Lebesgue类与实可构造函数的准备
Lebesgue classes and preparation of real constructible functions
摘要：
我们称一个函数是可构造的，如果它有一个整体次解析区域，并且可以表示为整体次解析函数与正值整体次解析函数的乘积之和。对于任何$q > 0$和构造函数$f$和$μ$上的$E\times\RR^n$，我们证明了一个定理描述的结构的集合的所有$(x,p)$在$E \times (0,\infty]$的$y \mapsto f(x,y)$是在$L^p(|μ|_{x}^{q})$，其中$|μ|_{x}^{q}$是积极的措施$\RR^n$的Radon-Nikodym导数关于勒贝格措施是$y\mapsto |μ(x,y)|^q$。我们还证明了一个密切相关的准备定理$f$和$μ$。这些结果将分析（$L^p$ -空间的研究）与几何（零轨迹的研究）联系起来。
英文摘要：
We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any $q > 0$ and constructible functions $f$ and $μ$ on $E\times\RR^n$, we prove a theorem describing the structure of the set of all $(x,p)$ in $E \times (0,\infty]$ for which $y \mapsto f(x,y)$ is in $L^p(|μ|_{x}^{q})$, where $|μ|_{x}^{q}$ is the positive measure on $\RR^n$ whose Radon-Nikodym derivative with respect to the Lebesgue measure is $y\mapsto |μ(x,y)|^q$. We also prove a closely related preparation theorem for $f$ and $μ$. These results relate analysis (the study of $L^p$-spaces) with geometry (the study of zero loci).
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