论文标题
霍尔德 - 平滑曲线上的zygmund课程
Hölder--Zygmund classes on smooth curves
论文作者
论文摘要
我们证明，在本地zygmund类$ \ mathcal z^{m，1} $中，仅当其复合曲线均具有每个平滑曲线的class $ \ mathcal z^{m，1} $时。这补充了本地Hölder--lipschitz类的众所周知的类似结果$ \ Mathcal c^{m，α} $，我们在此过程中依靠。我们证明，这些结果概括为在Banach空间之间进行映射，并使用它们来研究叠加操作员的规律性$ f_*：g \ mapsto f \ circ g $作用于全球Zygmund Space $λ_{M+1}（\ Mathbb r^d）$。我们证明，对于所有整数$ m，k \ ge 1 $，地图$ f_*：λ_{m+1}（\ Mathbb r^d）\toλ_{m+1}（\ Mathbb r^d）$是LipsChitz class $ \ Mathcal c^{k-1} $ if和仅$ if $ z^{m+k，1}（\ mathbb r）$。
We prove that a function in several variables is in the local Zygmund class $\mathcal Z^{m,1}$ if and only if its composite with every smooth curve is of class $\mathcal Z^{m,1}$. This complements the well-known analogous result for local Hölder--Lipschitz classes $\mathcal C^{m,α}$ which we reprove along the way. We demonstrate that these results generalize to mappings between Banach spaces and use them to study the regularity of the superposition operator $f_* : g \mapsto f \circ g$ acting on the global Zygmund space $Λ_{m+1}(\mathbb R^d)$. We prove that, for all integers $m,k\ge 1$, the map $f_* : Λ_{m+1}(\mathbb R^d) \to Λ_{m+1}(\mathbb R^d)$ is of Lipschitz class $\mathcal C^{k-1,1}$ if and only if $f \in \mathcal Z^{m+k,1}(\mathbb R)$.
