardy spaces onXdefined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide withLp(X) whenp∈ (1,∞] and with each other whenp∈ (n/(n+ 1), 1]. An atomic characterization forH∗,p(X) withp∈ (n/(n+ 1), 1] is also established; moreover, in the rangep∈ (n/(n+ 1),1], it is proved that the spaceH*,p(X), the Hardy spaceHp(X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operatorTuniquely extends to a bounded sublinear operator fromHp(X) to some quasi-Banach spaceBif and only ifTmaps all (p,q)-atoms whenq∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements ofB.