theoremroutine

Bochner–Riesz bounds from local smoothing

Bochner–Riesz means bounded on LpL^p for α>δ(p)\alpha>\delta(p), with maximal bounds
the headline application to Bochner–Riesz
Theorem·thm:bochner-riesz·§10 Bochner--Riesz means

Statement (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.

In plain English

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.

Why it's here

This is the paper's application of local smoothing to Bochner–Riesz summation. The multiplier is decomposed into dyadic shells of width δ\delta near the sphere, each treated using the shell maximal bound with loss δ−σ\delta^{-\sigma}. The gain δα\delta^{\alpha} from the multiplier's smoothness at the sphere makes the sum over shells converge once α\alpha exceeds the relevant exponent, and this gives the uniform bounds, the maximal bound, and the convergence statements.
Classified routine:
This follows the known implication from local smoothing/shell estimates to Bochner–Riesz bounds (as in Sogge's and Carbery's work). It decomposes the multiplier into dyadic shells near the sphere, applies the shell maximal lemma, and sums the resulting geometric series in δ^{α−σ}.

Uses

lemMaximal bound for thin shell multipliers

Used by

Nothing in the paper

Sources

PaperOpen
Section
§10 Bochner--Riesz means
Page
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Label
thm:bochner-riesz
Line
sections/consequences.tex:16
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