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Local Smoothing Conjecture

The local smoothing conjecture (Sogge 1991) predicts improved space-time bounds for wave evolution after integration over a bounded time interval. For the Euclidean half-wave operator in n>=2 dimensions, use the Fourier transform convention f^^(xi)=int_(R^n)f(x)e^(-ix·xi)dx and define

Uf(x,t)=(2pi)^(-n)int_(R^n)e^(i(x·xi+t||xi||))f^^(xi)dxi.

This evolution solves the unit-speed wave equation. For a Schwartz function f, the predicted strict estimates are

||Uf||_(L^p(R^n×[1,2]))<=C_(n,p,alpha)||J^alphaf||_(L^p(R^n))

for 2<p<infty and alpha>max{0,(n-1)/2-n/p}, where J^alpha=(I-Delta)^(alpha/2) defines the Bessel potential space norm, I is the identity operator, and Delta is the Laplacian. In three dimensions, the regularity threshold is max{0,1-3/p}. The positive loss above the threshold is part of the conjecture and does not assert its zero-loss endpoint (Beltran et al. 2021).

OpenAI (2026a) reported the Euclidean estimates in every spatial dimension n>=2, together with the strict-range bounds in the Bochner-Riesz conjecture and Fourier restriction conjecture, and the Kakeya maximal conjecture. The report also claims full Hausdorff dimension for Kakeya sets in every dimension. An internal OpenAI model supplied the reported proof and Lean formalization. As of Oct. 27, 2026, independent specialist confirmation of the complete argument had not been reported (OpenAI 2026b). The claim concerns the Euclidean wave equation, rather than arbitrary variable-coefficient wave operators or the zero-loss endpoint.

See also

Bessel Potential Space, Bochner-Riesz Mean, Sobolev Space, Wave Equation

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References

Beltran, D.; Hickman, J.; and Sogge, C. D. "Sharp Local Smoothing Estimates for Fourier Integral Operators." In Geometric Aspects of Harmonic Analysis (Ed. P. Ciatti and A. Martini). Cham, Switzerland: Springer, pp. 29-105, 2021. https://doi.org/10.1007/978-3-030-72058-2_2.OpenAI. "Local Smoothing for the Euclidean Wave Equation." Oct. 25, 2026a. https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Local-smoothing-for-the-Euclidean-wave-equation-October-25-2026/paper.pdf.OpenAI. "Working with Academic Communities to Build Scientific Understanding." Oct. 26, 2026b. https://openai.com/index/building-scientific-understanding-with-academic-communities/.Sogge, C. D. "Propagation of Singularities and Maximal Functions in the Plane." Invent. Math. 104, 349-376, 1991. https://doi.org/10.1007/BF01245080.

Cite this as:

Weisstein, Eric W. "Local Smoothing Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LocalSmoothingConjecture.html

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