Determine precisely the exponents 0<p<∞ for which there exists a constant C_p<∞ such that every Dirichlet polynomial P satisfies sup_theta∈Rint_theta^theta+1|P(frac12+it)|^p,dt≤ C_p lVert P rVert_mathscrH^p^p, with C_p independent of the number and choice of prime variables on which P depends. The claimed classification is that this holds if and only if p≥ 2.

Local embedding problem for Hardy spaces of Dirichlet series

Determine precisely the exponents $0<p<\infty$ for which there exists a constant $C_p<\infty$ such that every Dirichlet polynomial $P$ satisfies $\sup_{\theta\in\mathbb{R}}\int_{\theta}^{\theta+1}\left|P\left(\frac12+it\right)\right|^p\,dt\le C_p\lVert P\rVert_{\mathscr{H}^p}^p$ , with $C_p$ independent of the number and choice of prime variables on which $P$ depends. The claimed classification is that this holds if and only if $p\ge 2$ .

References

Primary source

arXiv

Additional references

The Local Embedding Problem for Hardy Spaces of Dirichlet Series — arXiv — Bonan Chen, Xiang Fang, Feng Guo, Shengzhao Hou, Yizhou Shao, Qi Zhou

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A new unrefereed preprint claims the problem is solved: the embedding holds from exponent two onward and fails below two.

The problem asks for the exact finite-exponent range in which the local embedding holds for Hardy spaces of Dirichlet series. Earlier results settled the case $p=2$ , but not the full classification.

Known results

The local embedding theorem was established for $\mathscr{H}^{2}$ in 2010.
A sharp weighted global embedding at $p=2$ was proved in 2016, with optimal constant $\sqrt{2}$ .

September 10, 2026 preprint

Bonan Chen, Xiang Fang, Feng Guo, Shengzhao Hou, Yizhou Shao, and Qi Zhou claim that the local embedding extends to every $p\ge2$ and fails for $p<2$ , giving the sharp threshold and the conjectural classification. This is a new unrefereed preprint, so the claimed resolution remains unverified.

Current status (as of September 2026): The $p=2$ case and earlier partial results are established, while the full classification for $p\ge2$ and failure for $p<2$ is claimed but unverified.

Sources

No solutions have been posted yet.