Sharp norm inequalities for commutators of classical operators

TL;DR

This paper establishes sharp weighted norm inequalities for commutators of classical harmonic analysis operators, revealing new bump conditions and bounds that improve understanding of their behavior under weighted norms.

Contribution

It introduces new sharp weighted inequalities for commutators, including double log bump conditions and precise bounds for fractional integral operators, advancing the theory of weighted harmonic analysis.

Findings

Sharp weighted norm inequalities for commutators of classical operators.

Introduction of double log bump conditions for commutators.

Sharp bounds for fractional integral operator commutators.

Abstract

We prove several sharp weighted norm inequalities for commutators of classical operators in harmonic analysis. We find sufficient $A_p$-bump conditions on pairs of weights $(u,v)$ such that $[b,T]$, $b\in BMO$ and $T$ a singular integral operator (such as the Hilbert or Riesz transforms), maps $L^p(v)$ into $L^p(u)$. Because of the added degree of singularity, the commutators require a "double log bump" as opposed to that of singular integrals, which only require single log bumps. For the fractional integral operator $I_\al$ we find the sharp one-weight bound on $[b,I_\al]$, $b\in BMO$, in terms of the $A_{p,q}$ constant of the weight. We also prove sharp two-weight bounds for $[b,I_\al]$ analogous to those of singular integrals. We prove two-weight weak-type inequalities for $[b,T]$ and $[b,I_\al]$ for pairs of factored weights. Finally we construct several examples showing our bounds are sharp.