Transfer from scalar to convex-body sparse bounds

Does a scalar (r,s)-sparse bound for a bilinear form imply a convex-body sparse bound for its coordinate-wise Cⁿ extension? The stated range is 1≤r,s<∞ and 1/r+1/s>1.

Why it matters and what counts as progress

Why it matters: The implication would transfer scalar control to a vector-valued geometric bound without a stronger generic estimate.

What counts as progress: A bound under the scalar hypothesis alone, with dimension and sparsity dependence quantified and the dyadic-to-general extension justified.

How the problem developed

2017 · Conjecture [1]
Fedor Nazarov, Stefanie Petermichl, Sergei Treil and Alexander Volberg Introduced convex-body sparse domination and proposed that scalar sparse domination should transfer to the coordinate-wise vector-valued extension.
25 Aug 2026 · Reported result [2]
Aapo Laukkarinen and Emiel Lorist Proved the transfer for 1≤r,s<∞ with 1/r+1/s>1, including the classical (1,1) case; the authors' theorem does not cover the complementary exponent range.

AI contribution

Sparse domination implies convex body domination

Aapo Laukkarinen, Emiel Lorist

The AI-associated dyadic randomization step underlies Theorem A: for 1≤r,s<∞ with 1/r+1/s>1, a bilinear form on compactly supported bounded scalar functions with an (r,s)-sparse bound has a convex-body sparse bound for its coordinate-wise C^n extension, with stated n^(1+1/r+1/s) and η^−1 dependence. Theorem 2.6 proves a dyadic version; the non-dyadic extension and final theorem were completed by the human authors.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull