论文标题
真实分析功能的对数模量的Sobolev的可不同性能
Sobolev Differentiability Properties of Logarithmic Modulus of Real Analytic Functions
论文作者
论文摘要
令$ f $为$ \ mathbb {r}^n $ in $ n \ geq 2 $中的原始分析函数的胚芽，并假设$ f $ at $ \ mathbf {0} $的零集合的零集的零集合至少为$ 2 $。我们表明，$ \ log | f | $是$ w^{1,1} _ {\ operatatorName {loc}} $附近$ \ mathbf {0} $。特别是，这意味着差异不平等$ | \ nabla f | \ leq v | f | $持有$ v \ in l^1 _ {\ operatatorName {loc}} $。作为一个应用程序，我们得出了与此类功能有关的不等式和唯一性指数的不平等。
Let $f$ be the germ of a real analytic function at the origin in $\mathbb{R}^n $ for $n \geq 2$, and suppose the codimension of the zero set of $f$ at $\mathbf{0}$ is at least $2$. We show that $\log |f|$ is $W^{1,1}_{\operatorname{loc}}$ near $\mathbf{0}$. In particular, this implies the differential inequality $|\nabla f |\leq V |f|$ holds with $V \in L^1_{\operatorname{loc}}$. As an application, we derive an inequality relating the Łojasiewicz exponent and singularity exponent for such functions.
