Three Observations on Commutators of Singular Integral Operators with BMO Functions

Three observations on commutators of Singular Integral Operators with BMO functions are exposed, namely 1. The already known subgaussian local decay for the commutator, namely 1/vertical bar Q vertical bar vertical bar{x is an element of Q : vertical bar[b, T](f chi Q)(x)vertical bar > M(2)f(x)t} <= ce(-root ct parallel to b parallel to BMO) is sharp, since it cannot be better than subgaussian. 2. It is not possible to obtain a pointwise control of the commutator by a finite sum of sparse operators defined by L log L averages. 3. Motivated by the conjugation method for commutators, it is shown the failure of the following endpoint estimate, if w is an element of A(p) \ A(1) then parallel to wM(f/w)parallel to(L1(Rn) -> L1,infinity(Rn)) = infinity.