||We prove that the square root of a uniformly complex elliptic operator L = − div(A∇) with bounded measurable coefficients in Rn satisfies the estimate kL 1/2fkp . k∇fkp for sup(1, 2n n+4 − ε) < p < 2n n−2 + ε, which is new for n ≥ 5 and p < 2 or for n ≥ 3 and p > 2n n−2. One feature of our method is a Calder´on-Zygmund decomposition for Sobolev functions. We make some further remarks on the topic of the converse Lp inequalities (i. e. Riesz transforms bounds), pushing the recent results of [BK2] and [HM] for 2n n+2 < p < 2 when n ≥ 3 to the range sup(1, 2n n+2 −ε) < p < 2+ε 0. In particular, we obtain that L1/2 extends to an isomorphism from W˙ 1,p(Rn) to Lp(Rn) for p in this range. We also generalize our method to higher order operators.