凸域上的边界加权傅里叶不等式
Boundary-Weighted Fourier Inequalities for Convex Domains
AI总结：
该研究针对无仿射直线的凸域上Paley-Wiener空间的边界加权傅里叶不等式，刻画了多面体与球域下使不等式成立的参数三元组，还关联了球域次临界不等式与Kakeya猜想及截断Hankel算子理论。
AI中文摘要：
我们研究无仿射直线的凸集$Ω\subset \mathbb{R}^n$（$n \geq 2$）上Paley--Wiener空间$\mathrm{PW}^q(Ω)$的一类自然傅里叶不等式，该空间由傅里叶支撑在$Ω\subset \mathbb{R}^n$内的$L^q$函数构成。不等式形式为：\\[\n\int_Ω\dfrac{|\hat{f}(x)|^p}{ω_Ω^d(x)}dx\leq C\\|f\\|_{L^q}^p,\quad f\in \mathrm{PW}^q(Ω).\n\\]其中$\hat{f}$是$f$的傅里叶变换，$1 \leq p, q < \infty$，$d \in \mathbb{R}$，$ω_Ω$是与$Ω$对应的Paley--Wiener空间相关的频率乘子权，定义为：\\[\nω_Ω(x)=m(Ω\cap (2x-Ω)), \qquad x \in Ω.\n\\]对任意多面体$P$，我们完全刻画了使傅里叶不等式成立的参数三元组$(p,q,d)$。对球域$B$，我们刻画了$p \geq 2$时的有效三元组。当$p < 2$时，球域的情形有所不同，自然的临界不等式不成立。但我们证明了球面限制猜想蕴含球域的一族次临界傅里叶不等式，而这些不等式又蕴含（闵可夫斯基形式的）Kakeya猜想。最后，我们将这类傅里叶不等式与作用在$Ω$对应的Paley--Wiener空间上的截断Hankel算子理论联系起来。
英文摘要：
We consider the natural family of Fourier inequalities for the Paley--Wiener space $\mathrm{PW}^q(Ω)$, consisting of $L^q$-functions with Fourier support in a convex set $Ω\subset \mathbb{R}^n$, $n \geq 2$, free of affine lines. Namely, \[ \int_Ω\dfrac{|\hat{f}(x)|^p}{ω_Ω^d(x)}dx\leq C\|f\|_{L^q}^p,\quad f\in \mathrm{PW}^q(Ω). \] Here $\hat{f}$ is the Fourier transform of $f$, $1 \leq p, q < \infty$, $d \in \mathbb{R}$, and $ω_Ω$ is the frequency multiplier weight associated with the Paley--Wiener space of $Ω$, \[ ω_Ω(x)=m(Ω\cap (2x-Ω)), \qquad x \in Ω. \] For an arbitrary polyhedron $P$, we completely characterize the triples $(p,q,d)$ which yield valid Fourier inequalities. For a ball $B$, we characterize the valid triples when $p \geq 2$. When $p < 2$, the situation is different for the ball, and natural critical inequalities fail. However, we show that the spherical restriction conjecture implies a family of subcritical Fourier inequalities for the ball, which in turn imply the Kakeya conjecture (in its Minkowski-form). Finally, we link our family of Fourier inequalities to the theory of truncated Hankel operators acting on the Paley--Wiener space of $Ω$.
