Convex Body Sparse Domination

• Convex body sparse domination is a framework that replaces classical scalar averages with convex bodies to retain directional information in vector-valued and matrix-weighted settings.
• It utilizes geometric tools such as John ellipsoids and Minkowski products to achieve basis-free sparse estimates, crucial for analysing rough singular integrals and Banach-space functions.
• The approach leads to sharp quantitative bounds and extends to complex operators including commutators and multi-scale Calderón–Zygmund operators by systematically integrating directional averages.

Convex body sparse domination is a vectorial refinement of sparse domination in which the scalar local averages that appear in classical sparse bounds are replaced by convex bodies encoding all directional averages of a vector-valued function. In place of a positive sparse form built from quantities such as $\langle |f|\rangle_{p,Q}$, one obtains sparse control by Minkowski products of convex body averages, and this geometric reformulation is particularly suited to matrix weights, Banach-space-valued functions, rough singular integrals, and commutators (Nazarov et al., 2017, Hytönen, 2023, Laukkarinen, 2024, Laukkarinen, 2023, Isralowitz et al., 10 Mar 2026).

1. Scalar sparse domination and the convex body replacement

In the scalar setting, sparse domination means that a linear operator $T$ satisfies an estimate of the form

$$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$

for a sparse family $\mathcal S$ of cubes. Sparsity is expressed by the existence of measurable sets $E_Q\subset Q$ with $|E_Q|\ge c|Q|$ that are pairwise disjoint. This scalar form is the backbone of many sharp weighted inequalities, but it discards directional information as soon as one passes to vector-valued data (Laukkarinen, 2024).

Convex body sparse domination replaces scalar averages by convex sets. In the complex-valued formulation used for rough singular integrals, if $\vec f=(f_1,\dots,f_n)$ is a $\mathbb C^n$-valued function on a cube $Q$, the associated $L^p$ convex body is

$T$0

Equivalently,

$T$1

For two such convex bodies one defines the Minkowski dot product

$T$2

with $T$3. In the complex case this set is compact, convex, symmetric, hence a closed disk, and one identifies it with its radius (Laukkarinen, 2024).

The same idea appears in a basis-free real Banach-space formulation. For a normed space $T$4 and $T$5,

$T$6

These convex bodies are convex, compact, and symmetric, and they retain the simultaneous directional information that scalar norms suppress. In the pointwise formulation introduced for vector-valued Calderón–Zygmund theory, the corresponding convex body sparse operator is

$T$7

where the sum is understood as a Minkowski sum of sets (Hytönen, 2023, Nazarov et al., 2017).

2. Geometric structure: John ellipsoids, basis-free control, and directional information

The central geometric device is the John ellipsoid of a symmetric convex body. If $T$8 is convex, compact, and symmetric, its John ellipsoid $T$9 is the maximal-volume ellipsoid contained in $$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$0 and satisfies

$$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$1

This relation is the key step that converts componentwise scalar estimates into basis-independent convex body estimates (Nazarov et al., 2017, Hytönen, 2023).

In the abstract framework, one chooses a linear map sending the John ellipsoid to the Euclidean unit ball and decomposes the vector-valued inputs along an orthonormal basis. The resulting estimate has the form

$$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$2

In the complex setting of sesquilinear forms, the corresponding lemma reads

$$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$3

This reduction is what makes the final sparse form basis-free and therefore compatible with matrix weights (Hytönen, 2023, Laukkarinen, 2024).

A recurrent point in the literature is that the finite-dimensional geometry lives on the component index side rather than on the Banach-space side. In particular, even for Bochner spaces $$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$4, the associated convex bodies remain subsets of $$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$5, so John ellipsoid methods and Euclidean convex geometry remain available. This is precisely why convex body domination extends naturally from finite-dimensional vector-valued functions to Banach-space-valued functions (Hytönen, 2023).

The geometric content is not ornamental. It expresses, in a single scalar quantity such as $$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$6 or $$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$7, the interaction of all directions that a matrix weight may detect. Scalar averages do not provide this information, and the papers repeatedly emphasize that they are no longer adequate in the matrix-weighted setting (Laukkarinen, 2024, Nazarov et al., 2017).

3. Abstract sparse domination mechanisms

At an abstract level, convex body sparse domination is obtained by lifting a local scalar decomposition to a vector-valued one and then iterating it through a stopping-time argument. In the Banach-space framework, if local bilinear forms on cubes satisfy a single-scale estimate of the type

$$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$8

then there exists a sparse family $$|\langle Tf,g\rangle| \lesssim \sum_{Q\in\mathcal S} |Q|\,\langle |f|\rangle_{p_1,Q}\,\langle |g|\rangle_{p_2,Q},$$9 such that

$\mathcal S$0

This furnishes a general passage from local norm control to global convex body sparse domination (Hytönen, 2023).

For multi-scale operators, Laukkarinen extends the scalar sparse domination of Beltran–Roos–Seeger to the convex body setting. If $\mathcal S$1 satisfies the structural conditions (T1)–(T6), then for $\mathcal S$2 and Banach spaces $\mathcal S$3, there is an $\mathcal S$4-sparse family of dyadic cubes such that

$\mathcal S$5

with

$\mathcal S$6

This places convex body sparse domination in a genuinely multi-scale environment rather than only in classical Calderón–Zygmund theory (Laukkarinen, 2023).

For rough singular integrals, the framework is formulated in terms of bounded sesquilinear forms

$\mathcal S$7

with a kernel decomposition

$\mathcal S$8

where $\mathcal S$9. Under the size condition $E_Q\subset Q$0, the truncation bound $E_Q\subset Q$1, and the local testing bounds $E_Q\subset Q$2–$E_Q\subset Q$3, one obtains the abstract domination

$E_Q\subset Q$4

The proof uses stopping collections $E_Q\subset Q$5, the localized spaces $E_Q\subset Q$6 and $E_Q\subset Q$7, and scale-separated representations of $E_Q\subset Q$8 adapted to these stopping collections (Laukkarinen, 2024).

4. Rough singular integrals, unbounded angular parts, and critical Bochner–Riesz means

A principal application is the rough homogeneous singular integral

$E_Q\subset Q$9

Its kernel admits the scale decomposition

$|E_Q|\ge c|Q|$0

with $|E_Q|\ge c|Q|$1 a fixed radial smooth function supported in an annulus. The case $|E_Q|\ge c|Q|$2 corresponds to an unbounded angular part, and the 2024 paper extends convex body sparse domination precisely into that regime (Laukkarinen, 2024).

The key local quantity is the Lorentz–Orlicz quasi-norm

$|E_Q|\ge c|Q|$3

together with its homogeneous variant $|E_Q|\ge c|Q|$4. The paper verifies that the kernel size condition satisfies

$|E_Q|\ge c|Q|$5

and that the truncations satisfy

$|E_Q|\ge c|Q|$6

This is the mechanism by which the roughness of the angular part is absorbed into the domination theorem (Laukkarinen, 2024).

The resulting convex body sparse domination for $|E_Q|\ge c|Q|$7 states that if $|E_Q|\ge c|Q|$8, $|E_Q|\ge c|Q|$9, and $\vec f=(f_1,\dots,f_n)$0, then there exists a sparse collection $\vec f=(f_1,\dots,f_n)$1 such that

$\vec f=(f_1,\dots,f_n)$2

where

$\vec f=(f_1,\dots,f_n)$3

By duality there is also a symmetric bound with the $\vec f=(f_1,\dots,f_n)$4 and $\vec f=(f_1,\dots,f_n)$5 roles reversed. These are vector-valued extensions of the scalar sparse domination of Conde-Alonso–Culiuc–Di Plinio–Ou, and they handle unbounded angular parts through the Lorentz–Orlicz norm (Laukkarinen, 2024).

The same paper also treats the Bochner–Riesz operator at the critical index

$\vec f=(f_1,\dots,f_n)$6

Using the Muller–Rivera-Ríos grand maximal truncation method and the work of Shrivastava–Shuin, it proves that for $\vec f=(f_1,\dots,f_n)$7 and $\vec f=(f_1,\dots,f_n)$8 there exists a sparse family $\vec f=(f_1,\dots,f_n)$9 such that

$\mathbb C^n$0

The formal type is the same as for $\mathbb C^n$1, but the norm dependence differs: there is no $\mathbb C^n$2-factor (Laukkarinen, 2024).

5. Matrix-weighted inequalities and quantitative consequences

The historical motivation of convex body domination is matrix-weighted harmonic analysis. In the foundational 2017 paper, convex body valued sparse operators were introduced and used to dominate Calderón–Zygmund operators with Dini modulus, Haar shifts, and paraproducts. Estimating these sparse operators yielded the one-weight bound

$\mathbb C^n$3

as well as two-weight $\mathbb C^n$4–$\mathbb C^n$5 estimates in the matrix setting (Nazarov et al., 2017).

The Banach-space framework later showed that the same mechanism persists for operator-valued kernels and $\mathbb C^n$6-valued functions. If $\mathbb C^n$7 is a Banach space and $\mathbb C^n$8 is a Dini–Calderón–Zygmund operator with operator-valued kernel, then for any matrix weight $\mathbb C^n$9,

$Q$0

In the UMD case this applies in particular to scalar Calderón–Zygmund operators with Hölder kernels, including the Hilbert transform, and gives boundedness on $Q$1 (Hytönen, 2023).

For rough operators, the weighted theory becomes more intricate. The 2024 rough singular integral paper introduces mixed matrix characteristics adapted to the range $Q$2, with

$Q$3

and a mixed class $Q$4. Using convex body sparse domination together with matrix-weighted extrapolation due to Kakaroumpas–Nguyen–Vardakis, it proves quantitative matrix-weighted bounds for $Q$5 when $Q$6, and corresponding bounds for commutators. For $Q$7, it recovers the previously known matrix-weighted rough bound of Muller–Rivera-Ríos, although not by simply passing to the limit $Q$8 (Laukkarinen, 2024).

The multi-scale theory gives analogous consequences in a broader class of operators. If a BRS operator satisfies convex body sparse domination and $Q$9, then Laukkarinen derives quantitative matrix-weighted norm estimates on $L^p$0 with explicit dependence on $L^p$1 and $L^p$2. The same framework also yields matrix-weighted bounds for commutators of these multi-scale operators (Laukkarinen, 2023).

Across these formulations, a consistent structural fact emerges: convex body sparse domination separates the operator-theoretic step from the weight-theoretic step. The sparse form retains enough directional information for matrix weights, while the quantitative norm estimates are recovered by combining the sparse form with matrix $L^p$3 or mixed $L^p$4–reverse Hölder technology (Nazarov et al., 2017, Laukkarinen, 2024).

6. Commutators, generalized commutators, and recent extensions

One of the most productive consequences of convex body sparse domination is its interaction with commutators. For a scalar symbol $L^p$5 and a rough singular integral $L^p$6, the commutator is

$L^p$7

Using Laukkarinen’s general principle that convex body domination of $L^p$8 implies sparse domination of the commutator, the 2024 rough paper proves that for $L^p$9, $T$00, and $T$01, there exists a sparse family $T$02 such that

$T$03

where

$T$04

Combining this with Lerner–Lorist–Ombrosi yields Bloom-type two-weight bounds for $T$05 with unbounded angular part, and the paper explicitly notes that these bounds are new even in the scalar case (Laukkarinen, 2024).

In the abstract Banach-space setting, Hytönen’s framework applies to generalized commutators of the form

$T$06

If $T$07 has $T$08 convex body domination and

$T$09

then $T$10 extends to a bounded operator on $T$11 for all $T$12. Classical commutators, iterated commutators, and certain nonstandard combinations such as

$T$13

fit into this scheme (Hytönen, 2023).

The multi-scale theory also carries commutators. For BRS operators, convex body domination leads to matrix-weighted commutator bounds when the symbol has BMO entries, again with explicit dependence on the weight characteristics and the BMO norms (Laukkarinen, 2023).

A further extension appears in the 2026 work on vector-valued commutators with matrix multi-symbols. There the base operator is assumed to satisfy an integral or pointwise convex body domination property, and the higher commutator with multi-symbol $T$14 is written explicitly as

$T$15

This identity is then combined with the domination hypothesis to obtain convex body domination formulas for the commutator itself and weighted strong-type estimates in matrix-weighted spaces. The resulting theory introduces matrix multi-symbol BMO spaces adapted to the commutator geometry and shows how convex body sparse domination can be lifted from an operator to its higher, noncommutative commutators (Isralowitz et al., 10 Mar 2026).

In the broader theory, convex body sparse domination is therefore not merely a reformulation of scalar sparse domination. It is a geometric framework in which sparse estimates, matrix weights, Banach-space-valued extensions, rough kernels, and commutator structures can be handled within a single directional formalism. The existing papers show that this formalism is already effective for Calderón–Zygmund operators, Haar shifts, paraproducts, BRS multi-scale operators, rough homogeneous singular integrals with unbounded angular part, critical Bochner–Riesz means, and higher commutators with matrix symbols (Nazarov et al., 2017, Hytönen, 2023, Laukkarinen, 2024, Laukkarinen, 2023, Isralowitz et al., 10 Mar 2026).