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 $Ω$.

Disclosure

“6 K. BAMPOURAS AND K.-M. PERFEKT 1.2. Acknowledgments. The second author was supported by grant no. 334466 of the Research Council of Norway, ”Fourier Methods and Multiplicative Analysis”. The authors used ChatGPT 5.4/5.5/5.6 by OpenAI as a tool in the development of this paper. However, as a project started quite a while ago, it was primarily produced in the old-school way. The authors independently wrote and verified all parts of the article.”

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Brainstorming or outlining

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2

Verified

2026-08-22T18:13:34.821593Z

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