高维Hausdorff算子在Hp(Rn)上的有界性
The Multidimensional Hausdorff Operators on Hp(Rn)
本文主要研究以下形式的Hausdorff算子Hφf(x)=∫Rnφ(u1,...,un)f(u1x1,...,unxn)du1 · · · dun,其中φ是Rn上的缓增分布.当n ≥ 2,0<p<1,若φ是Schwartz函数,我们得到Hφ在Hp(Rn)上有界当且仅当φ=0.进一步,当n ≥ 2,(n+1)/n<p<1,如果φ仅仅是连续函数,并且Hφ有合适定义,那么Hφ在Hp(Rn)上有界当且仅当φ是常数.这些结果都表明Hausdorff算子Hφ在Hp(Rn)上的有界性很复杂.此外,我们将Hφ转化成卷积型算子,得到Hφ在Lebesgue空间上有界的一些新的结果.
We consider the following Hausdorff operator Hφf(x)=∫Rnφ(u1,..., un) · f(u1x1,..., unxn)du1 · · · dun, where φ can be considered as a distribution on Rn. When n ≥ 2 and φ is a Schwartz function, we show that Hφ is bounded on Hp(Rn) for some p ∈ (0, 1) if and only if φ ≡ 0. Furthermore, when n ≥ 2, if φ is just a continuous function and Hφ can be defined suitable, then we can also prove that Hφ is bounded on Hp(Rn) for some p ∈ ((n+1)/n, 1) if and only if φ equals to a constant. These facts mean that Hφ is very complicated on Hp(Rn) (n ≥ 2). Moreover, we establish a result of the boundedness of Hφ on Lp(Rn), p > 1. The key idea used here is to reformulate Hφ as a convolution operator.
Hausdorff算子 / Lebesgue空间 / Hardy空间 {{custom_keyword}} /
Hausdorff operator / Lebesgue space / Hardy space {{custom_keyword}} /
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国家自然科学基金资助项目(11871436)
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