lemmaroutineinferred_dep

Maximal bound for thin shell multipliers

Bounds the maximal operator over thin shells of width δr\delta r by δ−σ\delta^{-\sigma}
shell-by-shell input for Bochner–Riesz
Lemma·lem:maximal-shell·§10 Bochner--Riesz means

Statement (paper)

For 0<δ≤1/80<\delta\le1/8 and ψ∈Cc∞((1/2,2))\psi\in C_c^\infty((1/2,2)), set Aδ,rf=ψ((1−∣D∣/r)/δ)f. A_{\delta,r}f=\psi\big((1-|D|/r)/\delta\big) f . For 2≤p<∞2\le p<\infty and σ>s+(p)\sigma>s_+(p), ∥sup⁡r>0∣Aδ,rf∣∥p≲δ−σ∥f∥p. \left\|\sup_{r>0}|A_{\delta,r} f|\right\|_p \lesssim \delta^{-\sigma}\|f\|_p . The constant is uniform for ψ\psi in bounded subsets of Cc∞((1/2,2))C_c^\infty((1/2,2)).

In plain English

For 0<δ≤1/80<\delta\le 1/8 and a bump ψ\psi supported in (1/2,2)(1/2,2), the multipliers Aδ,rA_{\delta,r} localize frequencies to a shell of width about δr\delta r around ∣ξ∣=r|\xi|=r. For 2≤p<∞2\le p<\infty and σ>s+(p)\sigma>s_+(p), the maximal function sup⁡r>0∣Aδ,rf∣\sup_{r>0}|A_{\delta,r}f| is bounded on LpL^p with norm ≲δ−σ\lesssim\delta^{-\sigma}, uniformly for ψ\psi in bounded subsets of Cc∞((1/2,2))C_c^\infty((1/2,2)).

Why it's here

This turns local smoothing into a maximal estimate over all radii for thin spherical shells, with a loss of δ−σ\delta^{-\sigma} just above the local smoothing exponent s+(p)s_+(p). The maximal bound is the shell-by-shell input that Bochner–Riesz theorem 10.1 needs. Summing over shells of width δ\delta near the sphere gives the Bochner–Riesz bound for α>δ(p)\alpha>\delta(p).
Classified routine:
Standard reduction of maximal Bochner–Riesz bounds to local smoothing via Fourier representation in t and Sobolev embedding in r, as in Mockenhaupt–Seeger–Sogge (1992) and Sogge's 'Fourier Integrals in Classical Analysis'; it follows from the paper's local smoothing theorem.

Uses

thmSobolev local smoothing for the wave equation

Used by

thmBochner–Riesz bounds from local smoothing

Sources

PaperOpen
Section
§10 Bochner--Riesz means
Page
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Label
lem:maximal-shell
Line
sections/consequences.tex:49
Leannot aligned

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Friction

  • info
    inferred_dep
    critique: The proof (consequences.tex:73-74) uses . Consequences.tex:34-35 introduces that estimate with "We rescale Theorem  to a unit frequency annulus".