Sharp $L^p$ estimates of powers of the complex Riesz transform

Abstract

Let $R_{1,2}$ be scalar Riesz transforms on ${\mathbb {R}}^2$. We prove that the $L^p$ norms of k-th powers of the operator $R_2+iR_1$ behave exactly as $\vert k\vert ^{1-2/p^*}(p^*-1)$, uniformly in $k\in {\mathbb {Z}}\backslash \{0\}$ and $1<p<\infty$, 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 $\Vert (R_2+iR_1)^k\Vert _p$. Furthermore, we establish the sharp behaviour of weak (1, 1) constants of $(R_2+iR_1)^k$ and an $L^\infty$ to BMO estimate that is sharp up to a logarithmic factor.