Multilinear Fractional Integral Operators: A counter-example

Pablo Rocha

REVIEW 3 major objections 3 minor 4 references

Multilinear Fractional Integral Operators: A counter-example

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A counterexample shows that the bilinear fractional integral operator $I_\gamma$ is not bounded from $H^1(\mathbb{R})\times H^p(\mathbb{R})$ into $H^q(\mathbb{R})$ when $0<p\le 1/\gamma$ and $1/q=1+1/p-\gamma$.

desk verdict The counterexample is right and the note is publishable after fixes; the printed sign error in Lemma 1 and the abstract's range typo are load-bearing but repairable from the paper's own proof. read the letter →

arxiv 1908.00668 v2 pith:6MM2DZQI submitted 2019-08-02 math.CA

Pablo Rocha This is my paper · ORCID

classification math.CA MSC 42B2042B30

keywords multilinearfractionalintegralHardyspacescounterexamplebilinearoperatormomentconditionsdistributionalFouriertransformunboundedness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs a counterexample to show that the bilinear fractional integral operator $I_\gamma$, defined by $I_\gamma(f_1,f_2)(x)=\int\int f_1(s)f_2(t)(|x-s|+|x-t|)^{\gamma-2}\,ds\,dt$, is not bounded from $H^1(\mathbb{R})\times H^p(\mathbb{R})$ into $H^q(\mathbb{R})$ when $1<\gamma<2$, $0

What carries the argument

The carrying mechanism is the distributional Fourier transform, in the $x$ variable, of the kernel $K_{s,t}^{\alpha}(x)=(|x-s|+|x-t|)^{\alpha-1}$. Lemma 1 computes this transform; feeding it through a limiting argument with a Schwartz function and using the moment condition of one input yields Corollary 2's identity, which reduces the integral of $I_{\alpha+1}(a_1,a_2)$ to a constant times $|t-s|^\alpha$ weighted by $a_1(s)a_2(t)$. This identity is what turns the abstract question of Hardy-space membership into the concrete question of whether that weighted double integral can be nonzero. The explicit test functions are chosen precisely so that the double integral is nonzero.

What would settle it

Numerically evaluate the identity of Corollary 2 for the paper's explicit functions $a_1(s)=\chi_{(-1,0)}(s)-\chi_{(0,1)}(s)$ and $a_2(t)=a_1(t-2)$ at $\alpha=1/2$: the formula predicts a nonzero value, so a direct quadrature of $\int\int a_1(s)a_2(t)|t-s|^{1/2}\,ds\,dt$ should match the closed form; if it instead matches the sign-altered version of Lemma 1, the counterexample collapses.

Watch

Extended reading notes

Core claim

For $\gamma=\alpha+1$ with $0<\alpha<1$, the paper's central claim is enforced by an exact identity: whenever $a_1,a_2$ are bounded and compactly supported and at least one has integral zero, $\int_{\mathbb{R}} I_{\alpha+1}(a_1,a_2)(x)\,dx = \frac{\alpha-1}{\alpha}\int_{\mathbb{R}^2} a_1(s)a_2(t)|t-s|^\alpha\,ds\,dt$. With the explicit pair $a_1(s)=\chi_{(-1,0)}(s)-\chi_{(0,1)}(s)$ and $a_2(t)=a_1(t-2)$, the double integral equals $[4\cdot 3^{\alpha+2}-4^{\alpha+2}-6\cdot 2^{\alpha+2}+4]/((\alpha+1)(\alpha+2))$, which is nonzero for every $0<\alpha<1$. The integral of the operator output is therefore nonzero, and by the Hardy-space moment conditions no element of $L^1\cap H^q$ with $q\le1$ can have a nonzero integral. The proof then approximates the special $a_2$ by functions in $H^p$, preserving the nonzero obstruction, and concludes that the operator is unbounded from $H^1(\mathbb{R})\times H^p(\mathbb{R})$ into $H^q(\mathbb{R})$ for $0<p\le1/\gamma$.

Load-bearing premise

The load-bearing premise is Corollary 2's exact identity, which depends on the distributional Fourier transform of the kernel; if Lemma 1 is read with its printed sign error, the identity's coefficient changes and the nonzero-integral conclusion fails.

Editorial extensions

If this is right

For every $1<\gamma<2$, the operator $I_\gamma$ is not bounded from $H^1(\mathbb{R})\times H^p(\mathbb{R})$ into $H^q(\mathbb{R})$ when $0<p\le1/\gamma$ and $1/q=1+1/p-\gamma$.
The earlier positive estimates from products of Hardy spaces into $L^q$ cannot be upgraded to a Hardy-space target at these endpoint parameters, even in the one-dimensional bilinear case.
For $p=1/\gamma$, where $q=1$, the failure is a genuine $H^1$-to-$H^1$ failure: the integral of the output is nonzero, so the output is not in $H^1$.
The obstruction is stable: the special second factor $a_2$ can be replaced by a dense class of $H^p$ functions without making the integral of the output vanish, so the unboundedness is not an isolated example.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The abstract and theorem print the range as $p\le\gamma-1$, but the proof establishes $p\le1/\gamma$; reading the printed exponent as $-1$ is necessary for the stated counterexample, and the paper's own proof follows the $1/\gamma$ version.
Lemma 1's displayed Fourier transform has a sign in its second term opposite to the one obtained in the proof; if the displayed sign were correct, the coefficient in Corollary 2's identity would change and the nonzero-integral conclusion would fail, although the proof's computation appears to be the correct one.
The identity suggests a necessary condition for any Hardy-space bound: the weighted double integral $\int\int a_1(s)a_2(t)|t-s|^\alpha\,ds\,dt$ must vanish for all admissible compactly supported inputs; test functions with separated supports and opposite signs are natural obstructions.
A similar moment-obstruction argument should extend to $m$-linear versions of $I_\gamma$, since the kernel's Fourier transform would produce an analogous weighted integral over all $m$ variables.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the counterexample works, and the paper deserves a referee, but not in its current printed form. The two things everyone will trip on — the sign error in Lemma 1's statement and the abstract's range p ≤ γ−1 — are real, and both are typos rather than structural cracks. Lemma 1's proof computes the interval term with the opposite sign from the printed formula, and the proof's version is the correct one. A reader who follows the proof gets Corollary 2's identity with the right coefficient; a reader who trusts the printed lemma gets a coefficient that would kill the counterexample. The abstract should read p ≤ γ^{−1}, matching the proof.

What the paper does genuinely well: the natural question after Lin-Lu and Cruz-Uribe-Moen-van Nguyen is whether the L^q (weighted Hardy) bounds upgrade to H^q in the range p ≤ 1/γ, where q ≤ 1. They do not, and this note shows it by computing the distributional Fourier transform of the kernel in x, extracting the zero-frequency contribution, and getting ∫ I_{α+1}(a1,a2) dx = ((α−1)/α) ∫∫ a1(s)a2(t)|t−s|^α dsdt under a moment condition. The explicit a1, a2 and the stated nonzero value of the double integral check out (I spot-checked numerically; it is positive inside (0,1), vanishing only at the endpoints). The density argument from the endpoint p = 1/γ down to all p < 1/γ is standard and goes through. No fitting, no circularity, and the citations — Gelfand-Shilov for the |x|^{α−1} transform, Stein for Hardy-space facts, the two positive-result papers — are appropriate. The kernel identity is a genuinely reusable tool for endpoint questions of this type.

Soft spots in order of size. One: the printed Lemma 1 sign error is load-bearing as printed but repaired by the paper's own proof; a referee should insist the statement match the proof. Two: the abstract range. Three, which neither the reader nor the stress-test flagged: Corollary 2's proof has two canceling Fourier-convention slips. The displayed equality ∫ K φ_ε dx = ∫ K̂ φ̂_ε dξ is off by 1/2π (the pairing identity runs the other way), and the limit is evaluated as if ∫ φ̂ = φ(0) = 1, whereas ∫ φ̂ = 2π. The two errors cancel, so the stated identity is correct, but the proof as written is not self-consistent on constants.

Who this is for: specialists in multilinear fractional integrals and Hardy spaces. It settles a boundary case in a short note; it is not a breakthrough and opens no new line. With the fixes above I would publish it in a mainstream harmonic analysis journal and would cite it for the endpoint failure. Recommendation: send it to peer review, with acceptance conditional on the typo and constants fixes.

Referee Report

3 major / 3 minor

Summary. The paper constructs an explicit counterexample to show that the bilinear fractional integral operator I_γ, for 1 < γ < 2, is not bounded from H^1(R) × H^p(R) into H^q(R) on the range 0 < p ≤ γ^{-1}, 1/q = 1 + 1/p − γ. The proof computes the distributional Fourier transform of the kernel (|x−s|+|x−t|)^{α−1}, uses it to derive the identity ∫_R I_{α+1}(a_1,a_2)(x) dx = ((α−1)/α) ∫∫_{R^2} a_1(s)a_2(t)|t−s|^α dsdt for compactly supported bounded a_1,a_2 with a vanishing moment, and then chooses explicit functions a_1 ∈ H^1 and a_2 ∈ H^{(α+1)^{-1}} for which the double integral is nonzero. The case of smaller p is obtained by a density argument that moves the example from H^{(α+1)^{-1}} to H^p.

Significance. If the corrected version of Lemma 1 and the identity in Corollary 2 are accepted, the paper resolves a natural question: while L^q estimates for multilinear fractional integrals were proved by Lin–Lu and Cruz-Uribe–Moen–van Nguyen, the corresponding Hardy-space target H^q fails, and the explicit counterexample demonstrates the optimality of the stated range. The argument is elementary and fully explicit, with no fitted parameters, and the nonzero double integral is computed directly; the negative result follows from an external boundedness theorem without assuming the target unboundedness. The main weaknesses are several typographical errors and a missing justification of a limit interchange, all of which are local and fixable.

major comments (3)

[Lemma 1 (Section 2)] The displayed statement of Lemma 1 contains a sign error in the second term: for s < t it reads |t−s|^{α−1} sgn(t−s) ∫_t^s e^{-ixξ} dx, but the proof computes II = (t−s)^{α−1} ∫_s^t e^{-ixξ} dx. Since ∫_t^s = −∫_s^t, the printed formula has the opposite sign. The proof's version is correct, but as printed the statement is inconsistent with its proof and would change the coefficient in Corollary 2 to −(α+1)/α instead of (α−1)/α. This must be fixed before the paper can be verified.
[Corollary 2 proof (Section 2)] The third equality in the displayed chain in Corollary 2 passes the limit ε → 0+ inside the double integral over s and t, after a change of variables in the inner distributional pairing. This interchange is not justified. The authors should state the regularized identity explicitly and give a dominated-convergence argument, showing that the ε^{-α}|ξ|^{-α} singular term is controlled by the moment condition on a_1 (or a_2) and that the remaining integrands converge uniformly on the compact supports of a_1 and a_2.
[Abstract and Theorem 3] The abstract states 0 < p ≤ γ−1, but the theorem and the proof require 0 < p ≤ γ^{-1} (since γ = α+1, the two ranges are not equivalent: for 1<γ<2, γ^{-1}<1 while γ−1 is in (0,1) and is larger than γ^{-1}). The abstract should read γ^{-1} to match Theorem 3 and the argument in Section 3.

minor comments (3)

[Section 2, proof of Lemma 1 and Corollary 2] The notation \widehat{φ_ε}(ξ) is not defined at first use; under the Fourier convention \widehat{f}(ξ)=∫ f(x)e^{-ixξ}dx, one has \widehat{φ_ε}(ξ)=ε^{-1}\widehat{φ}(ξ/ε), and this change of variables should be displayed explicitly.
[Section 3, density argument] The sentence beginning "For 0 < p < (α + 1)−1" should state explicitly that the approximating function b is chosen from the dense class of bounded, compactly supported functions with vanishing moments up to N while approximating a_2 in H^{(α+1)^{-1}}; the inclusion of that class in H^p then gives b ∈ H^p.
[Section 3, computation of the double integral] The numerator 4·3^{α+2} − 4^{α+2} − 6·2^{α+2} + 4 is asserted to be nonzero without proof; since this is a nontrivial claim controlling the whole counterexample, a brief justification (for instance, evaluating at α=0 and α=1 or showing monotonicity) would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the unboundedness counterexample is derived from an explicit kernel computation, standard Hardy-space facts, and an external boundedness estimate; the only apparent issue is a sign typo in Lemma 1, not a circular step.

full rationale

The paper's central claim is an unboundedness counterexample for Iγ. The load-bearing ingredients are: (1) Lemma 1, an explicit distributional Fourier transform of the kernel K_{s,t}(x)=(|x-s|+|x-t|)^{α-1}, computed from standard Fourier transforms cited to Gelfand–Shilov; (2) Corollary 2, which derives ∫ I_{α+1}(a1,a2) dx = ((α-1)/α)∫∫ a1(s)a2(t)|t-s|^α dsdt by passing a limiting test function and using Lemma 1 together with the moment condition ∫a1=0 or ∫a2=0; (3) explicit functions a1=χ_{(-1,0)}-χ_{(0,1)} and a2=a1(·-2) for which the double integral is computed to be nonzero; and (4) a density argument using Stein's Hardy-space characterization and a boundedness estimate cited to Cruz-Uribe–Moen–van Nguyen, used only to control the approximation error. None of these steps assumes the target unboundedness, fits a parameter to the conclusion, or imports a uniqueness theorem from the author's own prior work. The proof does contain a sign typo in the printed statement of Lemma 1, since the proof's computation of II has the opposite sign, but a typo is a correctness issue, not circularity. The derivation chain is self-contained and benchmarked against external results, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted numbers or invented entities. The paper's contribution is an explicit construction using standard tools; the only load-bearing inputs are the cited theorems and the Fourier computation.

assumptions (4)

standard math Hardy space definition and moment conditions (S1), (S2) from Stein [4].
Used to conclude that a function in L^1 ∩ H^q with q≤1 must have zero integral, the key disqualifier.
standard math Fourier transform formula χ̂_{|x|^{α−1}}(ξ) = −2 Γ(α) sin((α−1)π/2)|ξ|^{−α} from Gelfand-Shilov [2].
Used in Lemma 1 to compute the kernel's Fourier transform; with the sign convention in the paper.
domain assumption The full-range L^q boundedness theorem of Cruz-Uribe, Moen and van Nguyen [1], quoted as Theorem 1.1.
Used to control the error term ∫|I(a1,a2−b)| in the approximation step; if this theorem were false, the density argument would need a different bound.
standard math Density of bounded compactly supported functions with sufficiently many vanishing moments in H^r for p≤r≤1 (Stein [4]).
Allows replacement of the endpoint Hardy function a2 by an H^p function b while keeping the nonzero integral.

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Cite this review

Pith. "Pith review of Multilinear Fractional Integral Operators: A counter-example." pith.science (2026). https://pith.science/paper/6MM2DZQI

@misc{pith2026190800668,
  author       = {Pith},
  title        = {Pith review of: Multilinear Fractional Integral Operators: A counter-example},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MM2DZQI}},
  note         = {Machine review of arXiv:1908.00668}
}

read the original abstract

By means of a counter-example we show that the multilinear fractional operator is not bounded from a product of Hardy spaces into a Hardy space.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Multilinear fractional Calder\'on-Zygmund operators on weighted Hardy spaces

D. Cruz-Uribe, K. Moen and H. van Nguyen, Multilinear fractional Calder´ on-Zygmund ope- rators on weighted Hardy spaces. Preprint: arXiv:1903.01593v1 [math.CA] 4 Mar (2019)

work page Pith review arXiv 2019
[2]

I.M.Gelfand, G. E. Shilov: Generalized Functions, Prop erties and Operations. Vol. 1, Aca- demic Press Inc., (1964)

work page 1964
[3]

Lin and S

Y. Lin and S. Lu, Boundedness of multilinear singular integral operators on Hardy and Herz- type spaces. Hokkaido Math. J., 36(3):585-613, (2007)

work page 2007
[4]

Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory In- tegrals

E.M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory In- tegrals. Princeton Univ. Press, Princeton N. J., (1993). Departamento de Matem ´atica, Universidad Nacional del Sur, A v. Alem 1253 - Bah ´ıa Blanca 8000, Buenos Aires, Argentina. E-mail address : pablo.rocha@uns.edu.ar

work page 1993

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Multilinear fractional Calder\'on-Zygmund operators on weighted Hardy spaces

D. Cruz-Uribe, K. Moen and H. van Nguyen, Multilinear fractional Calder´ on-Zygmund ope- rators on weighted Hardy spaces. Preprint: arXiv:1903.01593v1 [math.CA] 4 Mar (2019)

work page Pith review arXiv 2019

[2] [2]

I.M.Gelfand, G. E. Shilov: Generalized Functions, Prop erties and Operations. Vol. 1, Aca- demic Press Inc., (1964)

work page 1964

[3] [3]

Lin and S

Y. Lin and S. Lu, Boundedness of multilinear singular integral operators on Hardy and Herz- type spaces. Hokkaido Math. J., 36(3):585-613, (2007)

work page 2007

[4] [4]

Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory In- tegrals

E.M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory In- tegrals. Princeton Univ. Press, Princeton N. J., (1993). Departamento de Matem ´atica, Universidad Nacional del Sur, A v. Alem 1253 - Bah ´ıa Blanca 8000, Buenos Aires, Argentina. E-mail address : pablo.rocha@uns.edu.ar

work page 1993