Sparse domination implies convex body domination
Any bilinear form with a sparse bound has a convex body sparse bound for its vector-valued extension, answering a question about the relation between the two notions.
- Lab
- Independent
- Model
- GPT-5.6 Sol Pro
- Field
- Mathematics
- Date
- 2026-08-25
- Human collaborators
- Aapo Laukkarinen, Emiel Lorist
- Problem posed
- 2017 · open 9 yrs
Sources
Original work
Independent commentary
What was found
If a bilinear form admits an (r,s)-sparse bound, then its coordinate-wise extension to functions valued in complex n-space admits an (r,s)-convex body sparse bound. The proof rests on a randomization argument, and the result is established both for sparse families in a fixed dyadic lattice and for sparse families of arbitrary cubes. As an application the authors deduce sparse domination for iterated commutators, with the local oscillations of the symbol appearing in the sparse form.
Novelty check
Convex body domination was introduced by Nazarov, Petermichl, Treil and Volberg, whom vibemathed records as posing the question in 2017, and sparse domination is a standard tool with its own literature, cited in the paper alongside the Bellman-function and sparse-revolution surveys. The implication between the two notions is what was open; the paper proves it in both the fixed-lattice and arbitrary-cube settings rather than one. No prior proof of the implication appears.
Caveats and known objections
A short preprint, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-assisted, the weakest reading consistent with the disclosure, because the paper's AI disclosure statement claims only prototyping and not discovery: GPT-5.6 Sol Pro, accessed through ChatGPT, was used to prototype proof strategies for this note. Nothing in the disclosure attributes the final argument to the model.
Also recorded at
vibemathedsparse-domination-implies-convex-body-domination
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-assisted.
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Cite this entry
whataifound.org. (2026). Sparse domination implies convex body domination. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-25-sparse-convex-body-dominationBibTeX
@misc{whataifound-independent-2026-domination,
title = {Sparse domination implies convex body domination},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-08-25-sparse-convex-body-domination}
}