Two weight bump conditions for compactness of commutators (Q2678887)

Let \(T\) be a Calderón-Zygmund operator and \(b\) be a BMO function. Define the commutator of \(b\) and \(T\) to be \[ [b,T]f=bTf-T(bf). \] In this paper, if \(u\) and \(v\) are weights that satisfy \[ \sup_{Q} \|u^{\frac{1}{p}}\|_{L^p(\log L)^{2p-1+\delta},Q} \sup_{Q} \|v^{-\frac{1}{p}}\|_{L^{p'}(\log L)^{2{p'}-1+\delta},Q} <\infty \] for some \(\delta>0\), the authors prove that the commutator \([b,T]\) is a compact operator from \(L^p(v)\) to \(L^p(u)\). The condition in this result is a strengthening of the classic two-weight \(A_p\) condition. It does not require individual assumptions on \(u\) and \(v\).