Boundary-Weighted Fourier Inequalities for Convex Domains

We consider the natural family of Fourier inequalities for the Paley–Wiener space ${\mathrm{P}\mathrm{W}}^q(\mathrm{\Omega })$, consisting of $L^q$-functions with Fourier support in a convex set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, free of affine lines. Namely, $${\int }_{\mathrm{\Omega }}{\displaystyle \frac{\mathrm{\mid }\widehat{f}(x){\mathrm{\mid }}^p}{{\omega }_{\mathrm{\Omega }}^d(x)}}dx\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^q}^p,f\in {\mathrm{P}\mathrm{W}}^q(\mathrm{\Omega })\mathrm{.}$$ Here $\widehat{f}$ is the Fourier transform of $f$, $1\le p,q<\mathrm{\infty }$, $d\in \mathbb{R}$, and ${\omega }_{\mathrm{\Omega }}$ is the frequency multiplier weight associated with the Paley–Wiener space of $\mathrm{\Omega }$, $${\omega }_{\mathrm{\Omega }}(x)=m(\mathrm{\Omega }\cap (2x-\mathrm{\Omega })),x\in \mathrm{\Omega }\mathrm{.}$$ 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 $p\ge 2$. 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 $\mathrm{\Omega }$.