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 prove that for every Banach space $Y$, the Besov spaces of functions from the $n$-dimensional Euclidean space to $Y$ agree with suitable local approximation spaces with equivalent norms. In addition, we prove that the Sobolev spaces of type $q$ ar
e continuously embedded in the Besov spaces of the same type if and only if $Y$ has martingale cotype $q$. We interpret this as an extension of earlier results of Xu (1998), and Martinez, Torrea and Xu (2006). These two results combined give the characterization that $Y$ admits an equivalent norm with modulus of convexity of power type $q$ if and only if weakly differentiable functions have good local approximations with polynomials.
We bring a precision to our cited work concerning the notion of Borel measures, as the choice among different existing definitions impacts on the validity of the results.
