Sharp local smoothing for the three-dimensional wave equation
For a Schwartz function $f$ on $\mathbb{R}^3$ let $u(t, x) = e^{it\sqrt{-\Delta}} f(x)$, that is, $\widehat{u}(t, \xi) = e^{it|\xi|}\widehat{f}(\xi)$. For fixed $t$ and $p > 2$, the sharp $L^p(\mathbb{R}^3)$ bound for $e^{it\sqrt{-\Delta}}$ loses $2(1/2 - 1/p)$ derivatives (Peral; Miyachi). Sogge's local smoothing conjecture predicts that averaging over $1 \le t \le 2$ recovers almost $1/p$ of a derivative when $p \ge 3$; among these exponents $p = 3$ is the strongest case, the others following from it by interpolation with the $L^2$ and $L^\infty$ estimates. The decoupling theorem of Bourgain and Demeter gives this gain for every $p \ge 4$.
Problem. For every $3 \le p < \infty$ and every $\varepsilon > 0$, is there a constant $C_{p,\varepsilon}$ such that, for every $\lambda \ge 1$ and every such $f$ with Fourier support in $\{\lambda \le |\xi| \le 2\lambda\}$, $$\left(\int_1^2 \int_{\mathbb{R}^3} |u(t, x)|^p \, dx \, dt\right)^{1/p} \le C_{p,\varepsilon}\, \lambda^{1 - 3/p + \varepsilon}\, \|f\|_{L^p(\mathbb{R}^3)}?$$
Source: Sogge (1991), the local smoothing conjecture for the Euclidean half-wave propagator, as stated by Beltran, Hickman and Sogge (2021), Conjecture 2.4. Stated by AIM — Open Applied Problems in three dimensions, for 3 ≤ p < ∞, in frequency-localised form.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, Fourier integral operators, local smoothing, wave equation
References
@article{sogge1991,
author = {Sogge, Christopher D.},
title = {Propagation of singularities and maximal functions in the plane},
journal = {Invent. Math.},
volume = {104},
number = {1},
year = {1991},
pages = {349--376},
doi = {10.1007/BF01245080},
}
@incollection{beltran2021,
author = {Beltran, David and Hickman, Jonathan and Sogge, Christopher D.},
title = {Sharp local smoothing estimates for {F}ourier integral operators},
booktitle = {Geometric Aspects of Harmonic Analysis},
series = {Springer INdAM Series},
publisher = {Springer},
address = {Cham},
year = {2021},
pages = {29--105},
doi = {10.1007/978-3-030-72058-2_2},
eprint = {1812.11616},
archivePrefix = {arXiv},
}
@article{peral1980,
author = {Peral, Juan C.},
title = {{$L^p$} estimates for the wave equation},
journal = {J. Funct. Anal.},
volume = {36},
number = {1},
year = {1980},
pages = {114--145},
doi = {10.1016/0022-1236(80)90110-X},
}
@article{miyachi1980,
author = {Miyachi, Akihiko},
title = {On some estimates for the wave equation in {$L^p$} and {$H^p$}},
journal = {J. Fac. Sci. Univ. Tokyo Sect. IA Math.},
volume = {27},
number = {2},
year = {1980},
pages = {331--354},
}
@article{bourgain2015,
author = {Bourgain, Jean and Demeter, Ciprian},
title = {The proof of the $l^2$ decoupling conjecture},
journal = {Ann. of Math. (2)},
volume = {182},
number = {1},
year = {2015},
pages = {351--389},
doi = {10.4007/annals.2015.182.1.9},
eprint = {1403.5335},
archivePrefix = {arXiv},
}
@misc{aim-euclidean-wave-local-smoothing,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp local smoothing for the three-dimensional wave equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/210-euclidean-wave-local-smoothing.md},
note = {Entry 210 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
Yes: ||e^{it sqrt(-Delta)} f||_{L^3(R^3 x [1,2])} <= C_eps ||(1-Delta)^{eps/2} f||_3 for every eps > 0, and hence for every 2 < p < infinity the bound with alpha > max{0, 1-3/p} derivatives, which gives AIM's annular estimate for every 3 <= p < infinity, eps > 0 and lambda >= 1..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00681, "Sharp local smoothing for the three-dimensional wave equation", clopen.solutions.