Sparse domination implies convex body domination

Nazarov, Petermichl, Treil and Volberg conjectured that scalar sparse domination should imply convex body domination for the corresponding coordinate-wise vector-valued extension. More precisely, if a bilinear form $\Lambda$ admits an $(r,s)$-sparse bound, then its extension to $\mathbb C^n$-valued functions should admit an $(r,s)$-convex body sparse bound. Laukkarinen and Lorist prove this implication for $1\leq r,s<\infty$ with $1/r+1/s>1$, in particular for the classical $(1,1)$-sparse setting.

If a bilinear form admits an $(r,s)$-sparse bound, its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound, for $1\le r,s<\infty$ with $\tfrac1r+\tfrac1s>1$. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form. On scope, two things separate. What Nazarov, Petermichl, Treil and Volberg proposed was the principle for the classical sparse setting, and that is settled outright, the hypothesis holding comfortably there. The $(r,s)$ range is the authors' own generalization beyond what was asked, and the restriction bites only inside it, leaving $\tfrac1r+\tfrac1s\le1$ open. Their summary: "we provide a proof of the full general statement". It also gives sparse domination for iterated commutators directly from that of the underlying form. The dependence on $n$ is not claimed optimal.