AI 中文总结
该文证明Engel群上次拉普拉斯算子在局部H^s(s>2)条件下的谱乘子定理，确定尖锐阈值为2，结合加权Plancherel估计与秩二谱子空间分解，并给出局部化核的渐近展开。
AI 中文摘要
我们在Engel群上证明了标准次拉普拉斯算子的一个谱乘子定理，该定理在尺度不变的局部$H^s$条件下成立，其中$s>2$。乘子算子$F(\mathcal L)$具有弱型$(1,1)$性质，并且对每个$1<p<\infty$在$L^p$上有界。结合已知的下界，这确定了尖锐阈值为$2$，即拓扑维数的一半。证明将四次振荡器的加权Plancherel估计与分解为秩二谱子空间相结合，后者在井逃逸至无穷远时一致地控制参数导数。对于紧支撑在$(0,\infty)$上的固定光滑乘子，我们还确定了局部化核的首项和第一修正项。在局部$H^2$条件下弱型$(1,1)$估计是否成立仍然开放。
英文摘要
We prove a spectral multiplier theorem for the standard sub-Laplacian on the Engel group under a scale-invariant local $H^s$ condition with $s>2$. The multiplier operators $F(\mathcal L)$ are of weak type $(1,1)$ and bounded on $L^p$ for every $1<p<\infty$. Together with the known lower bound, this identifies the sharp threshold as $2$, half the topological dimension. The proof combines a weighted Plancherel estimate for quartic oscillators with a decomposition into rank-two spectral subspaces, which controls parameter derivatives uniformly as one well escapes to infinity. For fixed smooth multipliers compactly supported in $(0,\infty)$, we also determine the leading term and first correction for the localized kernels. Whether the weak type $(1,1)$ estimate holds under the local $H^2$ condition remains open.
Comments35 pages, comments are welcome
