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arXiv 1002.3123math.FAmath-phmath.MP
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Besov函数迹的局部性态：普遍结果

Local behavior of traces of Besov functions: Prevalent results

Jean-Marie Aubry, Delphine Maman, Stéphane Seuret

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25 Pages, 3 figures
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摘要：

设$1 \leq d < D$和$(p,q,s)$满足$0 < p < \infty$，$0 < q \leq \infty$，$0 < s-d/p < \infty$ .本文研究了属于Besov空间$B^{s}_{p,q}(\R^D)$的函数的迹在$\R^D$的仿射子集上的全局和局部正则性。给定一个$d$维的子空间$H \subset \R^D$，对于几乎所有的功能在$B^{s}_{p,q}(\R^D)$（在流行的意义上），我们能够计算的奇异谱的轨迹$f_a$的$f$仿射子空间的形式$a+H$，勒贝格-几乎每$a \in \R^{D-d}$。特别是，我们证明了几乎每一个$a \in \R^{D-d}$，这些痕迹$f_a$更经常比可以预期从标准的跟踪定理，和$f_a$享有多重分形行为。

英文摘要：

Let $1 \leq d < D$ and $(p,q,s)$ satisfying $0 < p < \infty$, $0 < q \leq \infty$, $0 < s-d/p < \infty$. In this article we study the global and local regularity properties of traces, on affine subsets of $\R^D$, of functions belonging to the Besov space $B^{s}_{p,q}(\R^D)$. Given a $d$-dimensional subspace $H \subset \R^D$, for almost all functions in $B^{s}_{p,q}(\R^D)$ (in the sense of prevalence), we are able to compute the singularity spectrum of the traces $f_a$ of $f$ on affine subspaces of the form $a+H$, for Lebesgue-almost every $a \in \R^{D-d}$. In particular, we prove that for Lebesgue-almost every $a \in \R^{D-d}$, these traces $f_a$ are more regular than what could be expected from standard trace theorems, and that $f_a$ enjoys a multifractal behavior.

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