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

TL;DR

This paper precisely characterizes the asymptotic behavior of the $L^p$ norms of powers of the complex Riesz transform on $ ^2$, resolving a question posed in 1996 and providing new lower bounds.

Contribution

It offers the first complete asymptotic description of the $L^p$ norms of powers of the complex Riesz transform, including three different proofs for the lower estimates.

Findings

Exact asymptotic behavior of $ orm{(R_2+iR_1)^k}_p$ as $|k| o\infty$

Sharp bounds for weak $(1,1)$ constants of the operator

Sharp $L^\infty$ to BMO estimate for the operator

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 $|k|^{1-2/p}p$, uniformly in $k\in\mathbb{Z}\backslash\{0\}$, $p\geq2$. 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 $\|(R_2+iR_1)^k\|_p$. 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.