论文标题
具有控制衍生物的可微分功能的定量驯服性能
Quantitative tame properties of differentiable functions with controlled derivatives
论文作者
论文摘要
我们表明，在凸体上定义的可区分函数$ k \ subseteq \ mathbb r^d $，其衍生物的衍生物不超过合适的给定实数的序列，与多项式共享许多属性。多项式程度的作用是由与给定的真实序列，$ k $的直径以及链接到$ c^0 $ - 函数的$ C^0 $ norm相关的整数扮演的。我们提供有关零集大小的定量信息，表明它通过Sobolev函数允许局部参数化，并证明了Remez-Type的不等式。从后者来看，我们得出了几种后果，例如，限制了级别的集合的数量，并比较$ l^p $ -norms扭转了霍尔德的不平等。许多结果的有效性仅取决于衍生物达到一定有限顺序。可以根据给定数据指定订单。
We show that differentiable functions, defined on a convex body $K \subseteq \mathbb R^d$, whose derivatives do not exceed a suitable given sequence of positive real numbers share many properties with polynomials. The role of the degree of a polynomial is hereby played by an integer associated with the given sequence of reals, the diameter of $K$, and a real parameter linked to the $C^0$-norm of the function. We give quantitative information on the size of the zero set, show that it admits a local parameterization by Sobolev functions, and prove an inequality of Remez-type. From the latter, we deduce several consequences, for instance, a bound on the volume of sublevel sets and a comparison of $L^p$-norms reversing Hölder's inequality. The validity of many of the results only depends on the derivatives up to some finite order; the order can be specified in terms of the given data.
