We present a pair of joint conditions on the two functions $b_1,b_2$ strictly weaker than $b_1,b_2in operatorname{BMO}$ that almost characterize the $L^2$ boundedness of the iterated commutator $[b_2,[b_1,T]]$ of these functions and a Calderon-Zygmun
d operator $T.$ Namely, we sandwich this boundedness between two bisublinear mean oscillation conditions of which one is a slightly bumped up version of the other.
We show that, if $bin L^1(0,T;L^1_{mathrm{loc}}(mathbb{R}))$ has spatial derivative in the John-Nirenberg space $mathrm{BMO}(mathbb{R})$, then it generalizes a unique flow $phi(t,cdot)$ which has an $A_infty(mathbb R)$ density for each time $tin [0,T
]$. Our condition on the map $b$ is optimal and we also get a sharp quantitative estimate for the density. As a natural application we establish a well-posedness for the Cauchy problem of the transport equation in $mathrm{BMO}(mathbb R)$.
