A short proof of commutator estimates

The goal of this note is to give, at least for a restricted range of indices, a short proof of homogeneous commutator estimates for fractional derivatives of a product, using classical tools. Both $L^{p}$ and weighted $L^{p}$ estimates can be proved by the same argument. When the space dimension is 1, we obtain some new estimates in the unexplored range $1/3<r\le1/2$.