发表机构
Uniwersytet Wrocławski（弗罗茨瓦夫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对任意根系与非负重函数的Dunkl拉普拉斯算子，建立了Hardy空间H^p的Coifman-Weiss型原子刻画，证明其与满足消没条件的原子生成空间一致，并给出拟范数等价及L^∞尺寸条件的等价性。
AI 中文摘要
设 $\Delta_k$ 为与任意根系及非负重函数相关联的 Dunkl 拉普拉斯算子。对于每个 $0<p\leq1$，建立了 Hardy 空间 $H^p_{\mathrm{Dunkl}}$ 的 Coifman-Weiss 型原子刻画。更精确地，证明了最初由相关平方函数定义的空间 $H^p_{\mathrm{Dunkl}}$ 与由 $({\rm CW},p,2)$-原子生成的空间一致，这些原子支撑在欧氏球上，并满足对次数 $\leq s_p$ 的所有多项式的消没条件，其中 $s_p=\left\lfloor \mathbf N\left(\frac1p-1\right)\right\rfloor$，$\mathbf N$ 为底层 Dunkl 测度的齐次维数。相应的拟范数等价。我们还证明了当原子定义中的 $L^2$ 尺寸条件替换为 $L^\infty$ 尺寸条件时，得到相同的空间。证明策略是使用与 Dunkl 拉普拉斯算子相关的算子型原子分解，然后证明每个这样的算子原子可以写成 $({\rm CW},p,2)$-原子的线性组合。
英文摘要
Let $Δ_k$ be the Dunkl Laplacian associated with an arbitrary root system and a nonnegative multiplicity function. For every $0<p\leq1$, a Coifman-Weiss type atomic characterization of the Hardy space $H^p_{\mathrm{Dunkl}}$ is established. More precisely, it is proved that the space $H^p_{\mathrm{Dunkl}}$, which is originally defined by a relevant square function, coincides with the space generated by $({\rm CW},p,2)$-atoms, that is, atoms supported on Euclidean balls and satisfying cancellation conditions against all polynomials of degree $ \leq s_p$, where \[ s_p=\left\lfloor \mathbf N\left(\frac1p-1\right)\right\rfloor \] and $\mathbf N$ is the homogeneous dimension of the underlying Dunkl measure. The corresponding quasi-norms are equivalent. We also show that the same space is obtained when the $L^2$ size condition in the definition of atoms is replaced by the $L^\infty$ size condition. The strategy of the proof is to use an operator-type atomic decomposition associated with the Dunkl Laplacian, and then prove that each such operator atom can be written a linear combination of $({\rm CW},p,2)$-atoms.
Comments23 pages
