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Generalized Chang–Yang conjecture

Conjecture 1 (Generalized Chang--Yang conjecture). For every integer N≥1N\geq 1 and every α≥12\alpha\geq\frac{1}{2}, let

Jα,N(u):=α2∫SNu(PNu) dw+(N−1)!∫SNu dw−(N−1)!Nlog⁡∫SNeNu dw,J_{\alpha,N}(u):=\frac{\alpha}{2}\int_{\mathbb{S}^{N}}u(P_Nu)\,dw+(N-1)!\int_{\mathbb{S}^{N}}u\,dw-\frac{(N-1)!}{N}\log\int_{\mathbb{S}^{N}}e^{Nu}\,dw,

and

LN:={u∈HN2(SN):∫SNeNuξj dw=0, j=1,…,N+1}.\mathcal{L}_N:=\left\{u\in H^{\frac{N}{2}}(\mathbb{S}^{N}):\int_{\mathbb{S}^{N}}e^{Nu}\xi_j\,dw=0,\ j=1,\ldots,N+1\right\}.

Then

inf⁡u∈LNJα,N(u)=0.\inf_{u\in\mathcal{L}_N}J_{\alpha,N}(u)=0.

Progress summary

Solved

A new preprint claims the conjecture is proved in every dimension three or higher, but the result has not been independently verified.

The generalized Chang–Yang conjecture seeks the sharp Beckner inequality on spheres in all integer dimensions N≥3N\geq 3 and the full parameter range. Changfeng Gui, Tuoxin Li, Juncheng Wei, and Zikai Ye claim an affirmative answer.

Known results

  • Wei and Xu previously treated α<1\alpha<1 sufficiently close to 11; the full interval α∈[12,1)\alpha\in[\frac12,1) was reported as open.

August 2026 full-range preprint claim

The authors claim that for every N≥3N\geq 3 and α≥12\alpha\geq\frac12, the Beckner inequality holds for all admissible u∈HN2(SN)u\in H^{\frac N2}(\mathbb{S}^{N}), equivalently inf⁡u∈LNJα,N(u)=0\inf_{u\in\mathcal{L}_{N}}J_{\alpha,N}(u)=0. Their method uses an integral representation and rigidity of stable critical points. A separate preprint claims the axially symmetric case, but does not independently verify the unrestricted result.

Current status (as of August 2026): The conjecture has a claimed full proof for all N≥3N\geq 3 and α≥12\alpha\geq\frac12, while independent verification remains open.

Sources
Sources & referencesView supporting material

Primary source

arXiv

Additional references

conformal-geometryfunctional-inequalitiesgeometric-analysis

Solutions 0

No solutions have been posted yet.