论文标题
耐火空间会重新审视谐波重量
Hardy spaces meet harmonic weights revisited
论文作者
论文摘要
我们调查了与自动接合操作员$ l $相对应的hardy空间$ h^1_l（x）$。我们的主要目的是在原子分解方面获取$ h^1_l（x）$的描述，类似于经典的hardy空间的这种特征$ h^1（\ mathbb {r}^d）$。在适当的假设下，Yan和作者在[Trans。阿米尔。数学。 Soc。 375（2022），否。 9，6417-6451]，其中考虑了与$ l $ harmonic功能相关的原子。在这里，我们继续这项研究并修改原子的先前定义。 当生成操作员与线性独立谐波功能系统相关时，修改后的方法允许我们调查设置。在这种情况下，调整了原子的取消条件以适合该系统。在一个明确的示例中，我们考虑了一个具有末端的对称歧管，$ \ mathbb {r}^d \＃\ mathbb {r}^d $。对于此流形，有限的谐波函数的空间是二维的。 hardy空间中的任何元素$ h^1_l（x）$都必须与系统中的所有谐波函数正交。
We investigate Hardy spaces $H^1_L(X)$ corresponding to self-adjoint operators $L$. Our main aim is to obtain a description of $H^1_L(X)$ in terms of atomic decompositions similar to such characterisation of the classical Hardy spaces $H^1(\mathbb{R}^d)$. Under suitable assumptions, such a description was obtained by Yan and the authors in [Trans. Amer. Math. Soc. 375 (2022), no. 9, 6417-6451], where the atoms associated with an $L$-harmonic function are considered. Here we continue this study and modify the previous definition of atoms. The modified approach allows us to investigate settings, when the generating operator is related to a system of linearly independent harmonic functions. In this context, the cancellation condition for atoms is adjusted to fit this system. In an explicit example, we consider a symmetric manifold with ends $\mathbb{R}^d \# \mathbb{R}^d$. For this manifold the space of bounded harmonic functions is two-dimensional. Any element from the Hardy space $H^1_L(X)$ has to be orthogonal to all of the harmonic functions in the system.
