A simple proof that the Riesz projection is bounded on $L^p(\mathbb{T})$ for $1<p<\infty$

Ole Fredrik Brevig

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arXiv · 2606.05190

A simple proof that the Riesz projection is bounded on $L^p(\mathbb{T})$ for $1<p<\infty$

Ole Fredrik Brevig

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Abstract

Let $\mathbf{P}$ denote the Riesz projection on the unit circle $\mathbb{T}$ and suppose that $1<p<\infty$. We present a simple proof of the bound $\|\mathbf{P}f\|_p \leq \max(p,q) \|f\|_p$, where $f$ is in $L^p(\mathbb{T})$ and $p^{-1}+q^{-1}=1$. Our proof is a variation of a classical argument due to M. Riesz demonstrating that the Hilbert transform is bounded on $L^p(\mathbb{T})$.

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BibTeX RIS

Ole Fredrik Brevig. 2026-05-06. A simple proof that the Riesz projection is bounded on $L^p(\mathbb{T})$ for $1<p<\infty$. https://arxiv.org/abs/2606.05190

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