lemmaroutine

Local smoothing bound on small balls

Bounds

U(t)\Pi_Nf

B(x_0,r)

\min\{1,(Nr)^{n/p}\}L(Nr)^A

ball localization used in band and envelope bounds

Lemma·lem:angular-ball·§B.1 Observation on a ball

Statement (paper)

For

N\ge2

0<r\le1

and every

x_0\in\RR^n

\|U(t)\Pi_N f\|_{L^p(I\times B(x_0,r))} \lesssim \min\{1,(Nr)^{n/p}\}\,L(Nr)^A\|f\|_p .

In plain English

For

N\ge2

0<r\le1

and any center

x_0

, the

L^p

norm of

U(t)\Pi_N f

over

I\times B(x_0,r)

is bounded by

\min\{1,(Nr)^{n/p}\}\,L(Nr)^A\|f\|_p

up to a constant, so the bound improves with the size of the ball.

Why it's here

The lemma localizes the frequency-annulus estimate to a ball of radius

r

, showing a gain of

(Nr)^{n/p}

when

Nr<1

and only a

L(Nr)^A

loss otherwise. The proof splits into the trivial small-radius case, a comparison on radial shells with the profile operator

\mathcal T

(whose

L^p

bound comes from the main theorem), and a sum over shells with a negligible remainder. The ball estimate is then used for the angular-band bounds and the angular envelope lemma.

Classified routine:

A localization of the paper's main annulus local smoothing theorem to small balls. It combines a trivial Bernstein/volume bound for

Nr<1

with a decomposition into radial shells compared against the profile operator, which is standard bookkeeping following from the main theorem.

Uses

thmLocal smoothing on one frequency annulus

Used by

propAngular polynomial local smoothing boundlemAngular envelope bound and small-source estimateclaimStrong bound for unselected angular bands

Inside the proof

claimKernel asymptotics and small-radius caseclaimShell profile operator approximates propagatorclaimSumming shell estimates over radii

Sources

PaperOpen

Section: §B.1 Observation on a ball
Page: —
Label: lem:angular-ball
Line: sections/angular.tex:67

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