AI 中文总结
研究在芬斯勒测度空间上 $1 < p < n$ 的 $L^p$ -不确定性原理，通过曲率界等条件证明相关原理并刻画其尖锐性，推广了芬斯勒和黎曼情形下的相关结果。
AI 中文摘要
本文证明了在 $n(\geq 2)$ 维向前完备且非紧的芬斯勒测度空间 $(M, F, \mathfrak{m})$ 上，对于 $1 < p < n$ 的 $L^p$ -不确定性原理，包括经典的海森堡 - 泡利 - 外尔不等式等作为特殊情况。这些不等式在曲率有上下界常数时成立。此外，通过 $F$ 的可逆性以及由测度 $\mathfrak{m}$ 诱导的旗（或里奇）曲率和 S - 曲率的界来刻画 $L^p$ -不确定性原理的尖锐性，并得到一些刚性结果，推广了芬斯勒情形下[HKZ]和黎曼情形下[KKPZ]的相关结果。
英文摘要
In this paper, we prove the $L^p(p>1)$-uncertainty principles for any $1<p<n$, including the classical Heisenberg-Pauli-Weyl inequality, Caffarelli-Kohn-Nirenberg interpolation inequality and Hardy inequality in $\mathbb R^n$ as special cases, on $n(\geq 2)$-dimensional forward complete and noncompact Finsler measure spaces $(M, F, \mathfrak{m})$ with curvatures bounded from above or below by constants. Further, we characterize the sharpness of $L^p$-uncertainty principles in terms of the reversibility of $F$ and the bounds of flag (or Ricci) curvature and S-curvature induced by the measure $\mathfrak m$ and obtained some rigidity results, which generalize the related ones in [HKZ] in Finslerian case and [KKPZ] in Riemannian case.
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