Grushin型退化椭圆算子的Riesz变换及其相关不等式
Riesz transform and its related inequalities for degenerate elliptic operators of Grushin type
浏览论文内容
中文总结 AI 辅助
研究Grushin型退化椭圆算子的Riesz变换\(L^p\)有界性及反向Riesz不等式，通过显式构造泊松核、格林核及调和湮灭方法，证明其在不同退化情形下的有界性，阐明相关现象机制并得出精确不等式。
中文摘要 AI 辅助
我们研究了Grushin型退化椭圆算子的Riesz变换的\(L^p\)有界性和反向Riesz不等式。当退化变量的维数至少为二时，我们证明了Riesz变换的全范围\(L^p\)有界性，并在一维弱退化情况下得到了精确范围，包括端点障碍。在强退化的一维情形下，我们恢复了全范围有界性，揭示了奇异集行为的显著转变。证明过程为奇异集附近的Grushin调和函数发展了一个反向Hölder理论。主要成分是适应Friedrichs扩张的显式泊松核和格林核构造，以及分离Riesz核关键部分的调和湮灭方法。这些技术阐明了有界和无界现象背后的机制，并为同一类算子产生了基本精确的反向Riesz不等式。
英文摘要
We study the $L^p$ boundedness of the Riesz transform and the reverse Riesz inequality for degenerate elliptic operators of Grushin type. We prove full-range $L^p$ boundedness of the Riesz transform when the degenerate variable has dimension at least two, and obtain the sharp range in the one-dimensional weakly degenerate case, including the endpoint obstruction. In the strongly degenerate one-dimensional regime, we recover full-range boundedness, revealing a striking transition in the behavior of the singular set. The proof develops a reverse Hölder theory for Grushin harmonic functions near the singular set. The main ingredients are explicit Poisson and Green kernel constructions adapted to the Friedrichs extension and a harmonic annihilation method which isolates the critical part of the Riesz kernel. These techniques illuminate the mechanism behind both the boundedness and unboundedness phenomena, and yield essentially sharp reverse Riesz inequalities for the same class of operators.
