Let T be the unit circle, let P_+ denote the Riesz projection onto the nonnegative Fourier modes, and let 1<q<∞. The conjecture asserts the sharp contractive estimate |P_+f|_L^4(1-1/q)(T)≤ |f|_L^q(T) for every f∈ L^q(T). It also includes the corresponding logarithmic endpoint assertion as q downarrow 1.

Brevig–Ortega-Cerdà–Seip–Zhao Riesz projection conjecture

The conjecture asserts that the Riesz projection is contractive at the critical exponent in the stated range, with a logarithmic endpoint statement. The conjecture is associated with Brevig, Ortega-Cerdà, Seip, and Zhao.

Known results

An earlier paper established weaker contractive ranges by interpolation and showed that no positive $q$ works contractively from $L^1(\mathbb{T})$ to $H^q$ .
A 2022 paper proved the related Dirichlet–Hardy inequality $\|f\|_{H^p}\leq\|f\|_{D_{p/2}}$ for $p>2$ , presenting it as evidence rather than a proof of the Riesz projection conjecture.
A 2024 preprint proved the endpoint case $q=1$ and extended the conjectural framework, without claiming a full settlement.

September 2026 claimed completion

The preprint On the contractivity of the Riesz projection claims to prove the previously unresolved exponent range and thereby complete the conjecture. This is an unrefereed claim, with no independent verification reported in the retrieved sources.

Current status (as of September 2026): The conjecture is claimed completely proved by an unrefereed preprint, but the claimed settlement remains unverified.

Brevig–Ortega-Cerdà–Seip–Zhao Riesz projection conjecture

Primary source

Additional references

Progress summary

Solutions 0