Recently, Steinerberger (Potential Analysis, 2020) 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 L¹ 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.

中文翻译：

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LP拟范数的下界和一致的子集问题

最近，Steinerberger（电位分析，2020年）证明了拉普拉斯算子的统一不等式，可以作为标准的统一子集集不等式的对立点，而后者对于拉普拉斯算子是失败的。在本文中，我们观察到这种类型的许多不等式遵循L ¹范数上的一致下界，并且对于任何线性微分算子都给出了类似的结果，对于非线性算子可能会失败。我们认为${\mathrm{大号}}^p$ 准规范 $p<\mathrm{1个}$作为一个更强的属性，它仍然比统一的子级集不等式弱，并为拉普拉斯算子和热算子证明了这一点。我们以一些自然而然的问题作为结尾。