Uniform decoupling for convex curves

Hrit Roy

REVIEW 4 major objections 6 minor 22 references

Uniform decoupling for convex curves

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every convex plane curve, an ℓ²L⁶ decoupling holds uniformly with R^ε loss.

desk verdict Genuine new result—uniform R^ε decoupling for all convex curves—but the proof leans on three unproved overlap assertions that a referee should require before acceptance. read the letter →

arxiv 2505.02981 v1 pith:UYWFWGN3 submitted 2025-05-05 math.CA

Hrit Roy This is my paper · ORCID

classification math.CA MSC 42B20

keywords decouplingconvexcurveshigh/lowargumentidealpartitionaffinedimensionℓ²L⁶estimateslow-regularityFourierrestriction

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 establishes a uniform $\ell^2 L^6$ decoupling theorem for arbitrary convex curves in the plane. For every $\epsilon > 0$ and every $R \geq 1$, the graph of any convex function with slope between 0 and 1 carries an 'ideal partition' into canonical boxes such that any function Fourier-supported in the $R^{-1}$-neighbourhood of the curve satisfies an $L^6$ estimate with constant $C_\epsilon R^\epsilon$, where $C_\epsilon$ depends only on $\epsilon$ and not on the curve. This extends the classical parabolic decoupling theorem to curves that may have no curvature, no self-similarity, and no smoothness beyond convexity. The proof uses the high/low argument over a coarse sequence of scales, with a multi-scale algorithm that builds the partition and a dyadic pigeonholing that controls box lengths at every scale. The result matters because it shows that the essential cancellation behind parabolic decoupling is a feature of convexity itself, not of special algebraic or fractal structure.

What carries the argument

The load-bearing object is the ideal partition $\mathcal{J}$ of the parameter interval, whose pieces satisfy $(b-a)(\gamma'_L(b)-\gamma'_R(a)) \leq 2R^{-1}$, whose number is at most $c_\epsilon R^\epsilon N(\Gamma,R^{-1})$, and whose lengths are at least $R^{-1}$; each piece corresponds to a canonical box of size roughly $|J| \times R^{-1}$. The proof's engine is the high/low argument: the square function $g_k = \sum_{\tau_k} |f_{k+1,\tau_k}|^2 * \omega_{\tau_k}$ is split into a low-frequency part $g_k^{\mathrm{lo}}$ (frequencies near the origin) and a high-frequency part $g_k^{\mathrm{hi}}$; the High Lemma gives an $\ell^4 L^4$ orthogonality bound for $g_k^{\mathrm{hi}}$ using direction separation and bounded overlap of tubes, and the Low Lemma shows $g_k^{\mathrm{lo}}$ is pointwise controlled by the next finer square function $g_{k+1}$ up to an $R^{2\delta}$ factor. Iterating over the $O(1/\epsilon)$ coarse scales and pigeonholing lengths at each scale converts the high/low split into the desired $\ell^2 L^6$ decoupling.

What would settle it

For the Cantor-staircase curve of Example 1.9 at $R=3^{2K}$, compute the maximum number of sumset rectangles $\tau_k+\tau'_k$ that contain a common point when $\tau'_k$ is an exceptional box. If this multiplicity grows faster than $R^{C\epsilon}$ for every fixed $C$, then the local bilinear square-function estimate cannot hold with the stated $R^\epsilon$ loss, and the proof's overlap premise would be refuted; if it remains $R^{O(\epsilon)}$, the premise is verified.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the parabola's $\ell^2 L^6$ decoupling bound is universal across all convex curves: for each $\epsilon$ there is a constant $C_\epsilon$ such that $\|f\|_{L^6(\mathbb{R}^2)} \leq C_\epsilon R^\epsilon (\sum_{J \in \mathcal{J}} \|f_J\|_{L^6(\mathbb{R}^2)}^2)^{1/2}$ for every convex curve $\Gamma$ satisfying the slope condition, where $\mathcal{J}$ is an ideal partition chosen for $\Gamma$, $\epsilon$, and $R$. The constant does not depend on $\Gamma$. The theorem treats curves whose curvature may fail to exist on sets of Hausdorff dimension arbitrarily close to 1, so the proof cannot use non-vanishing curvature or parabolic rescaling. Instead, it constructs the partition by a multi-scale algorithm adapted to the affine geometry of each curve, then runs the high/low iteration to reduce the $L^6$ norm to an $\ell^2$ sum over the finest boxes.

Load-bearing premise

The proof assumes that after pigeonholing by box lengths, the high-frequency tubes and the sums of adjacent boxes overlap only $R^{O(\epsilon)}$ times for every convex curve, including exceptional boxes; this geometric overlap bound is stated as evident or left to a minor modification of the cited work [21], and if it failed the $\ell^2 L^4$ orthogonality driving the high/low iteration would collapse.

Editorial extensions

If this is right

By interpolation, the same partition yields $\ell^2 L^q$ decoupling for every $2 \leq q \leq 6$ with $C_\epsilon R^\epsilon$ loss, so the uniform bound covers the full classical range.
For $\gamma(t)=t^2$ the theorem recovers the parabolic $\ell^2 L^6$ decoupling theorem; for piecewise convex polynomial curves it recovers the polynomial-curve decoupling theorem of the cited work [22], with a constant depending on the number of pieces.
The estimate applies to curves such as graphs of integrals of devil's staircases, whose curvature is undefined on a set of Hausdorff dimension arbitrarily close to 1, even though those curves lack smoothness and self-similarity.
Because the iteration uses only $O(1/\epsilon)$ scales independent of $R$, the proof does not require any self-similarity or homogeneity of the curve, so the mechanism is available for curves with non-uniform behaviour across scales.
The $\epsilon$-dependence of the partition is stated to be harmless for applications; if removed, the theorem becomes a one-parameter family of partitions independent of $\epsilon$.

Reading between the lines

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

If the same ideal-partition/high-low scheme is stable under affine transformations, the theorem should extend to convex hypersurfaces in higher dimensions, where the role of 'convex' would be played by surfaces with no curvature regularity; the paper does not pursue this.
The overlap bound for exceptional boxes (Lemma 2.21, Case 2) is the natural place to probe uniformity: a careful computation of sumset overlaps for the Cantor-staircase curve at $R=3^{2K}$ would either confirm the geometrical assertion or reveal a need to modify the partition.
Since the theorem's constant is independent of the curve, it suggests that compactness arguments over families of convex curves could yield quantitative information about affine dimension, or that affine dimension controls only the box count and not the decoupling exponent within the high/low framework.
The proof yields $R^{\epsilon}$ rather than polylogarithmic losses; a natural next step is to refine the coarse-scale pigeonholing near exceptional scales to recover a log-power bound for curves without self-similarity, as the high/low method does for the parabola.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Dear —\n\nThe headline: this is a real advance. Theorem 1.5 removes regularity assumptions entirely and gives a curve-independent R^ε constant for ℓ²L⁶ decoupling, which answers Thiele's question. No prior work covered arbitrary convex curves with a uniform constant, so the main theorem is new, not a repackaging. The adaptation of the Guth–Maldague–Wang high/low scheme is careful: the ideal partition construction, the multi-scale pigeonholing, and the rescaling lemmas are worked out in genuinely useful detail. Constants depend only on ε, and there is no circularity; the cited external inputs are independent.\n\nThe soft spots are real and load-bearing. Three geometric overlap estimates are asserted rather than proved. In the High Lemma, the bound # {τ_k^◦ ∋ ξ} ≤ R^{ε+2δ} is called 'simple trigonometry' from Lemma 2.21, but that lemma controls slope increments across four neighbouring boxes, not the number of descendant boxes whose direction differences pass through a point. In the Low Lemma, the O(R^{2δ}) overlap of 2R^{2δ}τ_{k+1}+B(0,2λ_{k+1}) is called 'geometrically evident'—which it is not for devil's-staircase-type curves with exceptional chains. In Lemma 4.2, the finite-overlap of sumsets is delegated to a 'minor modification' of [21, Lemma 2.4], but the modification has to work in the non-uniform, pigeonholed setting, and that is precisely where the cited argument does not obviously apply.\n\nThe stress-test worry about exceptional chains is legitimate: if any of these overlap counts is actually R^c with c>0 independent of ε, the ℓ²L⁴ orthogonality fails and Theorem 3.2—and hence Theorem 1.5—is unsupported. I am not saying the central claim is false; the architecture is sound and the result is very plausible. But the paper currently asks the reader to accept three quantitative geometric assertions on faith, and they sit exactly where the low-regularity difficulty lives. A referee report should require full derivations or precise localized citations for these three bounds before publication. The reverse Hölder step from (2.51) to Proposition 2.15 is also delegated to [12, §5]; that one is more standard and less concerning, but the needed modifications should at least be sketched.\n\nBottom line: the paper deserves a serious referee and likely acceptance after substantive revision. I would cite it and would bring it to reading group, mainly to test whether the overlap assertions can be proved. My recommendation: send it to review, with instructions to make the overlap bounds a required revision rather than a cosmetic request.\n\nBest,

Referee Report

4 major / 6 minor

Summary. The paper claims a universal ℓ²L⁶ decoupling estimate for all convex curves in the plane with no regularity assumptions: for every ε>0 and every R≥1, there is an ideal partition J of the parameter interval and a constant C_ε, depending only on ε, such that ‖f‖_{L⁶} ≤ C_ε R^ε (Σ_J ‖f_J‖²_{L⁶})^{1/2} for all Schwartz f whose Fourier transform is supported in the R⁻¹-neighbourhood of any convex curve satisfying (1.4). The proof adapts the high/low argument of Guth–Maldague–Wang [12] to the non-smooth setting: it constructs a multi-scale partition of admissible curves, pigeonholes in the box lengths at every scale, applies wave-packet pruning, proves High and Low Lemmas, and combines them through a weak-type/broad-narrow iteration. The final passage from the L⁶-branch estimate (2.51) to Proposition 2.15 is delegated to [12, §5]. Section 5 discusses illustrative fractal examples.

Significance. If valid, this is a substantial advance: it extends Bourgain–Demeter decoupling from the parabola and C² curves of positive curvature to arbitrary convex curves, with a constant uniform over the whole class. The multi-scale construction of the ideal partition from the affine dimension is genuinely new, and the fact that the constant does not depend on the curve is a strong and clearly falsifiable claim. The paper also gives credit where due: the main external inputs, [12] and [21], are independent, and the author's own prior work [20] is used only for illustrative examples in Section 5. The proof is not circular: the partition is constructed from the curve's geometry and no constant is fitted to the target estimate. The main risk is not the overall strategy but several unproved geometric overlap assertions inside the High Lemma, Low Lemma, and Lemma 4.2, which are load-bearing for the orthogonality steps.

major comments (4)

[§3.3, Low Lemma] The sentence 'Geometrically, it is evident that the sets 2R^{2δ}·τ_{k+1}+B(0,2λ_{k+1}) are only O(R^{2δ})-overlapping' is used as the sole input that converts local L²-orthogonality (Lemma A.3) into the factor R^{2δ} in the estimate |g^lo_k| ≤ C_lo R^{2δ} g_{k+1}. This assertion is not proved, and it is not a formal consequence of the stated direction-separation lemma. By (2.44), the lengths of sibling boxes at level k+1 can vary by a factor R^ε, and by Lemma 2.13 a fixed parent has up to O(R^{ε/2}) children; hence many τ_{k+1} centres can lie within a ball of radius 2λ_{k+1}. Lemma 2.21 only controls the slope increment across four neighbouring boxes, so the required overlap bound is exactly the kind of geometric fact that must be written down for the pigeonholed, non-uniform setting. Please supply a proof of the overlap bound, or state and prove a variant with the exact quantitative loss it produces.
[§3.3, High Lemma] The High Lemma's orthogonality step asserts: 'By an application of Lemma 2.21, and simple trigonometry, it follows (see Figure 1) that # {τ̄^∘_k ∋ ξ} ≤ R^{ε+2δ} for all ξ.' This is the step that converts the high-frequency part into the ℓ⁴L⁴ expression ‖g^hi_k‖²_{L²} ≤ C R^{2ε} Σ ‖f_{k+1,τ_k}‖⁴_{L⁴}. Lemma 2.21 as stated bounds the increment of γ'_R between the first and fourth of four neighbouring boxes; it does not, by itself, bound the number of boxes whose dual tubes contain a fixed point. The counting argument must be made explicit, including the exceptional-box case of Lemma 2.21, because this is the precise point where unusual convex curves (e.g. devil's-staircase curves) could produce many boxes with almost parallel directions. Please write out the argument so that the R^{ε+2δ} factor is verifiable.
[§4.1, Lemma 4.2] The proof of Lemma 4.2 delegates the key overlap estimate to a 'minor modification' of [21, Lemma 2.4] and concludes that the sumsets R^{2δ}·τ_k + R^{2δ}·τ'_k + B(0,λ_k) are O(R^{3ε+4δ})-overlapping. The cited lemma is proved for the single-scale Seeger–Ziesler decomposition, not for the multi-scale pigeonholed collections T^Λ_k used here. In the present setting, boxes at a fixed level arise from different level-1 ancestors, have lengths spread by powers of R^ε, and can be linked by exceptional chains (Lemma 2.18) that force a box to share its interval with an earlier typical ancestor. This is precisely the situation in which the directional spread of descendants of a fixed level-1 box is not controlled by Lemma 2.21 alone. The overlap bound is load-bearing for the local bilinear square-function estimate and hence for the proof of (4.1) for all k≥1. Please provide a self-contained proof in the present setting, or state the exact modification of [21, Lemma 2.4] and verify all its hypotheses.
[§2.4, deduction of Proposition 2.15 from (2.51)] The paper states that Proposition 2.15 follows from (2.51) by a reverse Hölder argument whose details, 'mutatis mutandis, can be found in [12, §5]'. This step is not a cosmetic repetition: here the canonical boxes have variable aspect ratios and the partition depends on ε and R, whereas [12, §5] is written for the uniform parabolic boxes. The reverse Hölder inequality converts an averaged L⁶ estimate into the desired ℓ²L⁶ decoupling, so any mistake in its constants or hypotheses would invalidate Theorem 1.5 even if the high/low iteration is correct. Please either include the reverse Hölder lemma with its proof in the present variable-box setting, or give a precise statement of the version in [12] and verify that all hypotheses are satisfied when the partition is the ideal partition constructed in §2.3.

minor comments (6)

[Throughout] The manuscript consistently writes 'Schwarz function' where 'Schwartz function' is standard; please correct this spelling globally.
[§2.4] The symbol N is first used for 1/ε (the number of scales) and is then redefined to denote an arbitrary integer ≤1/ε; the paper acknowledges this, but the notation remains confusing in the statements of Theorem 2.20 and Proposition 2.15. Please use two distinct symbols (e.g. N_0 and N).
[§3.3, High Lemma] The proof refers to Figure 1 as the justification for the tube-overlap count, but the figure is not described quantitatively in the text; please add labels defining τ̄^∘_k, λ_k, R^{2δ}, and the relevant angular window, so that the 'simple trigonometry' can be checked without guessing.
[§2.3, Verifying (J2)] The bound #J_p ≤ C₁ N(Γ_p,R_p^{-1}) log(2+R_p) is obtained by combining (2.41) or (2.15) with [21, Lemma 2.3 (iii)]; this is a reasonable citation, but the sentence says 'using (2.41) ... to the left typical boxes' without explaining the role of the right and exceptional boxes. Please spell out the short argument.
[Appendix A.1] The appendix proves the essential Fourier-support bound with an error O(R^{-100(k+1)}), while the main text uses the uniform error O(R^{-100}) in Lemmas 3.18 and 3.20. The difference is harmless, but the exponents should be reconciled for readability.
[§1.4, Example 1.8] The sentence about the Hausdorff dimension of the set of points of non-differentiability of f_μ is correct as cited, but the exact statement needed for the example (the dimension of the non-differentiability set of γ'_μ) is not stated; please make the implication explicit or add the relevant reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the ideal partition is constructed from the curve, constants depend only on ε, and the author's own prior work appears only in illustrative examples.

full rationale

The derivation chain is self-contained with respect to the targeted decoupling estimate. The central object, the ideal partition, is produced by explicit single- and multi-scale algorithms in §2.2 from the geometry of the convex curve, and Theorem 1.5 then asserts the estimate for that constructed partition. No parameter in the proof is fitted to the right-hand side of (1.6), and the constant C_ε is shown to depend only on ε rather than on any curve-specific data. The main external inputs, Guth–Maldague–Wang [12] for the high/low method and Seeger–Ziesler [21] for the single-scale and local biorthogonality/square-function estimates, are independent prior works; the citation of [21, Lemma 2.4] supplies a result proved in an external setting, not a self-referential uniqueness or ansatz claim. The only self-citation is [20], used in §1.4 and §5 for illustrative non-sharp examples, and it is not load-bearing for the proof of Theorem 1.5. The skeptic's concern about asserted overlap bounds in the High Lemma, the Low Lemma, and Lemma 4.2 is a legitimate correctness gap about unproved geometric assertions, but it is not circularity: those assertions do not define the predicted quantity in terms of itself, do not rename an input as an output, and do not import a load-bearing conclusion solely from the author's own prior work. No circular step can therefore be exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central theorem depends on no fitted constants. The external inputs are the high/low scheme (Guth-Maldague-Wang), Seeger-Ziesler square function tools, and the John ellipsoid; the author's own prior work [20] appears only in Section 5's illustrative examples. The main unproved input is the claimed extension of [21, Lemma 2.4] to the pigeonholed low-regularity setting.

assumptions (3)

standard math Convex functions on [0,1] have left and right derivatives with one-sided continuity properties used throughout.
Invoked in Lemmas 2.3, 2.10 and in the initial partition (2.2)-(2.6) to take limits in the single-scale algorithm.
standard math John Ellipsoid Theorem (John 1948)
Used in Definition 3.9 to define polar bodies J(τ_k)^* and the weights φ_D, ρ_τ; standard convex geometry.
domain assumption The biorthogonality and overlap bound [21, Lemma 2.4] extends to the low-regularity pigeonholed boxes of this paper.
Lemma 4.2 and the High Lemma depend on finite overlap of sumsets τ_k+τ'_k that is asserted via 'minor modification' of [21, Lemma 2.4], a load-bearing external input not fully reproved.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Uniform decoupling for convex curves." pith.science (2026). https://pith.science/paper/UYWFWGN3

@misc{pith2026250502981,
  author       = {Pith},
  title        = {Pith review of: Uniform decoupling for convex curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UYWFWGN3}},
  note         = {Machine review of arXiv:2505.02981}
}

abstract

Using a high/low argument, we prove a universal $\ell^2L^6$ decoupling estimate with constant $C_\epsilon R^{\epsilon}$ for general convex curves in the plane. These curves have no additional regularity assumptions, and the constant $C_\epsilon$ is uniform across all such curves.

Figures

Figures reproduced from arXiv: 2505.02981 by the authors.

**Figure 1.** The high/low decomposition of gk. The square function gk is essentially Fourier supported in a union of tubes that forms a bush centred at the origin. The low part g lo k is Fourier supported in Bp0, 2λk`1q. The high part g hi k is essentially Fourier supported outside this ball, where these tubes have small overlap, as shown in this diagram. As a result, the terms |fk`1,τk | 2 ˚ ωk are essentially orthogonal in thi… view at source ↗

**Figure 2.** Sumsets of transversal boxs τk P T kpτ1q and τ 1 k P T kpτ 1 1 q. are at most OpR3ϵ`4δ q-overlapping. By local L 2 -orthogonality (Lemma A.3), we get ż R2 | ÿ τkăτ1,τ1 kăτ 1 1 pfk`1,τk ˚ρτk q¨ pfk`1,τ1 k ˚ρτ 1 k q|2ϕQk À R 4ϵ ż R2 ÿ τkăτ1,τ1 kăτ 1 1 |pfk`1,τk ˚ρτk q¨ pfk`1,τ1 k ˚ρτ 1 k q|2 |ϕQk |, where we used that R4δ ď Rϵ since ϵ ă 1{2. Now ÿ τkăτ1,τ1 kăτ 1 1 |pfk`1,τk ˚ ρτk q ¨ pfk`1,τ1 k ˚ ρτ 1 k q|2 “ ` ÿ τkPT… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 20 canonical work pages

[12]

Improved decoupling for the parabola

Larry Guth, Dominique Maldague, and Hong Wang. Improved decoupling for the parabola. J. Eur. Math. Soc. (JEMS) , 26(3):875–917, 2024

work page 2024
[21]

Riesz means associated with convex domains in the plane

Andreas Seeger and Sarah Ziesler. Riesz means associated with convex domains in the plane. Math. Z. , 236(4):643–676, 2001

work page 2001
[20]

Improved Lp bounds for Bochner-Riesz operators associated with rough convex domains in the plane

Hrit Roy. Improved Lp bounds for Bochner-Riesz operators associated with rough convex domains in the plane. Math. Z., 310(1):Paper No. 14, 2025

work page 2025
[1]

Bourgain

J. Bourgain. Bounded orthogonal systems and the Λ ppq-set problem. Acta Math., 162(3-4):227–245, 1989

work page 1989
[2]

The proof of the l2 decoupling conjecture

Jean Bourgain and Ciprian Demeter. The proof of the l2 decoupling conjecture. Ann. of Math. (2) , 182(1):351–389, 2015

2015
[3]

Bounds on oscillatory integral operators based on multilinear estimates

Jean Bourgain and Larry Guth. Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct. Anal., 21(6):1239–1295, 2011

work page 2011
[4]

Convex hypersurfaces and Fourier transforms

Joaquim Bruna, Alexander Nagel, and Stephen Wainger. Convex hypersurfaces and Fourier transforms. Ann. of Math. (2), 127(2):333–365, 1988

work page 1988
[5]

Decoupling for fractal subsets of the parabola

Alan Chang, Jaume de Dios Pont, Rachel Greenfeld, Asgar Jamneshan, Zane Kun Li, and Jos´ e Madrid. Decoupling for fractal subsets of the parabola. Math. Z. , 301(2):1851–1879, 2022

work page 2022

Show all 22 references

[6]

Small cap decouplings

Ciprian Demeter, Larry Guth, and Hong Wang. Small cap decouplings. Geom. Funct. Anal. , 30(4):989–1062, 2020. With an appendix by D. R. Heath-Brown

2020
[7]

Falconer

Kenneth J. Falconer. One-sided multifractal analysis and points of non-differentiability of devil’s staircases. Math. Proc. Cambridge Philos. Soc. , 136(1):167–174, 2004

2004
[8]

Shengwen Gan, Shaoming Guo, Larry Guth, Terence L. J. Harris, Dominique Maldague, and Hong Wang. On restricted projections to planes in R3. Preprint: arXiv:2207.13844, 2024

2024 arXiv
[9]

Square function estimates for conical regions

Shengwen Gan and Shukun Wu. Square function estimates for conical regions. Preprint: arXiv:2203.12155, 2022

2022 arXiv
[10]

Improved discrete restriction for the parabola

Shaoming Guo, Zane Kun Li, and Po-Lam Yung. Improved discrete restriction for the parabola. Math. Res. Lett. , 30(5):1375–1409, 2023

2023
[11]

A short proof of ℓ2 decoupling for the moment curve

Shaoming Guo, Zane Kun Li, Po-Lam Yung, and Pavel Zorin-Kranich. A short proof of ℓ2 decoupling for the moment curve. Amer. J. Math. , 143(6):1983–1998, 2021

1983
[13]

Incidence estimates for well spaced tubes

Larry Guth, Noam Solomon, and Hong Wang. Incidence estimates for well spaced tubes. Geom. Funct. Anal., 29(6):1844– 1863, 2019

2019
[14]

A sharp square function estimate for the cone in R3

Larry Guth, Hong Wang, and Ruixiang Zhang. A sharp square function estimate for the cone in R3. Ann. of Math. (2) , 192(2):551–581, 2020

2020
[15]

Restriction estimates for space curves with respect to general measures

Seheon Ham and Sanghyuk Lee. Restriction estimates for space curves with respect to general measures. Adv. Math. , 254:251–279, 2014

2014
[16]

Extremum problems with inequalities as subsidiary conditions

Fritz John. Extremum problems with inequalities as subsidiary conditions. In Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948 , pages 187–204. Interscience Publishers, New York, 1948

1948
[17]

Maximal estimates for averages over space curves

Hyerim Ko, Sanghyuk Lee, and Sewook Oh. Maximal estimates for averages over space curves. Preprint: arXiv:2105.01628, 2021

2021 arXiv
[18]

Effective l2 decoupling for the parabola

Zane Kun Li. Effective l2 decoupling for the parabola. Mathematika, 66(3):681–712, 2020. With an appendix by Jean Bourgain and Li

2020
[19]

An l2 decoupling interpretation of efficient congruencing: the parabola

Zane Kun Li. An l2 decoupling interpretation of efficient congruencing: the parabola. Rev. Mat. Iberoam. , 37(5):1761– 1802, 2021

2021
[22]

Uniform l2-decoupling in R2 for polynomials

Tongou Yang. Uniform l2-decoupling in R2 for polynomials. J. Geom. Anal. , 31(11):10846–10867, 2021

2021

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [12]

Improved decoupling for the parabola

Larry Guth, Dominique Maldague, and Hong Wang. Improved decoupling for the parabola. J. Eur. Math. Soc. (JEMS) , 26(3):875–917, 2024

work page 2024

[2] [21]

Riesz means associated with convex domains in the plane

Andreas Seeger and Sarah Ziesler. Riesz means associated with convex domains in the plane. Math. Z. , 236(4):643–676, 2001

work page 2001

[3] [20]

Improved Lp bounds for Bochner-Riesz operators associated with rough convex domains in the plane

Hrit Roy. Improved Lp bounds for Bochner-Riesz operators associated with rough convex domains in the plane. Math. Z., 310(1):Paper No. 14, 2025

work page 2025

[4] [1]

Bourgain

J. Bourgain. Bounded orthogonal systems and the Λ ppq-set problem. Acta Math., 162(3-4):227–245, 1989

work page 1989

[5] [2]

The proof of the l2 decoupling conjecture

Jean Bourgain and Ciprian Demeter. The proof of the l2 decoupling conjecture. Ann. of Math. (2) , 182(1):351–389, 2015

2015

[6] [3]

Bounds on oscillatory integral operators based on multilinear estimates

Jean Bourgain and Larry Guth. Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct. Anal., 21(6):1239–1295, 2011

work page 2011

[7] [4]

Convex hypersurfaces and Fourier transforms

Joaquim Bruna, Alexander Nagel, and Stephen Wainger. Convex hypersurfaces and Fourier transforms. Ann. of Math. (2), 127(2):333–365, 1988

work page 1988

[8] [5]

Decoupling for fractal subsets of the parabola

Alan Chang, Jaume de Dios Pont, Rachel Greenfeld, Asgar Jamneshan, Zane Kun Li, and Jos´ e Madrid. Decoupling for fractal subsets of the parabola. Math. Z. , 301(2):1851–1879, 2022

work page 2022