Matrix-Weighted Campanato Spaces

• Matrix-weighted Campanato spaces are vector-valued function spaces where local oscillation is measured relative to matrix Aₚ weights and reducing operators.
• They establish duality with matrix-weighted Hardy spaces, enabling atomic decompositions and a polynomial cancellation framework.
• The theory supports operator analysis via Calderón–Zygmund operators and connects to matrix-weighted BMO and Triebel–Lizorkin spaces.

Matrix-weighted Campanato spaces are vector-valued function spaces in which local oscillation is measured relative to a matrix $A_p$ weight and its reducing operators. For $p\in(0,\infty)$, $q\in[1,\infty)$, $s\in\mathbb Z_+$, and an $A_p$-matrix weight $W$, the space $\mathcal L_{p,q,s,W}$ consists of locally $L^q$ functions whose deviation from vector-valued polynomials of degree at most $s$ is uniformly controlled on cubes after normalization by a reducing operator $A_Q$ (Chen et al., 21 Aug 2025). In the scalar case this reproduces the classical weighted Campanato spaces. In the matrix setting, the role of reducing operators is central: they replace scalar averages, encode the local geometry and anisotropy of the weight, and provide the normalization needed for duality, atomic decompositions, and operator boundedness.

1. Definition, parameters, and normalization

Let $p\in(0,\infty)$0 denote the dimension of the underlying space $p\in(0,\infty)$1, let $p\in(0,\infty)$2 denote the dimension of the target space $p\in(0,\infty)$3, and let $p\in(0,\infty)$4 be an $p\in(0,\infty)$5 matrix $p\in(0,\infty)$6 weight, meaning a measurable function with values in non-negative definite, invertible, locally integrable matrices satisfying a matrix Muckenhoupt $p\in(0,\infty)$7 condition (Chen et al., 21 Aug 2025). For each cube $p\in(0,\infty)$8, one fixes a reducing operator $p\in(0,\infty)$9, a positive definite matrix associated with $q\in[1,\infty)$0.

The defining seminorm is

$q\in[1,\infty)$1

where $q\in[1,\infty)$2 is the Hölder conjugate of $q\in[1,\infty)$3, and $q\in[1,\infty)$4 is the space of vector-valued polynomials of degree at most $q\in[1,\infty)$5 (Chen et al., 21 Aug 2025). The infimum over $q\in[1,\infty)$6 implements Campanato’s polynomial subtraction mechanism; the matrix normalization $q\in[1,\infty)$7 adapts this oscillation to the local behavior of the weight.

Two structural facts are immediate in the theory. First, the norm does not depend on the particular choice of reducing operators $q\in[1,\infty)$8, up to equivalence. Second, if $q\in[1,\infty)$9, then $s\in\mathbb Z_+$0 (Chen et al., 21 Aug 2025). Both facts are standard signatures of Campanato-type spaces: the seminorm measures oscillation modulo low-degree polynomials rather than absolute size.

Symbol	Meaning	Role
$s\in\mathbb Z_+$1	Campanato/Hardy index	ranges $s\in\mathbb Z_+$2 for duality
$s\in\mathbb Z_+$3	local $s\in\mathbb Z_+$4 parameter	oscillation integrability
$s\in\mathbb Z_+$5	polynomial degree	cancellation order
$s\in\mathbb Z_+$6	matrix $s\in\mathbb Z_+$7 weight	determines geometry
$s\in\mathbb Z_+$8	reducing operator	cube-wise normalization

2. Reducing operators and matrix geometry

A family $s\in\mathbb Z_+$9 is a set of reducing operators of order $A_p$0 for $A_p$1 if

$A_p$2

uniformly in $A_p$3, for all $A_p$4 (Chen et al., 21 Aug 2025). This relation is the matrix analogue of scalar normalization by weighted averages. It allows the Campanato seminorm to measure oscillation in the local geometry induced by $A_p$5, rather than in a fixed Euclidean scale.

The same device appears across matrix-weighted analysis. In the Littlewood–Paley framework of Frazier and Roudenko, reducing operators satisfy

$A_p$6

and can be used interchangeably with $A_p$7 in the definitions of matrix-weighted Triebel–Lizorkin and sequence spaces (Frazier et al., 2019). In particular, for $A_p$8, one has $A_p$9, and for $W$0, $W$1 coincides with the matrix-weighted Sobolev space $W$2 (Frazier et al., 2019). These equivalences show that reducing operators are not ad hoc artifacts of Campanato theory; they are a recurrent normalization mechanism throughout the matrix-weighted function-space scale.

The operator-theoretic side exhibits the same geometry. In the matrix $W$3 theory of Bickel and Wick, the matrix Carleson Embedding Theorem is only known for matrix $W$4 weights, and the embedding constant depends on the weight through

$W$5

rather than only on a testing constant (Bickel et al., 2014). Their testing conditions, involving the action of an operator on bump functions $W$6, are described as reminiscent of Campanato/BMO-type requirements. This is one of the clearest points at which the geometry of matrix weights directly modifies classical scalar theory.

3. Duality with matrix-weighted Hardy spaces

For $W$7, matrix-weighted Campanato spaces arise as duals of matrix-weighted Hardy spaces. The Hardy space $W$8 is defined via the matrix-weighted grand non-tangential maximal function $W$9, and a $\mathcal L_{p,q,s,W}$0-atom is a vector-valued function $\mathcal L_{p,q,s,W}$1 supported on a cube $\mathcal L_{p,q,s,W}$2, satisfying moment cancellation

$\mathcal L_{p,q,s,W}$3

together with the size condition

$\mathcal L_{p,q,s,W}$4

(Chen et al., 21 Aug 2025).

The main duality theorem states that if $\mathcal L_{p,q,s,W}$5, $\mathcal L_{p,q,s,W}$6, $\mathcal L_{p,q,s,W}$7, and $\mathcal L_{p,q,s,W}$8, then

$\mathcal L_{p,q,s,W}$9

More precisely, each $L^q$0 defines a bounded linear functional

$L^q$1

on finite atomic combinations, and every bounded linear functional on $L^q$2 arises in this way for a unique $L^q$3 (Chen et al., 21 Aug 2025).

This duality stabilizes the parameter $L^q$4. If $L^q$5, then

$L^q$6

up to norm equivalence for all permissible $L^q$7 (Chen et al., 21 Aug 2025). The result places $L^q$8 in the same role that classical Campanato spaces occupy in the unweighted and scalar-weighted Hardy-space theory, but with polynomial cancellation and local scaling governed by matrix averages rather than scalar ones.

4. Equivalent norms, atomic testing, and John–Nirenberg-type structure

The space $L^q$9 admits several equivalent characterizations. One is projection-based: if $s$0 denotes the orthogonal projection onto polynomials of degree at most $s$1 over $s$2, then

$s$3

(Chen et al., 21 Aug 2025). This removes the explicit infimum over all polynomials and replaces it with a canonical local projection.

Another is a John–Nirenberg-type formulation. For any $s$4, membership in $s$5 is equivalent to finiteness of a norm built from

$s$6

inside the cube-wise $s$7 oscillation functional, with equivalent norms (Chen et al., 21 Aug 2025). This asserts that the space is robust under a nontrivial change in the local integrability exponent, provided the normalization is adjusted by the weight and the reducing operator.

The atomic viewpoint gives a dual reformulation. The Campanato space norm is equivalent to testing against atoms, and the dual norm of a functional on $s$8 is equivalent to its action on atoms (Chen et al., 21 Aug 2025). These equivalent descriptions are essential in applications: they connect local polynomial approximation, weighted oscillation, and Hardy-space duality in a single framework.

5. Calderón–Zygmund operators on matrix-weighted Campanato spaces

The operator theory for $s$9 is formulated in terms of modified Calderón–Zygmund operators. The modification is needed because Campanato and Hardy spaces, especially for small $A_Q$0, are not invariant under translation or polynomials. Accordingly, if $A_Q$1 is an $A_Q$2-standard kernel, the modified operator $A_Q$3 is defined by a local part plus a principal value integral with subtracted Taylor expansion, so that

$A_Q$4

(Chen et al., 21 Aug 2025).

The central boundedness criterion is exact. If $A_Q$5, $A_Q$6, $A_Q$7, $A_Q$8, and $A_Q$9 is an $p\in(0,\infty)$00-standard kernel, then

$p\in(0,\infty)$01

(Chen et al., 21 Aug 2025). The operator must vanish on polynomials of degree at most $p\in(0,\infty)$02; no weaker cancellation condition is stated.

Via duality, this yields the corresponding Hardy-space theorem. For $p\in(0,\infty)$03, one has

$p\in(0,\infty)$04

under the same structural conditions on $p\in(0,\infty)$05, $p\in(0,\infty)$06, and the kernel (Chen et al., 21 Aug 2025). The Campanato and Hardy theories therefore form a matched pair: boundedness on one side is characterized by polynomial cancellation of $p\in(0,\infty)$07, and boundedness on the dual side by polynomial cancellation of $p\in(0,\infty)$08.

6. Relation to matrix-weighted BMO and little BMO

Matrix-weighted Campanato spaces are closely tied to matrix-weighted BMO. In the one-parameter setting, the space $p\in(0,\infty)$09 is defined by oscillation conditions of the form

$p\in(0,\infty)$10

with reducing operators $p\in(0,\infty)$11 adapted to the weight (Isralowitz et al., 2015). That paper states that its BMO-type condition is essentially a matrix-weighted Campanato condition, and that the explicit formulas can be considered as matrix-weighted Campanato seminorms for the case $p\in(0,\infty)$12, i.e. BMO (Isralowitz et al., 2015). It also proves that, for $p\in(0,\infty)$13, the commutator $p\in(0,\infty)$14 with any Riesz transform is bounded on $p\in(0,\infty)$15 if and only if $p\in(0,\infty)$16.

The multi-parameter extension is provided by S. Kakaroumpas and O. Soler i Gibert, who develop two-matrix weighted little BMO in two parameters (Kakaroumpas et al., 2023). For biparameter matrix $p\in(0,\infty)$17 weights $p\in(0,\infty)$18, the two-matrix weighted little BMO norm is

$p\in(0,\infty)$19

with the supremum over rectangles $p\in(0,\infty)$20 and $p\in(0,\infty)$21 a reducing operator (Kakaroumpas et al., 2023). The main structural theorem gives a variable-by-variable uniform equivalence between this two-parameter space and the corresponding one-parameter matrix-weighted BMO spaces on slices.

The same work makes the Campanato connection explicit at the level of interpretation. It states that little BMO spaces are matrix-weighted analogues of Campanato spaces; that the matrix-weighted BMO space $p\in(0,\infty)$22 plays the role of a matrix-weighted Campanato space; and that the two-parameter little BMO space is a product or rectangular variant, more restrictive because the rectangles can be highly eccentric (Kakaroumpas et al., 2023). It also proves lower and upper bounds for commutators with product Riesz transforms and Journé operators in terms of the little BMO norm. A plausible implication is that matrix-weighted Campanato theory has both a polynomial-approximation formulation and a genuinely product-rectangular formulation, depending on whether the underlying geometry is one-parameter or multi-parameter.

7. Analytical infrastructure and broader function-space context

Several adjacent developments supply the analytic infrastructure for matrix-weighted Campanato spaces. The Littlewood–Paley theory of matrix-weighted Triebel–Lizorkin spaces furnishes wavelet and $p\in(0,\infty)$23-transform characterizations, together with discrete sequence spaces in which cube-wise coefficients are normalized by $p\in(0,\infty)$24 or by reducing operators $p\in(0,\infty)$25 (Frazier et al., 2019). The paper emphasizes that these sequence-space norms control the distribution of local oscillations and moments, a mechanism that is crucial for Campanato theory. It also observes that, for appropriate smoothness parameters, the Triebel–Lizorkin scale interpolates between $p\in(0,\infty)$26 and matrix-weighted Sobolev spaces, while Campanato spaces arise as limiting or endpoint cases.

On the operator side, the matrix $p\in(0,\infty)$27 theorems for well-localized operators show that boundedness between matrix-weighted $p\in(0,\infty)$28 spaces is governed by testing conditions and a matrix Carleson Embedding Theorem, with constants depending on the geometry of the weight (Bickel et al., 2014). The construction of a Haar basis adapted to a matrix weight and the dependence of the embedding constant on $p\in(0,\infty)$29 underline a persistent theme: noncommutativity changes not only definitions but also the quantitative structure of the theory.

Taken together, these results place matrix-weighted Campanato spaces at the intersection of local polynomial approximation, weight-adapted normalization, atomic Hardy-space duality, and weighted singular integral theory. The framework developed in the recent literature shows that reducing operators are the common language linking these domains, and that Campanato, BMO, Hardy, Triebel–Lizorkin, and commutator theories become mutually interpretable once the local geometry of a matrix weight is built into the norms (Chen et al., 21 Aug 2025).