Euclidean wave-equation local smoothing conjecture

Prove the local smoothing conjecture for the half-wave evolution on \(\mathbb{R}^n\): for every \(\varepsilon>0\), establish the estimate \(\|e^{it\sqrt{-\Delta}}f\|_{L^p(\mathbb{R}^n\times[1,2])}\lesssim_{\varepsilon}\|f\|_{L^p_{s_p-1/p+\varepsilon}(\mathbb{R}^n)}\) whenever \(p\ge 2n/(n-1)\), where \(s_p=(n-1)|1/2-1/p|\).

Background

The paper frames local smoothing as the gain in regularity obtained by averaging solutions of dispersive equations over time. For the half-wave evolution eit−Δfe^{it\sqrt{-\Delta}}f on Rn\mathbb{R}^n, fixed-time estimates lose sp=(n−1)∣1/2−1/p∣s_p=(n-1)|1/2-1/p| derivatives in LpL^p. Sogge’s local smoothing conjecture predicts an additional gain of almost $1/p$ derivatives after averaging over a compact time interval.

The conjecture remains broader than the results established in this paper. The paper improves the wave packet density method in high even dimensions and constructs Euclidean configurations showing that this method encounters an obstruction near the exponent $2+8/(3n-2)$; it does not establish the full conjectured range in all dimensions.

References

The $n$-dimensional local smoothing conjecture is difficult and interesting because it implies the other principal conjectures for the same dimension.

— The Work of Hong Wang  (2609.40161 - Sogge, 30 Sep 2026) in Section 4, immediately before equation (4.3)

Sogge observed that averaging over a compact time interval produces additional regularity and formulated the local smoothing conjecture . In the range $p>2$, the conjecture predicts that, for every $\varepsilon>0$,

|e{it\sqrt{-\Delta}f|_{Lp(\mathbb{R}n\times[1,2])} \lesssim_{\varepsilon} |f|{Lp{s_p-\frac1p+\varepsilon}(\mathbb{R}n)}

whenever

p\geq \frac{2n}{n-1}.

— Wave Packet Density in Local Smoothing: Refinements and Limitations  (2609.34675 - Wang, 28 Sep 2026) in Section 1, Introduction