论文标题
傅里叶扩展运算符和应用的层析成像范围
Tomography bounds for the Fourier extension operator and applications
论文作者
论文摘要
我们探讨了在$ \ Mathbb {r}^n $中支撑在球体上的$ l^p $密度的傅立叶变换在多大程度上可以在仿射子空间上具有很大的质量，从而特别强调了线条和超级平面。这涉及建立$ x（| \ wideHat {gdσ} |^2）$和$ \ Mathcal {r}（| \ wideHat {gdσ} |^2）$的界限，其中$ x $ and $ x $和$ x $ and $ x $ and $ \ m nathcal {r} $表示x ray and Radon and Radon and Radon Transolssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss;这里$dσ$表示单位球上的lebesgue度量$ \ mathbb {s}^{n-1} $，而$ g \ in l^p（\ mathbb {s}^{n-1}）$。我们还确定了这种类型的一些猜想界限，这些界限位于经典的傅立叶限制和Kakeya猜想之间。最后，我们为$ \ wideHat {gdσ} $的加权规范不等式的理论提供了一些应用，从1970年代开始，建立了一些自然的猜想变体。我们的方法起源于Planchon和Vega的作品，通过Plancherel的仿射子空间定理取消利用，避免了传统使用波包和固定相位方法。
We explore the extent to which the Fourier transform of an $L^p$ density supported on the sphere in $\mathbb{R}^n$ can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing bounds on quantities of the form $X(|\widehat{gdσ}|^2)$ and $\mathcal{R}(|\widehat{gdσ}|^2)$, where $X$ and $\mathcal{R}$ denote the X-ray and Radon transforms respectively; here $dσ$ denotes Lebesgue measure on the unit sphere $\mathbb{S}^{n-1}$, and $g\in L^p(\mathbb{S}^{n-1})$. We also identify some conjectural bounds of this type that sit between the classical Fourier restriction and Kakeya conjectures. Finally we provide some applications of such tomography bounds to the theory of weighted norm inequalities for $\widehat{gdσ}$, establishing some natural variants of conjectures of Stein and Mizohata--Takeuchi from the 1970s. Our approach, which has its origins in work of Planchon and Vega, exploits cancellation via Plancherel's theorem on affine subspaces, avoiding the conventional use of wave-packet and stationary-phase methods.
