Boundary-Weighted Fourier Inequalities for Convex Domains

Abstract

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 $Ω$.