Sharp Lp estimates of powers of the complex Riesz transform

Let R-1,R-2 be scalar Riesz transforms on R-2. We prove that the L-p norms of k-th powers of the operator R-2 + i R-1 behave exactly as vertical bar k vertical bar(1-2/p)* (p* - 1), uniformly in k is an element of Z\{0} and 1 < p < infinity, where p* is the bigger number between p and its conjugate exponent. This gives a complete asymptotic answer to a question suggested by Iwaniec and Martin in 1996. The main novelty are the lower estimates, of which we give three different proofs. We also conjecture the exact value of parallel to(R-2 + i R-1)(k)parallel to(p). Furthermore, we establish the sharp behaviour of weak (1, 1) constants of (R-2+ i R-1)(k) and an L-infinity to BMO estimate that is sharp up to a logarithmic factor.