论文标题
切片和良好的属性，用于有限的$ \ mathcal a $变量的功能
Slicing and fine properties for functions with bounded $\mathcal A$-variation
论文作者
论文摘要
我们研究了$ \ mathrm {bv}^{\ Mathcal a} $中功能的切片和精细属性，具有有限的$ \ Mathcal a $ variation的功能空间。在这里，$ \ Mathcal a $是具有恒定系数（任意顺序）的同质线性差分运算符。我们的主要结果是满足以下一维结构定理的所有$ \ mathcal a $的表征：\ mathrm {bv}^{\ mathcal a} $的每个$ u \ in in \ mathrm {bv}^{\ mathcal a} $都可以切成一维$ \ mathrm {bv} $ - 节。此外，将$ \ Mathcal a u $分解为绝对连续的部分$ \ Mathcal a^a u $，一个cantor part $ \ mathcal a^c u $和跳跃部分$ \ mathcal a^j u $，可以从相应的经典$ d^a，d^c $ d^c $ d^c $ d^j $ $ $ $ bv itsivivim ytivim ytivim sondivim sondivim sodivim sodivim sodivim sodivim sodivim y y $中恢复。通过这种结果，我们能够分析Lebesgue点的集合以及这些功能具有近似单方面限制的跳跃点集。因此，在$ \ mathrm {bv}^{\ mathcal a} $中证明结构和细性定理。我们的结果扩展了$ \ mathrm {bv} $的大多数经典罚款属性（以及所有以$ \ mathrm {bd} $而闻名的属性）。特别是，我们为$ \ mathscr {bv}^k，\ mathrm {bd}^k $建立了一个切片理论和精细属性，以及一个$ \ mathrm {bv}^{\ mathcal a} $ - 未被现有理论覆盖的空间。
We study the slicing and fine properties of functions in $\mathrm{BV}^{\mathcal A}$, the space of functions with bounded $\mathcal A$-variation. Here, $\mathcal A$ is a homogeneous linear differential operator with constant coefficients (of arbitrary order). Our main result is the characterization of all $\mathcal A$ satisfying the following one-dimensional structure theorem: every $u \in \mathrm{BV}^{\mathcal A}$ can be sliced into one-dimensional $\mathrm{BV}$-sections. Moreover, decomposing $\mathcal A u$ into an absolutely continuous part $\mathcal A^a u$, a Cantor part $\mathcal A^c u$ and a jump part $\mathcal A^j u$, each of these measures can be recovered from the corresponding classical $D^a,D^c$ and $D^j$ $BV$-derivatives of its one-dimensional sections. By means of this result, we are able to analyze the set of Lebesgue points as well as the set of jump points where these functions have approximate one-sided limits. Thus, proving a structure and fine properties theorem in $\mathrm{BV}^{\mathcal A}$. Our results extend most of the classical fine properties of $\mathrm{BV}$ (and all of those known for $\mathrm{BD}$). In particular, we establish a slicing theory and fine properties for $\mathscr {BV}^k, \mathrm{BD}^k$ and a whole class of $\mathrm{BV}^{\mathcal A}$-spaces that is not covered by the existing theory.
