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Optimal Sparse Bounds and Commutator Characterizations Without Doubling

Francesco D'Emilio, Yongxi Lin, Nathan A. Wagner, Brett D. Wick

TL;DR

This work advances nonhomogeneous harmonic analysis by establishing pointwise sparse domination for dyadic paraproducts with symbols in $\mathrm{BMO}(\mu)$ without the Lacey packing condition, enabling sharp weighted bounds for related commutators. It then develops a refined weighted theory for commutators with dyadic shifts via balanced Haar systems, yielding complexity- and weight-dependent estimates and extending to the dyadic Hilbert transform. A central achievement is the complete $L^p(\mu)$ characterization of when the dyadic Hilbert transform commutator $[\mathcal{H},b]$ is bounded, revealing a genuine $p$-dependent hierarchy between $\mathrm{BMO}(\mu)$, $[\mathrm{BMO}]_p(\mu)$, and $\mathrm{bmo}_p(\mu)$, with explicit necessary and sufficient conditions involving a $p$-dependent $\mathrm{bmo}$-type space and a bounded sequence $\beta_Q$. The results demonstrate that nonhomogeneous settings require new principles beyond classical BMO theory and provide explicit examples showing that packing and martingale BMO can diverge when $\mu$ is not doubling, significantly enriching the toolbox for nonuniform sampling, probabilistic models, and PDEs in irregular geometries.

Abstract

We examine dyadic paraproducts and commutators in the non-homogeneous setting, where the underlying Borel measure $μ$ is not assumed to be doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related operators with symbols $b \in \textrm{BMO}(μ)$, improving upon an earlier result of Lacey, where the symbol $b$ was assumed to satisfy a stronger Carleson-type condition, that coincides with $\textrm{BMO}$ only in the doubling setting. As an application of this result, we obtain sharpened weighted inequalities for the commutator of a dyadic Hilbert transform $\mathcal{H}$ previously studied by Borges, Conde Alonso, Pipher, and the third author. We also characterize the symbols for which the commutator $[\mathcal{H},b]$ is bounded on $L^p(μ)$ for $1<p<\infty$ and provide some interesting examples to prove that this class of symbols strictly depends on $p$ and is nested between symbols satisfying the $p$-Carleson packing condition and symbols belonging to martingale BMO (even in the case of absolutely continuous measures).

Optimal Sparse Bounds and Commutator Characterizations Without Doubling

TL;DR

This work advances nonhomogeneous harmonic analysis by establishing pointwise sparse domination for dyadic paraproducts with symbols in without the Lacey packing condition, enabling sharp weighted bounds for related commutators. It then develops a refined weighted theory for commutators with dyadic shifts via balanced Haar systems, yielding complexity- and weight-dependent estimates and extending to the dyadic Hilbert transform. A central achievement is the complete characterization of when the dyadic Hilbert transform commutator is bounded, revealing a genuine -dependent hierarchy between , , and , with explicit necessary and sufficient conditions involving a -dependent -type space and a bounded sequence . The results demonstrate that nonhomogeneous settings require new principles beyond classical BMO theory and provide explicit examples showing that packing and martingale BMO can diverge when is not doubling, significantly enriching the toolbox for nonuniform sampling, probabilistic models, and PDEs in irregular geometries.

Abstract

We examine dyadic paraproducts and commutators in the non-homogeneous setting, where the underlying Borel measure is not assumed to be doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related operators with symbols , improving upon an earlier result of Lacey, where the symbol was assumed to satisfy a stronger Carleson-type condition, that coincides with only in the doubling setting. As an application of this result, we obtain sharpened weighted inequalities for the commutator of a dyadic Hilbert transform previously studied by Borges, Conde Alonso, Pipher, and the third author. We also characterize the symbols for which the commutator is bounded on for and provide some interesting examples to prove that this class of symbols strictly depends on and is nested between symbols satisfying the -Carleson packing condition and symbols belonging to martingale BMO (even in the case of absolutely continuous measures).

Paper Structure

This paper contains 10 sections, 29 theorems, 159 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Paraproducts and sparse domination
  4. Weighted Inequalities for Commutators with dyadic shifts
  5. Haar shifts: modified sparse domination and weighted inequalities
  6. Improved weighted inequalities for commutators
  7. Dyadic Hilbert Transform: refined commutator bounds
  8. $L^p$ boundedness of $[\mathcal{H},b]$: necessary and sufficient conditions.
  9. Proof of \ref{['THM: sharp containment']}
  10. Final remarks and open questions

Key Result

Theorem 1

Let $\mu$ be an atomless Radon measure in $\mathbb{R}^n$ with $0< \mu(Q)< \infty$ for every $Q \in \mathcal{D}$, and $b \in \mathrm{BMO}$. Then any $T \in \{\Pi_b, \Pi^\ast_b, \Delta_b\}$ satisfies the following: for every $f \in L^1(\mu)$ compactly supported on $Q_0 \in \mathcal{D}$, there exists a where the implicit constant depends on $T$ and $n$. Consequently, for $T \in \{ \Pi_b, \Pi_b^\ast,

Figures (3)

  • Figure 1: The construction of $\mu$ on $[0,1)$
  • Figure 2: A visualization of $f_2$.
  • Figure 3: Values and averages of $q$

Theorems & Definitions (58)

  • Theorem 1: Sparse domination with BMO symbols
  • Corollary 2: Sharp weighted inequalities for Haar shifts
  • Corollary 3
  • Theorem 4: Characterization of Dyadic Hilbert Transform Commutator Bounds
  • Corollary 5
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: John-Nirenberg inequality
  • Proposition 2.4: TREIL2010
  • Definition 2.5
  • ...and 48 more