arxiv: 2604.26096 · v1 · submitted 2026-04-28 · 🧮 math.CA

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Uncertainty Principle for distributions with Fourier transform in L_{p,q}(mathbb{R}^d)

Nikita Dobronravov

Pith reviewed 2026-05-07 13:44 UTC · model grok-4.3

classification 🧮 math.CA

keywords uncertainty principleLorentz spaceFourier transformNetrusov-Hausdorff capacityHausdorff measurereal analysisendpoint case

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The pith

Nonzero functions exist in Lorentz space L_{p,q} with Fourier transform supported on zero (2d/p, β)-Netrusov-Hausdorff capacity precisely when β > q/(2(q-1)).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The classical uncertainty principle asserts that no nonzero function in L_p(R^d) can have its Fourier transform supported on a set of finite α-Hausdorff measure when α is less than 2d/p. This statement fails at the endpoint α = 2d/p. The paper replaces the L_p integrability with the weaker Lorentz space L_{p,q} and substitutes the critical Hausdorff measure with Netrusov-Hausdorff capacity. It establishes the sharp threshold: such nonzero functions exist if and only if the second capacity parameter β exceeds q over 2(q-1). This yields the precise form of the uncertainty principle in the endpoint case.

Core claim

The paper proves that there exists a non-zero function in the Lorentz space L_{p,q}(R^d) such that its Fourier transform is supported by a set of zero (2d/p, β)-Netrusov-Hausdorff capacity if and only if β > q/(2(q-1)). This supplies the sharp uncertainty principle at the endpoint dimension where the version with ordinary Hausdorff measure breaks down.

What carries the argument

The (2d/p, β)-Netrusov-Hausdorff capacity on R^d, which serves as the refined size measure that distinguishes existence of nonzero Fourier-supported functions in L_{p,q} exactly at the stated β threshold.

If this is right

The endpoint uncertainty principle requires replacement of Hausdorff measure by Netrusov-Hausdorff capacity to recover sharpness.
The critical value of β is determined by the second Lorentz index q and equals q/(2(q-1)).
For β below or equal to the threshold, every function in L_{p,q} with Fourier transform supported on such a zero-capacity set must be zero.
The result recovers the known failure of the L_p uncertainty principle at α = 2d/p as a special case when q approaches p.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar capacity thresholds may govern uncertainty principles in other intermediate spaces such as Orlicz or variable-exponent Lebesgue spaces.
Explicit constructions could use fractal sets tuned to have exactly the critical capacity zero while supporting nonzero Fourier transforms in L_{p,q}.
The statement suggests that relaxing integrability from L_p to L_{p,q} enlarges the class of sets that can serve as Fourier support without forcing the function to vanish.

Load-bearing premise

The classical uncertainty principle is assumed to hold for sets of finite α-Hausdorff measure whenever α is strictly less than 2d/p, allowing the endpoint behavior to be captured by switching to the capacity.

What would settle it

An explicit nonzero function in L_{p,q}(R^d) whose Fourier transform is supported on a set of zero (2d/p, β)-Netrusov-Hausdorff capacity for some β ≤ q/(2(q-1)), or a demonstration that no such function exists for β strictly larger than that value.

Figures

Figures reproduced from arXiv: 2604.26096 by Nikita Dobronravov.

**Figure 1.** Figure 1: Tree T2 for M0 = 3, M1 = 4, M2 = 6 13 view at source ↗

**Figure 2.** Figure 2: Graph of fr for r = 0.1 Let Fr(x) = Q d i=1 fr(xi) (here x = (x1, . . . , xd)). Then, EµM,r = λ[0,r] d ∗ λ[0,1−r] d = Fr(x)dx (5.12) and µˆM,r(x) D= 1 M X M j=1 e −2πi<βr,j ,x>λˆ 0(xr), (5.13) Where {βr,j}M j=1 is the sequence of independent vectors that are uniformly distributed on the cube [0, 1 − r] d . The notation D= means equality of distributions. Let νM,r = µM,r(ω) be the value of µM,r at some poin… view at source ↗

**Figure 3.** Figure 3: Construction Remark 5.3. The cube sequence {Q0, Q1, . . . } corresponds to a tree T . For this cube sequence the recurrence relations (5.9) turn into µk+1 = µk − b(Qk)λQk + b(Qk)νMk, rk l(Qk) ,Qk . (5.16) 5.2 Estimation of example The following lemma was proved in [9, Lemma 3.2]. Lemma 5.4. For all M ∈ N, r < 1 2 and p > 1 the inequality Z Rd E|µˆM,r(x) − EµˆM,r(x)| p dx ≲ M− p 2 r −d (5.17) is true. Fix p… view at source ↗

read the original abstract

A version of the Uncertainty Principle says: There does not exist a non zero function in $L_p(\mathbb{R}^d)$ if its Fourier transform is supported by a set of finite $\alpha$-Hausdorff measure with $\alpha<2d/p$. This UP does not hold at the endpoint $\alpha=2d/p$. We find the sharp form of the UP in the limit case. We prove that there exists a non-zero function in the Lorentz space $L_{p,q}(\mathbb{R}^d)$ such that its Fourier transform is supported by a set of zero $(\frac{2d}{p},\beta)$-Netrusov--Hausdorff capacity if and only if $\beta>\frac{q}{2(q-1)}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper sharpens the classical uncertainty principle at the endpoint by giving an if-and-only-if condition for Lorentz spaces using a tuned Netrusov-Hausdorff capacity.

read the letter

The main point is that the usual uncertainty principle stops working at the critical dimension alpha = 2d/p, and this paper replaces Hausdorff measure with Netrusov-Hausdorff capacity to recover a sharp statement in L_{p,q} spaces. It proves existence of a non-zero function whose Fourier transform sits on a zero-capacity set exactly when the capacity parameter beta exceeds q over 2(q-1). That threshold looks like the natural interpolation point coming from the Lorentz exponent q, and the if-and-only-if form is what makes the result precise rather than just an extension of the subcritical case.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a sharp endpoint version of the uncertainty principle. It states that there exists a non-zero function in the Lorentz space L_{p,q}(R^d) whose Fourier transform is supported on a set of zero (2d/p, β)-Netrusov-Hausdorff capacity if and only if β > q/(2(q-1)). This is presented as the precise threshold replacing the classical α-Hausdorff measure statement, which holds for α < 2d/p but fails at the endpoint α = 2d/p.

Significance. If the result holds, it supplies the sharp form of the uncertainty principle at the critical exponent by combining Lorentz-space interpolation with the Netrusov-Hausdorff capacity. The paper builds directly on the classical subcritical case and gives an explicit, falsifiable threshold for β, which is a natural refinement at the endpoint.

minor comments (2)

The title refers to 'distributions' while the abstract and theorem statement concern functions in L_{p,q}(R^d). Clarify whether the result extends to tempered distributions or is restricted to functions.
The abstract states the theorem without indicating the range of p and q (e.g., 1 < p < ∞, 1 ≤ q ≤ ∞). Adding the precise parameter regime in the statement would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment, including the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states and proves an if-and-only-if existence result for non-zero functions in Lorentz space L_{p,q} whose Fourier transforms are supported on sets of zero (2d/p, β)-Netrusov-Hausdorff capacity precisely when β exceeds q/(2(q-1)). This refines the classical uncertainty principle for L_p (which the paper invokes only as a known subcritical fact for α-Hausdorff measure when α < 2d/p) by replacing the endpoint measure with the appropriate capacity. No step reduces by definition to its own output, no parameter is fitted and then renamed as a prediction, and no load-bearing premise rests on a self-citation chain or an imported uniqueness theorem from the same author. The argument is presented as a direct proof using standard properties of Fourier transforms, Lorentz spaces, and capacities; the central claim therefore remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard background from Fourier analysis and function space theory without introducing new free parameters or invented entities. The central claim rests on prior definitions of Lorentz spaces and capacities.

axioms (2)

standard math Standard properties of the Fourier transform on R^d and its support conditions
The uncertainty principle framework assumes the Fourier transform maps L_p to distributions with support properties.
domain assumption Definitions and basic properties of Lorentz spaces L_{p,q} and Netrusov-Hausdorff capacity
These are invoked as the ambient spaces and the measure of set size in the endpoint case.

pith-pipeline@v0.9.0 · 5421 in / 1567 out tokens · 85829 ms · 2026-05-07T13:44:24.785643+00:00 · methodology

discussion (0)

Reference graph

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