The LpL^p decoupling conjecture for the hyperbolic paraboloid

About 1 year old · traced to

Let

Ef(x1,x2,x3):=∫[−1,1]2ei(x1ξ1+x2ξ2+x3ξ1ξ2)f(ξ1,ξ2)dξ1dξ2.E f(x_1,x_2,x_3):=\int_{[-1,1]^2}e^{i(x_1\xi_1+x_2\xi_2+x_3\xi_1\xi_2)}f(\xi_1,\xi_2)d\xi_1d\xi_2.

Hyperbolic-paraboloid decoupling conjecture. For p>3p>3, for every ε>0\varepsilon>0 there exists Cε>0C_\varepsilon>0 such that for all R≥1R\geq1,

∥Ef∥Lp(BR)p≤CεRε∥f∥pp.\|Ef\|_{L^p(B_R)}^p\leq C_\varepsilon R^\varepsilon\|f\|_p^p.

This is presented as a standard formulation reducing the restriction conjecture for the truncated hyperbolic paraboloid in R3\mathbb{R}^3; the supplied text does not state whether it is resolved.

References

Primary source

Ciprian Demeter and Shukun Wu, “Restriction and decoupling estimates for the hyperbolic paraboloid in R^3”, arXiv:2505.09037 (2026).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.