论文摘要
本文以四元Heisenberg群为研究对象。主要研究了以下三个方面的内容： 首先，鉴于欧氏空间中Laplace算子的平均值定理和Hardy不等式在偏微分方程和相关学科中所起的重要作用，我们在四元Heisenberg群上建立次Laplace算子的平均值定理。作为对平均值定理的运用，我们得到p为2的Hardy不等式和不确定原理。利用picone恒等式的方法，我们得到四元Heisenberg群上一般p的Hardy不等式。 其次，我们通过考察球面函数的技巧，得到四元Heisenberg群上次Laplace算子的唯一延拓性的若干结果。 最后，我们建立四元Heisenberg群上p一次Laplace算子的基本解，并证明四元Heisenberg群是一个可极化Carnot群。
论文目录
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