Maximal and variational triangular Hilbert transforms

Source and verification

This article explains family 082 of the October 6, 2026 OpenAI collection. The research conclusions are attributed to its manuscripts. The source review covers statements, introductions, and the proof mechanisms described below; it is not an independent verification of the complete proofs.

Overview

The triangular Hilbert transform couples two coordinate directions with the same singular integration parameter. Family 082 contains a maximal theorem, a stronger variation theorem, and a dyadic companion. Its matrix-energy argument is designed to control cancellation uniformly across arbitrarily many scales.

🏷️ Preliminaries: truncation and entanglement

For $F,G:{\mathbb{R}}^2\to \mathbb{C}$ and $0<\epsilon <R$, put

$$\begin{array}{c}{K}_{F,G}(x,y;t)\\ =F(x+t,y)G(x,y+t),\\ B_{\epsilon ,R}(F,G)(x,y)\\ ={\int }_{\epsilon <|t|<R}{K}_{F,G}(x,y;t)\frac{dt}{t}.\end{array}$$

The odd kernel $1/t$ requires cancellation near zero. The functions move in different coordinate directions, but with the same $t$. This coupling prevents a direct reduction to two independent one-dimensional Hilbert transforms.

The pointwise maximal truncation is $B_{\ast }(F,G)={\sup }_{0<\epsilon <R}|B_{\epsilon ,R}(F,G)|$. For $r>2$, define the annular variation by

$$\begin{array}{c}{A}_j=B_{t_{j-1},t_j}(F,G),\\ V_r(F,G)\\ =\underset{0<t_0<\cdots <t_J}{\sup }{\left(\sum _{j=1}^J|A_j{|}^r\right)}^{1/r}.\end{array}$$

The supremum can be restricted to rational endpoints. The partition and its length may depend on $(x,y)$. Variation measures cumulative oscillation across scales rather than just the largest truncation.

🏷️ The family claims

Family 082, as stated in the manuscripts

For complex $F,G\in L^3({\mathbb{R}}^2)$ and every $r>2$,

$$\Vert V_r(F,G){\Vert }_{3/2}\le C_r\Vert F{\Vert }_3\Vert G{\Vert }_3.$$

In particular $\Vert B_{\ast }(F,G){\Vert }_{3/2}\le C\Vert F{\Vert }_3\Vert G{\Vert }_3$, and the joint principal value as $\epsilon \downarrow 0$, $R\uparrow \infty$ exists almost everywhere and in $L^{3/2}$. The family also contains a uniform real-input dyadic triangular-form estimate. (OpenAI, 2026) (OpenAI, 2026a) (OpenAI, 2026c)

Taking $J=1$ shows directly why the variation bound contains the maximal estimate. The endpoint $r=2$ is not claimed. The dyadic theorem is a separate model result; it is not obtained merely by substituting dyadic notation in the continuous operator.

🏷️ The scalar triangular form

Pair the output against $H\in L^3({\mathbb{R}}^2)$. Hölder gives

$$\begin{array}{c}\left|\int H{B}_{\epsilon ,R}(F,G)\right|\\ \le \Vert H{\Vert }_3\Vert B_{\epsilon ,R}(F,G){\Vert }_{3/2}.\end{array}$$

The substitution $x=-a$, $y=-b$, $t=a+b+c$ has absolute Jacobian one. Set $G_0(a,b)=H(-a,-b)$, $G_1(b,c)=F(b+c,-b)$, and $G_2(c,a)=G(-a,c+a)$. Each two-variable coordinate change preserves the $L^3$ norm. The form becomes

$$\begin{array}{c}T(a,b,c)\\ =\frac{G_0(a,b)G_1(b,c)G_2(c,a)}{a+b+c},\\ \underset{\epsilon <|a+b+c|<R}{\iiint }T(a,b,c)d(a,b,c).\end{array}$$

Each input sees two of the three variables, forming a triangle of dependencies. This exact calculation explains both the terminology and the symmetric exponent relation $1/3+1/3+1/3=1$.

🏷️ Matrix energy and scale cancellation

On a finite mesh, each two-variable input becomes a matrix between the spaces associated with its variables. At a vertex, the source joins the two incident matrices into a row operator $R$ and assigns it the energy

$$\mathcal{E}(R)=\mathrm{tr}((R^{\ast }R{)}^{3/2}).$$

This is the cube of a Schatten norm. The exponent matches the cubic input norms. Matrix trace estimates bound the triangular interaction by changes of this energy under averaging.

In the dyadic companion, an admissible parent triple has four admissible children. The local form is controlled by a parent-to-child energy increment. The factor attached to a child is one quarter of the parent’s factor, so summing over the tree telescopes. A bound independent of the number of scales comes from a single total energy budget rather than paying the same estimate at every scale.

For the continuous maximal theorem, Gaussian averaging plays the role of scale evolution. Selecting different truncations at different output points switches individual entries of the dual matrix on or off. The energy may jump at those switching scales. The source therefore proves a separate bound on the sum of these jumps; monotonicity between switches would not suffice. The variation companion extends the control to many annular increments.

🏷️ Principal values by approximation

For smooth compactly supported inputs, local cancellation and tail decay yield the expected principal value. The maximal bound controls the error when approximating general $L^3$ inputs by such functions. A vanishing tail-oscillation argument then gives convergence with both endpoints varying. This explains why a pointwise supremum is useful: estimates for each fixed truncation alone would not control arbitrary point-dependent choices of endpoints.

🏷️ Key steps and proof outlines

Matrix curvature controls a cyclic product

The finite model associates matrices to the three edges of a triangle. At a vertex, the two incident edge matrices are combined into one row operator $R$, and its positive Gram matrix is $P=R^{\ast }R$. The energy is $\Phi (P)=\mathrm{tr}(P^{3/2})$. Its curvature measures how much energy is gained when averaging or splitting changes the incident data.

To differentiate the square root, set $A=P^{1/2}$. If $K$ is a perturbation of $P$, the derivative $X$ solves the Sylvester equation

$$AX+XA=K.$$

In an eigenbasis of $P$, this means $X_{ij}=K_{ij}/(\sqrt{p_i}+\sqrt{p_j})$ on the support. Differentiating the trace gives $d\Phi (P)[K]=(3/2)\mathrm{tr}(P^{1/2}K)$. The corresponding second derivative supplies weighted quadratic costs. The source proves a mixed trace inequality bounding the cyclic interaction by these costs. Rank-changing perturbations require a separate extension argument; one cannot divide by zero eigenvalues without first restricting to the support or regularizing. (OpenAI, 2026a)

Dyadic cancellation becomes an exact energy telescope

For an admissible parent triple, the four admissible children reflect the triangular relation among the dyadic coordinates. The child edge matrices include square-root probability weights, so their ordinary average is not the parent matrix. Instead, fix the child choice at one vertex and average the neighboring squared multipliers; their conditional expectations are the identity. This recovers the parent row Gram inside the fixed vertex compression. Convexity and the curvature estimate then compare the averaged child energy to the parent energy, while the trace inequality bounds the local signed interaction by this increase.

If $E_Q$ denotes the appropriately normalized energy at a parent, the schematic local inequality is

$$|{\Lambda }_Q|\lesssim \frac{1}{4}\sum _{Q^{\prime }\text{ child of }Q}{E}_{Q^{\prime }}-E_Q.$$

Multiplying by the scale weights and summing over a finite tree cancels every intermediate generation. Only boundary energies remain, controlled by the cubic input norms. The factor $1/4$ matches the four-child averaging and is therefore structural. A bound by the sum of child energies without the negative parent term would accumulate once per generation.

Continuous truncations create jumps in the energy

Gaussian smoothing replaces the discrete averaging operation in the continuous model. Differentiating the star energies along the heat scale gives dissipation terms whose signs and sizes are controlled by the matrix inequalities. Gaussian integration identities on the plane $a+b+c=d$ convert those terms into the triangular kernel interaction.

A maximal truncation is linearized by choosing endpoints depending on the output pair $(a,b)$. Consequently some entries of the dual edge matrix are switched on and off at selected scales. Between switches the heat identity applies. At a switch, the source interpolates the changed entries and bounds the resulting energy jump by maximal averages of the original inputs. The total jump bound is a separate lemma. Ignoring the jumps would leave the proof valid only for fixed, output-independent truncations. (OpenAI, 2026c)

Hard annuli and variation need two additional estimates

A smoothed annular kernel differs from the hard indicator kernel by endpoint errors. The variation companion estimates these errors through Gaussian bounds, frequency blocks, and cancellation of the first moment of an odd kernel. The first-moment subtraction improves frequency decay; the remaining low-frequency contribution is handled separately.

For $n$ disjoint selected annuli, let $S_n$ be the supremum of the sum of the magnitudes of those increments. The count estimate bounds its $L^{3/2}$ norm by $C\sqrt{n}\log (2+n)\Vert F{\Vert }_3\Vert G{\Vert }_3$. For any finite partition, order its increment magnitudes as $d_{(1)}\ge d_{(2)}\ge \cdots$. Then $d_{(2^j)}\le 2^{-j}{S}_{2^j}$. Grouping ranks into dyadic blocks gives

$$V_r{\lesssim }_r\sum _{j\ge 0}{2}^{j/r-j}{S}_{2^j}.$$

Minkowski’s inequality and the count estimate therefore reduce the norm bound to

$$\sum _{j\ge 0}(1+j)2^{j(1/r-1/2)}<\infty ,r>2.$$

The exponent is negative precisely above two. This is a norm-level deduction; the count theorem is not being used as a uniform pointwise bound. (OpenAI, 2026b)

Finally, finite menus of measurable endpoint choices are approximated by choices constant on rectangular meshes. The finite matrix estimates are uniform in the mesh, allowing passage to general measurable choices and then general $L^3$ inputs by density. The maximal consequence controls approximation errors, while variation controls tail oscillation. Together they yield the joint principal-value conclusion with both truncation endpoints moving.

See Also

• Family 083: Lipschitz directions claims a supremum of operator norms, a weaker placement of the supremum than the maximal theorem here.
• Family 086: trilinear Hilbert transform concerns three one-variable inputs, and is a different operator despite the similar name.

🏷️ Manuscripts and verification scope

• Annular variation of the triangular Hilbert transform at the symmetric point — the pinned manuscript; its directory supplies the source and citation metadata.
• An L³ bound for the dyadic triangular Hilbert form — the pinned manuscript; its directory supplies the source and citation metadata.
• The maximal triangular Hilbert transform at the symmetric point — the pinned manuscript; its directory supplies the source and citation metadata.

The repository provides a formalization-scope page. Its scope and comparator list describe the maximal estimate. They do not document the later full annular-variation or dyadic absolute-sum claims. These are statements about the repository’s documentation: no Lean build, axiom audit, or comparator-equivalence check was performed for this article.

OpenAI collection gives the reading sequence and source policy; coverage register records which families have articles and which remain to be written.

🐻  OpenAI 2026a. An L³ bound for the dyadic triangular Hilbert form.

🐻  OpenAI 2026b. Annular variation of the triangular Hilbert transform at the symmetric point.

🐻  OpenAI 2026c. The maximal triangular Hilbert transform at the symmetric point.