论文标题
$ l^p $ quasi-norms和统一级子套装问题的下限
Lower bounds on $L^p$ quasi-norms and the Uniform Sublevel Set Problem
论文作者
论文摘要
最近，斯坦纳伯格（Steinerberger）证明，拉普拉斯（Laplacian）是与标准统一统一的损失套件不等式的对立面的统一不等式，该套件众所周知，这因拉普拉斯（Laplacian）而失败。在本文中，我们观察到，这种类型的许多不平等来自$ l^1 $ norm的均匀下限，并为任何线性差分运算符给出类似的结果，这对于非线性运算符可能会失败。我们将$ p <1 $的$ l^p $ quasi-norms上的下限视为一个比统一的Sublevel套件不平等的弱财产，并为Laplacian和Heat Operators证明了这一点。我们以一些自然出现的问题得出结论。
Recently, Steinerberger proved a uniform inequality for the Laplacian serving as a counterpoint to the standard uniform sublevel set inequality which is known to fail for the Laplacian. In this paper, we observe that many inequalities of this type follow from a uniform lower bound on the $L^1$ norm, and give an analogous result for any linear differential operator, which can fail for non-linear operators. We consider lower bounds on the $L^p$ quasi-norms for $p<1$ as a stronger property that remains weaker than a uniform sublevel set inequality and prove this for the Laplacian and heat operators. We conclude with some naturally arising questions.
