Sharp norm inequalities for commutators of classical operators

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 $I_{\text{al}}$ we find the sharp one-weight bound on $[b,I_{\text{al}}]$, $b\in BMO$, in terms of the $A_{p,q}$ constant of the weight. We also prove sharp two-weight bounds for $[b,I_{\text{al}}]$ analogous to those of singular integrals. We prove two-weight weak-type inequalities for $[b,T]$ and $[b,I_{\text{al}}]$ for pairs of factored weights. Finally we construct several examples showing our bounds are sharp.