Convex body domination for the commutator of vector valued operators with matrix multi-symbol

We provide convex body domination results for the generalized vector-valued commutator of those operators that admit specific forms of convex body domination themselves. We also prove some strong type estimates and other consequences of these results, and we study the BMO spaces that appear naturally in this context.

💡 Research Summary

This paper develops a comprehensive convex‑body domination framework for commutators of vector‑valued operators that involve an arbitrary number of matrix symbols. Convex‑body domination, introduced in earlier work as a vector‑valued analogue of sparse domination, controls an operator pointwise by a sum of symmetric, convex, compact sets associated with a sparse family of cubes. The authors formalize this property as condition (P₁): for any function f∈Lʳ supported on a measurable set Ω, there exist a constant C, a sparse family 𝒮, and kernels K_Q(x,·) with ‖K_Q(x,·)‖{Lʳ(Q)}≤1 such that
 T f(x)=C ∑{Q∈𝒮}χ_Q(x)⟨K_Q(x,·), f⟩_Q,
where ⟨·,·⟩_Q denotes the average over Q.

The main result (Theorem 1) shows that if a linear operator T satisfies (P₁), then its higher‑order commutator with a multi‑symbol vector B=(B₁,…,B_m) also satisfies a convex‑body domination estimate of the same type. The commutator is defined recursively as
 T_B f=