AI 中文总结
该研究针对拉普拉斯算子本征函数的$L^p$界问题，提出了比非正曲率更通用的微局部Kakeya-Nikodym平均改进条件，利用相空间变换与高斯束近似拓展了相关理论。
AI 中文摘要
作者与Sogge的前期工作[BS17]、[BS18]已证明，微局部Kakeya-Nikodym平均在改进高频极限下拉普拉斯算子（近似）本征函数的$L^p$界方面具有重要意义。这类平均通过以下方式构造：利用伪微分算子将相空间中测地线段附近、与频率相关的小管区域内的本征函数局域化，再取其$L^2$范数。前者工作表明，当$p$小于Stein-Tomas指数时，$L^p$范数受限于这些平均的上确界；后者工作则证明，当$(M,g)$具有非正截面曲率时，这类平均存在对数增益。两项工作结合，在该几何场景下改进了本征函数的$L^p$理论，超越了Sogge的通用界。本工作中，我们为改进这类平均建立了比非正曲率更通用的充分条件，这些条件源于切丛上测地流的动力学，考虑流在至少某些方向上拉伸和压缩切向量的情况，例如部分双曲流。我们利用流形上的高斯波包（相空间）变换，以充分理解这些假设对微局部平均的增益，过程中还以坐标不变的方式进一步发展了波动方程的高斯束近似。
英文摘要
Previous works of the author and Sogge [BS17], [BS18] showed the significance of microlocal Kakeya-Nikodym averages in improving $L^p$ bounds on (approximate) eigenfunctions of the Laplacian in the high frequency limit. These averages are formed by taking the $L^2$ norm of an eigenfunction when localized in phase space to a small, frequency-dependent tube about a geodesic segment via a pseudodifferential operator. The former work showed that for values of $p$ beneath the Stein-Tomas exponent, $L^p$ norms are controlled by a supremum over these averages. The latter work then showed that when $(M,g)$ has nonpositive sectional curvatures, there is a logarithmic gain in the averages. In combination, these two works improved the $L^p$ theory for eigenfunctions over the universal bounds of Sogge in this geometric setting. In the present work, we develop sufficient conditions for improving these averages which are more general than nonpositive curvature. Instead our sufficient conditions are rooted in the dynamics of the geodesic flow on the tangent bundle, considering cases where the flow expands and contracts tangent vectors in at least some directions, e.g. partially hyperbolic flows. We make use of Gaussian wave packet (phase space) transforms on the manifold in order to fully appreciate the gain these hypotheses impart on the microlocal averages. In the process, we further develop Gaussian beam approximations to the wave equation in a coordinate invariant manner.
Comments59 pages
