Annular variation and dyadic absolute bounds for the triangular Hilbert transform

1 · The essentials

These manuscripts study the triangular Hilbert transform, a mathematical operation that combines two input functions. They ask whether its output stays controlled as the smallest and largest included scales change. The main manuscript shows in the manuscript that not only the largest output, but also its accumulated changes across scales, can be bounded. This control of changes holds for every parameter $r > 2$ , under the stated size conditions on the inputs. A companion result gives control over both scale cutoffs and says that removing them together produces a limiting output, both away from an exceptional set of measure zero and in an overall size measure. Pairing that output with a third input yields the estimate at the symmetric point, where all three inputs satisfy the same size condition. The two-cutoff maximal result has a Lean formalization; the supplied source does not identify the stronger control of changes as formalized.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine listening to a recording while adjusting both the lowest and highest frequencies you allow through. One bound controls the largest response across all settings, while the stronger variation bound also controls how much the response changes along sequences of settings. This is only an image for changing scale cutoffs, not a description of the transform itself.

Why it matters

The source presents these estimates as resolving the scalar triangular Hilbert transform problem at the symmetric point. The variation estimate gives more information than a largest-output bound: it controls changes across scales even when the chosen partitions differ from point to point. The convergence conclusions also specify a limiting output when both truncation cutoffs are removed.

2 · Going further

The continuous problem concerns complex inputs in $L^{3} (R^{2})$ and an output in $L^{3/2} (R^{2})$ . Manuscript 1 shows in the manuscript an annular $r$ -variation estimate for every $r > 2$ . Its partitions range over all positive scales and may depend on the output point. Thus the stated estimate concerns more than one fixed, globally chosen sequence of scales.

The implication described in the abstracts runs from annular variation to a two-endpoint maximal bound and then to joint principal-value convergence. Manuscript 3 gives the maximal estimate with a supremum over both hard truncation endpoints, for arbitrary complex $L^{3}$ inputs. As the lower endpoint tends to zero and the upper endpoint tends to infinity, the truncated outputs converge jointly almost everywhere and in $L^{3/2}$ . Pairing with a third $L^{3}$ input gives the symmetric $L^{3} \times L^{3} \times L^{3}$ scalar estimate. The abstracts state these implications but do not describe the underlying proof mechanism.

Manuscript 2 treats the dyadic triangular Hilbert form with unrestricted real inputs. It shows in the manuscript that the sum of absolute local contributions over any finite set of scales is at most forty times the product of the three $L^{3}$ norms. Consequently, coefficients of modulus at most one may vary independently among admissible interval triples. This is a companion absolute-sum estimate in the dyadic setting; the source does not specify how it enters the continuous arguments. Manuscript 3 is the Lean-formalized companion, whereas no formalization is specified for the annular variation or dyadic absolute-sum claims.

Useful notions

Functions of two real variables
Complex-valued and real-valued inputs
$L^{p}$ spaces and norms
Scale cutoffs and truncations
Almost-everywhere and norm convergence

What was known before / what is new

Before

The source describes the symmetric scalar estimate as conjectured and refers to the symmetric scalar triangular Hilbert transform problem. It supplies no account of earlier partial estimates or historical development, so a more detailed prior-state comparison is not available.

New

Manuscript 1 shows in the manuscript annular $r$ -variation bounds for every $r > 2$ , allowing point-dependent partitions over all positive scales. The Lean-formalized companion establishes the two-endpoint maximal estimate and states joint almost-everywhere and norm convergence, together with the symmetric scalar estimate. The dyadic companion shows in the manuscript an absolute local-contribution bound with constant forty and independently varying coefficients of modulus at most one.

Limits and assumptions

The continuous estimates stated here are at the symmetric exponents: complex $L^{3} (R^{2}) \times L^{3} (R^{2})$ inputs and $L^{3/2} (R^{2})$ output. The annular variation claim requires $r > 2$ ; the source states no conclusion for $r \leq 2$ or other exponent choices. The dyadic absolute-sum estimate is stated for real inputs and finite sets of scales, with coefficients attached to admissible interval triples. The main maximal result has a Lean formalization in Manuscript 3, whose stated scope includes both hard truncation endpoints and the associated convergence and symmetric scalar conclusions; this should not be extended to the variation and dyadic claims. The source lists no remaining open problems.

Mini glossary

$L^{p}$ norm: A measure of a function's overall size based on integrating its absolute value raised to the power $p$ .
Hard truncation endpoints: The lower and upper scale cutoffs used to restrict the transform.
Annular $r$ -variation: A measure of accumulated changes across scale annuli, with the parameter $r$ governing how those changes are combined.
Two-endpoint maximal bound: An estimate controlling the supremum over choices of both truncation endpoints.
Joint principal-value convergence: Convergence as the lower cutoff tends to zero and the upper cutoff tends to infinity together.
Symmetric point: The exponent choice at which all three inputs in the scalar estimate belong to $L^{3}$ .

Keywords : Triangular Hilbert transform · Annular variation · Dyadic absolute bounds

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves maximal and annular r-variation bounds, for every r > 2, from complex $L^{3} (R^{2}) \times L^{3} (R^{2})$ to $L^{3/2} (R^{2})$ . The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^{3} \times L^{3} \times L^{3}$ point.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/082.md

Manuscripts in this family

★

Annular variation of the triangular Hilbert transform at the symmetric point

05/10/2026

To be verified

We prove the annular r-variation estimate for the triangular Hilbert transform from complex $L^{3} \times L^{3}$ to L^3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.

PDF GitHub folder

An L³ bound for the dyadic triangular Hilbert form

05/10/2026

To be verified

We prove a uniform $L^{3} \times L^{3} \times L^{3}$ estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.

PDF GitHub folder

The maximal triangular Hilbert transform at the symmetric point

24/09/2026

Verified by Lean

For arbitrary complex inputs in $L^{3} (R^{2})$ , we prove the pointwise maximal $L^{3} \times L^{3} \to L^{3/2}$ estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L^3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.

PDF GitHub folder Lean formalization