Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Commutators of Fractional Integrals with $\operatorname{BMO}^β$ Functions

You-Wei Benson Chen, Alejandro Claros

Abstract

We study commutators of the Riesz potential $I_α$ with functions $b$ in the capacitary space $\mathrm{BMO}^β(\mathbb{R}^n)$, defined through the Hausdorff content $\mathcal{H}^β_\infty$. We prove a Chanillo-type theorem characterising $\mathrm{BMO}^β(\mathbb{R}^n)$ via the boundedness of the commutator $[b,I_α]$ on capacitary Lebesgue spaces. In addition, we obtain the endpoint estimate in the form of a capacitary modular weak-type inequality. These results follow from a pointwise estimate for the $β$-dimensional sharp maximal function of the commutator, together with a capacitary Fefferman-Stein inequality recently proved in [CC24].

Commutators of Fractional Integrals with $\operatorname{BMO}^β$ Functions

Abstract

We study commutators of the Riesz potential with functions in the capacitary space , defined through the Hausdorff content . We prove a Chanillo-type theorem characterising via the boundedness of the commutator on capacitary Lebesgue spaces. In addition, we obtain the endpoint estimate in the form of a capacitary modular weak-type inequality. These results follow from a pointwise estimate for the -dimensional sharp maximal function of the commutator, together with a capacitary Fefferman-Stein inequality recently proved in [CC24].
Paper Structure (9 sections, 26 theorems, 151 equations)

This paper contains 9 sections, 26 theorems, 151 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and known results
  3. Hausdorff content and Choquet integrals
  4. On $\beta$-dimensional bounded mean oscillation spaces
  5. Capacitary Orlicz spaces and maximal functions
  6. Riesz potential and Hausdorff content
  7. Key pointwise estimate
  8. Characterization of $\operatorname{BMO}^\beta$ in terms of commutator of fractional integrals
  9. Endpoint modular weak-type inequality

Key Result

Theorem 1.1

Let $B(t)=t\log(e+t)$. Fix $0<\beta\le n$ and $0<\alpha<\beta$. Let $b$ be a measurable function in $\operatorname{BMO}^\beta ({\mathbb R}^n)$ and let $f$ be a nonnegative measurable function. Then there exists a constant $C=C(\alpha,\beta,n)>0$ such that, for every $s>1$ and every $x\in{\mathbb R}^ where $\mathcal{M}_{\alpha,B,\mathcal{H}^\beta_\infty}$ denotes the fractional Orlicz maximal opera

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 45 more