Dyadic Coarsening Operators

• Dyadic coarsening operators are positive, sublinear averaging operators that project functions over hierarchical dyadic grids to reduce analytic complexity.
• They play a key role in harmonic analysis by facilitating sparse domination for Calderón–Zygmund operators and yielding sharp weighted norm inequalities.
• Utilizing wavelet decompositions, these operators establish norm equivalences in function spaces and serve as building blocks in reducing PDE and combinatorial problems.

Dyadic coarsening operators are a class of positive, sublinear averaging operators that act on functions by projecting or averaging them over hierarchical structures of dyadic intervals or cubes. They play a foundational role in modern harmonic analysis, particularly in the study of singular integral operators, function space theory, and the derivation of sharp weighted inequalities. Key families of dyadic coarsening operators include conditional expectation operators, martingale difference operators, dyadic potential and maximal operators, and sparse averaging operators indexed by families of dyadic cubes. These operators provide both technical insight and structural decomposition, enabling reduction of analytic problems to combinatorial and measure-theoretic arguments over dyadic grids.

1. Formal Definitions and Main Types

The core construction of a dyadic coarsening operator begins with a dyadic grid $\mathcal{D}$ in $\mathbb{R}^n$, where each dyadic cube $Q \in \mathcal{D}$ has side length $2^k$, $k \in \mathbb{Z}$, and the grid forms a nested partition of $\mathbb{R}^n$. For a function $f$ integrable on $Q$, its average is $f_Q = |Q|^{-1} \int_Q f$.

Key operators are defined as follows:

Operator	Formula	Description
Dyadic averaging $\mathbb{E}_N$	$$\mathbb{E}_N f(x) = \sum_{\mu} 1_{I_{N,\mu}}(x) \, 2^N \int_{I_{N,\mu}} f(t)\,dt$$	Conditional expectation on $2^{-N}$ intervals
Martingale difference $\mathbb{D}_N$	$$\mathbb{D}_N f = \mathbb{E}_{N+1} f - \mathbb{E}_N f$$	Finer-scale difference operator
Summation (potential) $T_\lambda$	$$T_\lambda(f \sigma)(x) = \sum_{Q \in \mathcal{D}} \lambda_Q \langle f \rangle_Q^\sigma 1_Q(x)$$	Weighted sum over dyadic cubes
Maximal $M_\lambda$	$$M_\lambda(f \sigma)(x) = \sup_{Q\in\mathcal{D}} \lambda_Q \langle f \rangle_Q^\sigma 1_Q(x)$$	Pointwise supremum of scaled averages
Sparse averaging $A_{\mathcal{D},\mathcal{S}}$	$$A_{\mathcal{D},\mathcal{S}}f(x) = \sum_{Q\in\mathcal{S}} f_Q \chi_Q(x)$$	Averaging on a sparse subcollection

Here, $I_{N,\mu} = [2^{-N}\mu, 2^{-N}(\mu+1))$ denotes the standard dyadic interval, $\lambda_Q$ a non-negative weight, $\mathcal{S} \subset \mathcal{D}$ a sparse family, and $\langle f \rangle_Q^\sigma$ an average with respect to measure $\sigma$.

2. Boundedness in Function Spaces and Wavelet Characterizations

The action of dyadic coarsening operators on Triebel–Lizorkin $F^s_{p,q}(\mathbb{R})$, Hardy–Sobolev $H^{s,p}$, and Besov $B^s_{p,q}$ spaces can be characterized via orthogonal compactly supported Daubechies wavelet systems. Specifically, for dyadic averaging operators, uniform boundedness is established in the range

$$\frac{1}{2} < p < \infty, \quad 0<q\le\infty, \quad \frac{1}{p} - 1 < s < \min\left\{\frac{1}{p},1\right\},$$

with the maximal bound

$$\sup_{N\geq0}\;\big\| \mathbb{E}_N f \big\|_{F^s_{p,q}} \leq C\,\|f\|_{F^s_{p,q}}.$$

Equivalent quasi-norms are obtained via wavelet coefficients $\lambda_{j,\nu}(f)$: $$\|f\|_{F^s_{p,q}} \asymp \left\| \left(\sum_{j=0}^{\infty}|2^{js} \sum_\nu \lambda_{j,\nu}(f) 1_{I_{j,\nu}}(x)|^q\right)^{1/q} \right\|_{L^p_x}.$$ The proof leverages decompositions of $f$ into wavelet projections, leading to norm equivalences and control over martingale differences and projections, with estimates on matrix coefficients of the wavelet expansions. This framework allows for analysis of how dyadic coarsening commutes with or disrupts multiscale decompositions, critical in harmonic analysis and PDE theory (Garrigós et al., 2016).

3. Sparse Domination and Calderón-Zygmund Theory

Sparse averaging operators $A_{\mathcal{D},\mathcal{S}}$ form the analytic core of recent developments in the sparse domination of Calderón–Zygmund operators (CZOs). The main sparse domination theorem states that, for any Banach function space $X$,

$$\| T_{\natural} f \|_X \leq C(n, T) \sup_{\mathcal{D},\mathcal{S}} \|A_{\mathcal{D},\mathcal{S}}f\|_X,$$

where $T_{\natural}$ is the maximal Truncation of a CZO and the supremum is over all dyadic grids $\mathcal{D}$ and all sparse $\mathcal{S} \subset \mathcal{D}$. The proof proceeds by representing $T_{\natural}$ as an expectation of Haar-shifts on random grids and showing that every Haar-shift is pointwise dominated by a linear combination of sparse dyadic averages, enabled by median-based stopping-time and local-oscillation decompositions (Lerner, 2012). This reduction facilitates sharp weighted norm inequalities for general CZOs by transferring the core analytic estimates to positive sparse models.

4. Weighted Inequalities and Two-Weight Problems

Dyadic coarsening operators serve as primary tools in addressing weighted norm inequalities, both in one- and two-weight settings. For the summation and maximal coarsening operators, sharp norm inequalities

$$\| T_\lambda(f \sigma) \|_{L^q(\omega)} \leq C \|f\|_{L^p(\sigma)}$$

are characterized precisely depending on the relationships between $p$ and $q$.

• For $0$\sigma \in A_\infty(\omega)$, sufficiency and necessity are controlled by a single explicit integral over dyadic cubes involving the weights and measures (Fujii–Wilson characteristic), closing classical conjectures.
• For maximal operators, a scale of integrability conditions parameterized by $\gamma$ interpolates between necessary and sufficient testing criteria, with an arbitrarily small gap (Hänninen et al., 2018).
• If coefficient weights $\{\lambda_Q\}$ display dyadic logarithmic bounded oscillation (DLBO), the necessary and sufficient conditions on the two measures and weights reduce to a single supremum of expression over dyadic cubes.

This framework yields the solution to the two-weight conjecture of Cruz-Uribe and Pérez and directly enables the proof and sharp constants in the $A_2$ theorem and related sharp weighted inequalities, extending to mixed $A_p$–$A_r$ and $A_1$ bounds for both maximal and singular integral operators (Lerner, 2012).

5. Interplay with Carleson Measures and Duality

A crucial structural property enabling two-weight inequalities is the mapping of Carleson sequences under multipliers associated to dyadic coarsening operators. Given the family $\mu_R = \frac{\lambda_R\,\omega(R)}{\sigma(R)}$, boundedness of $T_\lambda$ reduces to the requirement that $\{\mu_R\}$ sends $\ell^1_\omega$-Carleson sequences into $\ell^1_\sigma$-Carleson sequences. Duality in discrete Littlewood–Paley spaces, linearization of maxima (Sawyer-type arguments), and factorization lemmas enable the translation of operator inequalities into Carleson testing over combinatorial data tied to dyadic cubes (Hänninen et al., 2018).

6. Applications, Basis Properties, and Building Block Phenomenon

Dyadic coarsening operators underpin the Schauder basis property for the Haar system in Hardy–Sobolev and Triebel–Lizorkin spaces, and they guarantee convergence properties of such expansions within precise ranges on $(p, q, s)$. More fundamentally, the analytic structure of general singular integrals is efficiently approximated (and sometimes replaced) by sums, maxima, or projections over sparse collections of dyadic cubes. Their elementary nature means that proving norm inequalities for coarsening operators through combinatorial, probabilistic, or wavelet-based arguments suffices to establish boundedness for a wide swath of more complex operators. This "building block" character highlights the reduction of apparently analytic or oscillatory questions in Calderón–Zygmund theory to discrete, dyadic, and measure-theoretic core operations (Lerner, 2012, Garrigós et al., 2016, Hänninen et al., 2018).