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 $R_2+i{R}_1$ behave exactly as $|k{|}^{1-2/p}p$, uniformly in $k\in \mathbb{Z}\backslash \{0\}$, $p\ge 2$. 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 $\Vert (R_2+i{R}_1{)}^k{\Vert }_p$. Furthermore, we establish the sharp behaviour of weak $(1,1)$ constants of $(R_2+i{R}_1{)}^k$ and an $L^{\infty }$ to $BMO$ estimate that is sharp up to a logarithmic factor.

32 pages

References in corpus (1)

• Analysis of the Ahlfors-Beurling operator (lecture notes for the summer school at the University of Seville, 2013)

Cited by in corpus (1)

• Asymptotic behavior of $L^p$ estimates for a class of multipliers with homogeneous unimodular symbols