A uniform L3 estimate for the triangular Hilbert transform
For Schwartz functions $f, g : \mathbb{R}^2 \to \mathbb{C}$ and $0 < \varepsilon < R < \infty$ let $$H_{\varepsilon,R}(f,g)(x,y) = \int_{\varepsilon < |t| < R} f(x+t, y)\, g(x, y+t)\, \frac{dt}{t}.$$ Pairing $H_{\varepsilon,R}(f,g)$ with a third function $h$ and changing variables linearly gives a truncation of Thiele's triangular Hilbert form $$\Lambda_3(f,g,h) = \operatorname{p.v.} \iiint f(x,y)\, g(y,z)\, h(z,x)\, \frac{dx\, dy\, dz}{x+y+z}.$$ Kovač, Thiele and Zorin-Kranich proved bounds for a dyadic model of this form when one of the three functions has additional structure. Durcik, Kovač and Thiele bounded the form truncated to $1 < |x+y+z| < N$ by $C(\log N)^{1/2} \|f\|_{L^3} \|g\|_{L^3} \|h\|_{L^3}$, where the trivial bound grows like $\log N$.
Problem. Does there exist an absolute constant $C$ such that $$\|H_{\varepsilon,R}(f,g)\|_{L^{3/2}(\mathbb{R}^2)} \le C \|f\|_{L^3(\mathbb{R}^2)} \|g\|_{L^3(\mathbb{R}^2)}$$ for all Schwartz functions $f, g$ and all $0 < \varepsilon < R < \infty$? The constant must be independent of both truncation scales.
Source: Thiele, in Grafakos, Oliveira e Silva, Pramanik, Seeger and Stovall (2017), Section 9, Problem 13, for the trilinear form; stated for the truncated bilinear operator by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CA — Classical Analysis and ODEs
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, multilinear singular integrals, triangular Hilbert transform
References
@misc{grafakos2017,
author = {Grafakos, Loukas and Oliveira e Silva, Diogo and Pramanik, Malabika and Seeger, Andreas and Stovall, Betsy},
title = {Some problems in harmonic analysis},
year = {2017},
eprint = {1701.06637},
archivePrefix = {arXiv},
note = {Section 9, contributed by Christoph Thiele, Problem 13},
}
@article{kovac2015,
author = {Kova{\v{c}}, Vjekoslav and Thiele, Christoph and Zorin-Kranich, Pavel},
title = {Dyadic triangular {H}ilbert transform of two general functions and one not too general function},
journal = {Forum Math. Sigma},
volume = {3},
year = {2015},
pages = {e25},
doi = {10.1017/fms.2015.25},
eprint = {1506.00861},
archivePrefix = {arXiv},
}
@article{durcik2019,
author = {Durcik, Polona and Kova{\v{c}}, Vjekoslav and Thiele, Christoph},
title = {Power-type cancellation for the simplex {H}ilbert transform},
journal = {J. Anal. Math.},
volume = {139},
number = {1},
year = {2019},
pages = {67--82},
doi = {10.1007/s11854-019-0052-4},
eprint = {1608.00156},
archivePrefix = {arXiv},
}
@misc{aim-triangular-hilbert-transform-l3,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A uniform L3 estimate for the triangular Hilbert transform}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/447-triangular-hilbert-transform-l3.md},
note = {Entry 447 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
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 …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00729, "A uniform L3 estimate for the triangular Hilbert transform", clopen.solutions.