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Sharp L^p estimates of powers of the complex Riesz transform
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Sharp L^p estimates of powers of the complex Riesz transform
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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.
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Cited by 1 Pith paper
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Cotlar martingale transforms and related singular integrals
Cotlar's identity extends to martingale transforms, revealing that the vector of Riesz transforms in odd dimensions has the same L^p structure as the Hilbert transform, with matching asymptotic norms as p tends to infinity.
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