Ornstein-Uhlenbeck算子Hardy型空间的比较
Comparison of spaces of Hardy type for the Ornstein-Uhlenbeck operator
摘要：
用g表示R^n上的高斯测度，用L表示Ornstein-Uhlenbeck算子。本文引入了Goldberg型的局部Hardy空间h^1（g），并将它与Mauceri和Meda在前一篇文章中引入的Hardy空间H^1（g）进行了比较.我们证明了对于每个正的r，算子rI+L的虚幂从h^1（g）到L^1（g）是无界的。这个结果与以下两个事实形成了鲜明的对比：虚幂从$H^1(g}$到L^1（g）是有界的;对于欧几里得拉普拉斯算子Δ和勒贝格测度λ），rI-Δ的虚幂从Goldberg空间h^1（λ）到L^1（λ）是有界的。我们还考虑具有黎曼测度m的黎曼流形M的情况。我们证明了在M上的某些几何假设下，一个在L^2（m）上有界的算子T，其核满足某些解析假设，是从H^1（m）到L^1（m）有界的当且仅当它是从h^1（m）到L^1（m）有界的。这里H^1（m）表示作者在arXiv：0808.0146中引入的局部加倍度量测度空间上的Hardy空间，h^1（m）是M上的Goldberg型Hardy空间，等价于M最近引入的一个空间. Taylor.还考虑了齐次树上的平移不变算子的情形。
英文摘要：
Denote by g the Gauss measure on R^n and by L the Ornstein-Uhlenbeck operator. In this paper we introduce a local Hardy space h^1(g) of Goldberg type and we compare it with the Hardy space H^1(g) introduced in a previous paper by Mauceri and Meda. We show that for each each positive r the imaginary powers of the operator rI+L are unbounded from h^1(g) to L^1(g). This result is in sharp contrast both with the fact that the imaginary powers are bounded from $H^1(g}$ to L^1(g), and with the fact that for the Euclidean laplacian Δand the Lebesgue measure λ) the imaginary powers of rI-Δare bounded from the Goldberg space h^1(λ) to L^1(λ). We consider also the case of Riemannian manifolds M with Riemannian measure m. We prove that, under certain geometric assumptions on M, an operator T, bounded on L^2(m), and with a kernel satisfying certain analytic assumptions, is bounded from H^1(m) to L^1(m) if and only if it is bounded from h^1(m) to L^1(m). Here H^1(m) denotes the Hardy space on locally doubling metric measure spaces introduced by the authors in arXiv:0808.0146, and h^1(m) is a Goldberg type Hardy space on M, equivalent to a space recently introduced by M. Taylor. The case of translation invariant operators on homogeneous trees is also considered.
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