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Local Smoothing Conjecture

The local smoothing conjecture predicts improved space-time bounds for wave evolution after integration over a bounded time interval. For the Euclidean half-wave operator in three dimensions, use the Fourier transform convention f^^(xi)=int_(R^3)f(x)e^(-ix·xi)dx and define

Uf(x,t)=(2pi)^(-3)int_(R^3)e^(i(x·xi+t||xi||))f^^(xi)dxi.

This evolution solves the unit-speed wave equation. The predicted strict estimates are

||Uf||_(L^p(R^3×[1,2]))<=C_(p,alpha)||J^alphaf||_(L^p(R^3))

for 2<p<infty and alpha>max{0,1-3/p}, where J^alpha=(I-Delta)^(alpha/2) defines the Bessel potential space norm.

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.

Cite this as:

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

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