Boundary-Weighted Fourier Inequalities for Convex Domains

Abstract: We consider the natural family of Fourier inequalities for the Paley--Wiener space $\mathrm{PW}<sup>q(Ω)$, consisting of $L<sup>q$-functions with Fourier support in a convex set $Ω\subset \mathbb{R}<sup>n$, $n \geq 2$, 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 $f$, $1 \leq p, q &lt; \infty$, $d \in \mathbb{R}$, and $ω</em>Ω$ 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 $p &lt; 2$, 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 $Ω$.

• The paper completely characterizes valid exponents for polyhedra, proves the ball case for p≥2, and connects critical failures for p<2 to Kakeya-type geometry.
• Boundary weights defined through Macbeath regions automatically provide the geometric admissibility needed for Paley–Wiener Besov embeddings, density results, and scaling-based necessity arguments.
• The results yield applications to truncated Hankel operators, including Nehari’s theorem for Hilbert–Schmidt Hankel operators on the ball, while leaving higher-dimensional subcritical cases open.

Overview

This paper by Bampouras and Perfekt studies a natural scale of Fourier inequalities for Paley–Wiener spaces $\PW^q(\Omega)$, where $\Omega \subset \mathbb{R}^n$, $n \geq 2$, is an open convex set containing no affine lines. The inequalities take the form

$\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$

where the frequency weight is defined via Macbeath regions,

$\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$

The authors call these $(p,q,d)$ boundary-weighted Fourier inequalities. Two classical endpoints motivate the theory: the triple $(2,1,1)$ recovers Helson's inequality (a higher-dimensional analogue of Carleman's inequality, used to show that Hilbert–Schmidt Hankel operators have bounded symbols), while $(1,1,1)$ gives a Paley–Wiener analogue of Hardy's inequality for $H^1$. The paper delivers three main contributions: a complete characterization of valid triples $(p,q,d)$ for polyhedra; a partial characterization for balls, revealing a sharp dichotomy between flat and curved boundaries tied to Kakeya phenomena; and applications to Nehari-type theorems for truncated Hankel operators.

Admissibility of arbitrary convex sets

The paper opens with a foundational geometric result: every open convex set free of affine lines is $\Omega \subset \mathbb{R}^n$0-admissible for every $\Omega \subset \mathbb{R}^n$1. Admissibility requires a cover of $\Omega \subset \mathbb{R}^n$2 by parallelepipeds satisfying five axioms controlling their size relative to $\Omega \subset \mathbb{R}^n$3, bounded overlap, and bounded cardinality of additive-interaction index sets. This hypothesis was previously imposed by hand in the construction of Besov spaces of Paley–Wiener type; here it is shown to be automatic.

The proof combines John's ellipsoid theorem applied to Macbeath regions $\Omega \subset \mathbb{R}^n$4 with two geometric lemmas: a stability estimate showing that $\Omega \subset \mathbb{R}^n$5 varies by at most a factor $\Omega \subset \mathbb{R}^n$6 under dilation of Macbeath regions, and a nesting lemma stating that if $\Omega \subset \mathbb{R}^n$7 then $\Omega \subset \mathbb{R}^n$8. A maximal separated selection of centers yields the required cover. As corollaries, the authors establish the expected Besov embedding $\Omega \subset \mathbb{R}^n$9 when $n \geq 2$0, and density of $n \geq 2$1 in $n \geq 2$2 for $n \geq 2$3. These tools are used throughout the rest of the paper.

Necessary conditions

For general $n \geq 2$4, the paper establishes sharp necessary conditions on the triple $n \geq 2$5. The inequality can only hold when

$n \geq 2$6

with a strict defect $n \geq 2$7 required when $n \geq 2$8. If $n \geq 2$9 is unbounded, necessarily $\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$0 and $\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$1 exactly. The obstruction for $\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$2 comes from a randomized Khintchine-inequality argument on disjoint spectral bumps; the condition $\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$3 follows by testing against wave packets localized in admissibility parallelepipeds, where both sides scale as powers of $\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$4.

The case $\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$5 is more delicate. When $\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$6, a normalized packet argument forces a contradiction as the number of packets grows, since $\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$7. For sets of positive affine surface area, Schmuckenschläger's asymptotics for the measure of level sets of $\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$8 yield the quantitative refinement

$\int_{\Omega}\frac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}\,dx \leq C\|f\|_{L^q}^p, \qquad f\in\PW^q(\Omega),$9

showing that curvature degrades the admissible exponent even below the formal critical value. This quantitative gap plays no role for polytopes but is essential context for the ball.

Complete characterization for polyhedra

For polyhedra the necessary conditions are also sufficient, giving a complete answer. For $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$0 and $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$1, the $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$2 inequality holds for a bounded polyhedron if and only if $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$3, and for an unbounded polyhedron exactly when $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$4. For $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$5, it holds for bounded $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$6 precisely when $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$7, and fails for all $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$8 when $\omega_\Omega(x) = m(\Omega \cap (2x - \Omega)).$9 is unbounded.

The key structural input is a complex interpolation theorem:

$(p,q,d)$0

proved by lifting the classical product Hardy space interpolation through the Riesz projection. Since the polyhedral Fourier multiplier $(p,q,d)$1 factors into a composition of one-dimensional Hilbert transforms (with norm $(p,q,d)$2 for $(p,q,d)$3 facets), $(p,q,d)$4 is bounded on $(p,q,d)$5 for $(p,q,d)$6 — in stark contrast to Fefferman's theorem that the ball multiplier is unbounded unless $(p,q,d)$7. The endpoint $(p,q,d)$8 cannot be reached this way, so the authors triangulate $(p,q,d)$9 into pieces affinely equivalent to $(2,1,1)$0 via the Minkowski–Weyl theorem, use partition-of-unity multipliers to localize onto orthants where product Hardy interpolation applies, and reassemble. Combined with the $(2,1,1)$1 Hardy inequality for polyhedra (extending earlier results from polytopes to unbounded polyhedra by exhaustion), Hölder interpolation then produces the full critical range.

The authors explicitly note they do not know whether $(2,1,1)$2 sits on an interpolation scale for any domain other than a polyhedron — not even between $(2,1,1)$3 and $(2,1,1)$4 for the ball.

The ball: positive results for $(2,1,1)$5

For the ball $(2,1,1)$6, $(2,1,1)$7, one has $(2,1,1)$8. The authors construct, nearly explicitly, an $(2,1,1)$9 function whose distributional Fourier transform agrees with $(1,1,1)$0 on $(1,1,1)$1: writing $(1,1,1)$2 as a smooth function times $(1,1,1)$3 near the boundary, they pull back the tempered distributions $(1,1,1)$4 with $(1,1,1)$5 and verify membership in $(1,1,1)$6 via the half-integer Hankel function formulas. This proves Helson's inequality for the ball.

To reach the full range $(1,1,1)$7, $(1,1,1)$8, the authors build an analytic family of distributions $(1,1,1)$9 agreeing with $H^1$0 on $H^1$1, with exponential-in-$H^1$2 bounds on $H^1$3 and $H^1$4. Stein–Weiss interpolation of the associated multipliers yields the $H^1$5 inequality, and Hölder interpolation with the boundedness of the Fourier transform completes the characterization: for $H^1$6, the $H^1$7 inequality holds for the ball if and only if $H^1$8 and $H^1$9.

Notably, this interpolation argument does not require knowing whether the spaces $(p,q,d)$0 themselves interpolate — circumventing the obstruction caused by Fefferman's disc-conjecture counterexample.

Failure at criticality and the Kakeya connection

The situation changes qualitatively for $(p,q,d)$1. The central negative result is that for every $(p,q,d)$2, the critical diagonal inequality

$(p,q,d)$3

fails for the ball in every dimension $(p,q,d)$4. The mechanism is Besicovitch/Kakeya: testing against wave packets with Fourier support in $(p,q,d)$5-separated caps inside boundary annuli $(p,q,d)$6 (where $(p,q,d)$7), the assumed inequality would force tube unions $(p,q,d)$8 over $(p,q,d)$9-separated directions to satisfy $\Omega \subset \mathbb{R}^n$00 — contradicting the existence of measure-zero Besicovitch sets via thickening and rescaling.

More precisely, the paper shows that subcritical inequalities imply lower Minkowski dimension bounds for Kakeya sets: if the $\Omega \subset \mathbb{R}^n$01 inequality holds for some $\Omega \subset \mathbb{R}^n$02, then every union of tubes satisfies

$\Omega \subset \mathbb{R}^n$03

and consequently every compact Kakeya set has full lower Minkowski dimension $\Omega \subset \mathbb{R}^n$04. Thus the family of subcritical diagonal inequalities sits logically between the spherical restriction conjecture and the Kakeya conjecture in its Minkowski form.

In the converse direction, the spherical restriction conjecture implies the full subcritical range: assuming restriction holds, the $\Omega \subset \mathbb{R}^n$05 inequality for $\Omega \subset \mathbb{R}^n$06 holds for every $\Omega \subset \mathbb{R}^n$07 and $\Omega \subset \mathbb{R}^n$08, where $\Omega \subset \mathbb{R}^n$09. The proof dilates the restriction estimate radially, integrates in polar coordinates against $\Omega \subset \mathbb{R}^n$10, and interpolates with Plancherel. Since Zygmund proved the restriction conjecture for the circle, the planar case is unconditional: for the unit disc, the $\Omega \subset \mathbb{R}^n$11 inequality holds whenever $\Omega \subset \mathbb{R}^n$12 and $\Omega \subset \mathbb{R}^n$13. The authors leave open whether the hypothesis of the Kakeya implication is ever satisfied for $\Omega \subset \mathbb{R}^n$14.

Applications to Hankel operators and Nehari's theorem

The final section connects the $\Omega \subset \mathbb{R}^n$15 inequality to Nehari's problem for truncated Hankel operators $\Omega \subset \mathbb{R}^n$16. Two implications are established. First, if the $\Omega \subset \mathbb{R}^n$17 inequality holds for $\Omega \subset \mathbb{R}^n$18 with $\Omega \subset \mathbb{R}^n$19, then Nehari's theorem holds for $\Omega \subset \mathbb{R}^n$20; the proof embeds $\Omega \subset \mathbb{R}^n$21 into the dual Besov space $\Omega \subset \mathbb{R}^n$22 using the admissibility cover, then applies Hahn–Banach. Second, conversely, assuming the Schatten-to-Besov bound $\Omega \subset \mathbb{R}^n$23 (known for $\Omega \subset \mathbb{R}^n$24 always, for all $\Omega \subset \mathbb{R}^n$25 on simple polytopes, and for $\Omega \subset \mathbb{R}^n$26 on the ball), Nehari's theorem for $\Omega \subset \mathbb{R}^n$27 implies the $\Omega \subset \mathbb{R}^n$28 inequality.

Combining these with Helson's inequality for the ball answers an open question from prior work: every Hilbert–Schmidt Hankel operator on $\Omega \subset \mathbb{R}^n$29 is generated by a bounded symbol, i.e., Nehari's theorem holds for $\Omega \subset \mathbb{R}^n$30.

At the endpoint $\Omega \subset \mathbb{R}^n$31, the paper develops the analogy with the classical Hilbert matrix via the integral operator $\Omega \subset \mathbb{R}^n$32, whose quadratic form satisfies $\Omega \subset \mathbb{R}^n$33. Under the full Nehari theorem (equivalently, weak factorization of $\Omega \subset \mathbb{R}^n$34), the $\Omega \subset \mathbb{R}^n$35 Hardy inequality is equivalent to boundedness of $\Omega \subset \mathbb{R}^n$36. Since $\Omega \subset \mathbb{R}^n$37 is bounded (by the symbol construction) while the $\Omega \subset \mathbb{R}^n$38 inequality fails for the ball, the full Nehari theorem must fail for balls in $\Omega \subset \mathbb{R}^n$39, $\Omega \subset \mathbb{R}^n$40 — recovering, by a different route, a known negative result.

Limitations and open questions

Several limitations are stated plainly in the paper. The positive characterization for the ball covers only $\Omega \subset \mathbb{R}^n$41; for $\Omega \subset \mathbb{R}^n$42 the critical inequalities fail, and the validity of subcritical ones is established only conditionally (via the restriction conjecture) or in dimension two. Whether the subcritical $\Omega \subset \mathbb{R}^n$43 inequalities hold for $\Omega \subset \mathbb{R}^n$44 is unknown, though their truth would resolve the Minkowski form of the Kakeya conjecture. Complex interpolation of the scale $\Omega \subset \mathbb{R}^n$45 is proved only for polyhedra, and the authors state they do not know whether it holds for any other domain, including the ball. The reverse Schatten–Besov estimate $\Omega \subset \mathbb{R}^n$46 is known only in restricted ranges of $\Omega \subset \mathbb{R}^n$47 and classes of $\Omega \subset \mathbb{R}^n$48; the authors conjecture it holds for all $\Omega \subset \mathbb{R}^n$49 and all admissible $\Omega \subset \mathbb{R}^n$50. Finally, domains with zero affine surface area other than polyhedra (e.g., cylinders) fall outside the sharp quantitative analysis and require case-by-case treatment.

Conclusion

The paper provides a complete solution of the boundary-weighted Fourier inequality problem for polyhedra, a sharp positive theory for the ball in the range $\Omega \subset \mathbb{R}^n$51, and a precise identification of the curved-boundary obstruction with Kakeya geometry. The equivalence between Helson's inequality and the $\Omega \subset \mathbb{R}^n$52-Nehari theorem, combined with the explicit symbol construction for the ball, settles the Hilbert–Schmidt symbol question for $\Omega \subset \mathbb{R}^n$53, while the failure of the critical diagonal inequality for $\Omega \subset \mathbb{R}^n$54 delineates exactly where flat and curved frequency domains diverge. The remaining open cases — subcritical inequalities for the ball in dimensions $\Omega \subset \mathbb{R}^n$55, and interpolation of the $\Omega \subset \mathbb{R}^n$56 scale beyond polyhedra — are now formulated as concrete analytic questions with quantifiable consequences.