Proofmap
Family 079Real and complex analysisLean59 pp · Oct 25, 2026

Local smoothing for the Euclidean wave equation

What this paper proves

For n≥2n\ge2 and 1≤p≤∞1\le p\le\infty, the Bochner–Riesz means BrαB_r^\alpha are bounded on LpL^p uniformly in r>0r>0 whenever α>max⁡{0,n∣1/p−1/2∣−1/2}\alpha>\max\{0,n|1/p-1/2|-1/2\}, with LpL^p convergence for p<∞p<\infty; for 2≤p<∞2\le p<\infty and α\alpha above a further threshold, the maximal operator is bounded and convergence holds almost everywhere. The key input is a local smoothing estimate for the wave equation at p=2n/(n−1)p=2n/(n-1) that loses only a power of log⁡\log of the frequency, proved via a determinant energy built from a conserved current for superpositions of waves. The LpL^p range is exactly the one predicted by the Bochner–Riesz conjecture, so the theorem confirms it in every dimension.

Main theorem

theoremroutineBochner–Riesz bounds from local smoothingthm:bochner-riesz
For n≥2n\ge2 and α>δ(p)=max⁡{0,n∣1/p−1/2∣−1/2}\alpha>\delta(p)=\max\{0,n|1/p-1/2|-1/2\}, the Bochner–Riesz means BrαB_r^\alpha are bounded on Lp(Rn)L^p(\RR^n) uniformly in r>0r>0, and Brαf→fB_r^\alpha f\to f in LpL^p as r→∞r\to\infty when p<∞p<\infty. For 2≤p<∞2\le p<\infty and α>s+(p)\alpha>s_+(p), the maximal function B∗αfB_*^\alpha f is bounded on LpL^p and Brαf→fB_r^\alpha f\to f almost everywhere.
Theorem 10.1 · as stated in the paper
Let n≥2n\ge2, 1≤p≤∞1\le p\le\infty, and put δ(p)=max⁡{0,n∣1/p−1/2∣−1/2}\delta(p)=\max\{0,n|1/p-1/2|-1/2\}. If α>δ(p)\alpha>\delta(p), then sup⁡r>0∥Brαf∥Lp(Rn)≤Cn,p,α∥f∥Lp(Rn). \sup_{r>0}\|B_r^\alpha f\|_{L^p(\RR^n)} \le C_{n,p,\alpha}\|f\|_{L^p(\RR^n)}. In this range Brαf→fB_r^\alpha f\to f in LpL^p as r→∞r\to\infty if p<∞p<\infty. If 2≤p<∞2\le p<\infty and α>s+(p)\alpha>s_+(p), then ∥B∗αf∥Lp(Rn)≤Cn,p,α∥f∥Lp(Rn), \|B_*^\alpha f\|_{L^p(\RR^n)}\le C_{n,p,\alpha}\|f\|_{L^p(\RR^n)}, and Brαf→fB_r^\alpha f\to f almost everywhere as r→∞r\to\infty.

Where the new idea lives · 5 novel steps

Theorem 1.2Local smoothing on one frequency annulus
Bounds U(T)PNU(T)P_N on LpL^p at pcp_c with loss [log⁡(2+N)]2/p[\log(2+N)]^{2/p}
Proposition 5.1Energy barrier on a fixed fiber
Energy barrier: small input, large output, and ΔzF≤V(s)F\Delta_z\mathcal F\le V(s)\mathcal F
Proposition 6.3Periodic cap estimate with logarithmic loss
Bounds periodic cap waves in LpcL^{p_c} with loss [log⁡(2/h)]2/pc[\log(2/h)]^{2/p_c}
Theorem B.1Sharp logarithm with angular regularity
Angular regularity cuts the log loss to exponent 1/p1/p; weak type needs no log
Theorem E.1Local smoothing for symmetric pencil waves
Pencil waves obey LpcL^{p_c} smoothing with loss [log⁡(2/h)]2/pc[\log(2/h)]^{2/p_c}

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