凸性、傅里叶变换与格点偏差
Convexity, Fourier transforms, and lattice point discrepancy
AI总结：
本文将 Bruna–Nagel–Wainger 的傅里叶变换衰减估计推广到非有限线型凸实解析曲面和凸 $C^2$ 曲面，在振荡指数小于1时给出精确指数，并应用于格点偏差问题。
摘要：
在 Bruna、Nagel 和 Wainger [BNW] 的一篇著名论文中，针对界定凸区域的有限线型光滑超曲面，证明了傅里叶变换衰减估计。本文通过以下方式推广了他们的结果。首先，对于局部为凸实解析函数图像的超曲面，我们证明即使该曲面不是有限线型的，一个自然的类似结论仍然成立。其次，对于局部为凸 $C^2$ 函数图像的一般曲面，或由实解析方程定义的此类曲面的一部分，我们证明了一个结果，该结果在振荡指数小于 $1$ 的情形下蕴含类似的傅里叶变换衰减定理。在这些情形下，对于紧致曲面，所提供的指数是精确的。该结果对格点偏差问题具有启示意义，我们将对此进行描述。
英文摘要：
In a well-known paper by Bruna, Nagel and Wainger [BNW], Fourier transform decay estimates were proved for smooth hypersurfaces of finite line type bounding a convex domain. In this paper, we generalize their results in the following ways. First, for a surface that is locally the graph of a convex real analytic function, we show that a natural analogue holds even when the surface in question is not of finite line type. Secondly, we show a result for a general surface that is locally the graph of a convex $C^2$ function, or a piece of such a surface defined through real analytic equations, that implies an analogous Fourier transform decay theorem in situations where the oscillatory index is less than $1$. In such situations, for a compact surface the exponent provided is sharp. This result has implications for lattice point discrepancy problems, which we describe.
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