论文标题
切片的切片和全球平滑度常规和多序分析函数
Slice-by-slice and global smoothness of slice regular and polyanalytic functions
论文作者
论文摘要
在Quaternions的真实代数$ \ Mathbb {H} $上，将常规函数的概念概括为复杂变量的全态函数概念的概括。令$ω$为$ \ mathbb {h} $的一个开放子集，它与$ \ mathbb {r} $相交，并且在$ \ mathbb {h} $ ail $ \ mathbb {r} $的旋转中不变。函数$ f：ω\ to \ mathbb {h} $是定期的，如果它是$ \ mathscr {c}^1 $，对于所有复杂的平面$ \ mathbb {c} _i $跨度$ 1 $，$ 1 $，而quaternioniciminic timinic timinicaliminiagy Unit $ i $ i $ $ f $ $ f $ of $ f $ of $ f $ of $ f $ of $ f $ of $ f $ of $ f $ $ω_i=ω\ cap \ mathbb {c} _i $满足与$ i $相关的cauchy-riemann方程$ \ OVERLINE {\ partial} _i = \ frac {1} {2} {2} \ big（\ frac {\ partial} {\partialα}+i \ frac {\ partial} {\ partial} {\partialβ} \ big）$。给定任何正面的自然数$ n $，a函数$ f：ω\ to \ mathbb {h} $称为$ n $的slice polyanalytic，如果是$ \ mathscr {c}^n $和$ \ overline {\ partial} _i^_i^_i^{\，n} f_i = 0 $ $ $ $ $ $ $ $ $ $我们将订单$ n $的全局切片分析函数定义为函数$ f：ω\ to \ mathbb {h} $，它承认表格$ f（x）= \ sum_ {h = 0}^{n-1}^{n-1} \ sum_ {n-1} \ edline {x}}^}}^}^hf_h（x）$的分解$ f_0，\ ldots，f_ {n-1} $。任何订单$ n $的全球切片多芯片分析函数是相同订单$ n $的切片多序分析。相反是不正确的：对于每个$ n \ geq2 $，我们给出了$ n $的切片多动态分析功能的示例，这些功能不是全局。 本文的目的是研究片段定期和全球切片的连续性和差异性多种序列函数，被视为切片的解决方案$ \ overline {\ partial} _i^{\ partial} _i^{\，n} f_i = 0 $ω\ setMinus \ mathbb {r} $。我们的Quaternionic结果扩展到单基因。
The concept of slice regular function over the real algebra $\mathbb{H}$ of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let $Ω$ be an open subset of $\mathbb{H}$, which intersects $\mathbb{R}$ and is invariant under rotations of $\mathbb{H}$ around $\mathbb{R}$. A function $f:Ω\to\mathbb{H}$ is slice regular if it is of class $\mathscr{C}^1$ and, for all complex planes $\mathbb{C}_I$ spanned by $1$ and a quaternionic imaginary unit $I$, the restriction $f_I$ of $f$ to $Ω_I=Ω\cap\mathbb{C}_I$ satisfies the Cauchy-Riemann equations associated to $I$, i.e., $\overline{\partial}_I f_I=0$ on $Ω_I$, where $\overline{\partial}_I=\frac{1}{2}\big(\frac{\partial}{\partialα}+I\frac{\partial}{\partialβ}\big)$. Given any positive natural number $n$, a function $f:Ω\to\mathbb{H}$ is called slice polyanalytic of order $n$ if it is of class $\mathscr{C}^n$ and $\overline{\partial}_I^{\,n} f_I=0$ on $Ω_I$ for all $I$. We define global slice polyanalytic functions of order $n$ as the functions $f:Ω\to\mathbb{H}$, which admit a decomposition of the form $f(x)=\sum_{h=0}^{n-1}\overline{x}^hf_h(x)$ for some slice regular functions $f_0,\ldots,f_{n-1}$. Global slice polyanalytic functions of any order $n$ are slice polyanalytic of the same order $n$. The converse is not true: for each $n\geq2$, we give examples of slice polyanalytic functions of order $n$, which are not global. The aim of this paper is to study the continuity and the differential regularity of slice regular and global slice polyanalytic functions viewed as solutions of the slice-by-slice differential equations $\overline{\partial}_I^{\,n} f_I=0$ on $Ω_I$ and as solutions of their global version $\overline{\vartheta}^nf=0$ on $Ω\setminus\mathbb{R}$. Our quaternionic results extend to the monogenic case.
