Statement copy in the gpt 6 astra pro collection (Math-205)

Local Smoothing for the Euclidean Wave Equation

Definitions and mathematical statement

For a Schwartz function on ${\mathbb{R}}^n$, the half-wave propagator is the Fourier multiplier $\widehat{e^{it\sqrt{-\Delta}}f}(\xi)=e^{it|\xi|}\widehat f(\xi)$, using the Fourier convention in which $-\Delta$ has symbol $|\xi|^2$. Define the Bessel-potential norm $\|f\|_{W^{s,p}}=\|(1-\Delta)^{s/2}f\|_{L^p}$ and put $s_p=(n-1)(1/2-1/p)$. The assertion is \[ \left(\int_1^2\int_{{\mathbb{R}}^n}|e^{it\sqrt{-\Delta}}f(x)|^p\,\mathrm d x\,\mathrm d t\right)^{1/p} \le C_{n,p,\varepsilon}\|f\|_{W^{s_p-1/p+\varepsilon,p}} \] for $n\ge3$, $p>2n/(n-1)$, and every $\varepsilon>0$. Constants are independent of $f$. This is a spacetime estimate with an arbitrarily small derivative loss; the excluded endpoint and a zero-loss bound are not asserted.

Formulation and concept references: [S1].

Short English statement

Does averaging the wave evolution over time save almost one-over-p derivatives throughout the conjectured range of exponents?

Sources

• S1 Formal statement, Introduction, local smoothing conjecture. https://www.sciencedirect.com/science/article/pii/S0022123623003786. Catalogue formulation source.
• E1 J. Bourgain and C. Demeter, The proof of the $\ell^2$ Decoupling Conjecture, Annals of Mathematics $2$ 182 $2015$, no. 1, 351--389, Theorem 1.2 (cone decoupling) and the discussion immediately following it. https://annals.math.princeton.edu/2015/182-1/p09; author manuscript https://arxiv.org/pdf/1403.5335v3. Published result; consulted 24 September 2026.
• E2 S. Gan, D. He, X. Li, and S. Wu, On local smoothing estimates for wave equations, arXiv:2502.05973v2, 4 January 2026, Theorem 1.3, equation $1.4$, and Sections 1.1, 1.3, and 5.1. https://arxiv.org/html/2502.05973v2. Recent manuscript theorem, not independently audited here; version 2 has four authors and an improved main result. Consulted 24 September 2026.
• E3 D. Beltran, J. Roos, A. Rutar, and A. Seeger, A fractal local smoothing problem for the wave equation, Bulletin of the London Mathematical Society 57 $2025$, no. 12, 3667--3690; author manuscript arXiv:2501.12805v1, 22 January 2025, Theorem 1.1, equation $1.6$, and Section 5 (radial inputs). https://arxiv.org/html/2501.12805v1. The radial theorem, not its January 2025 general-frontier summary, is used here. Consulted 24 September 2026.