lemmaroutine

Local smoothing implies local spherical extension

Turns annulus loss

L(N)^A

into extension on

B(z,R)

with

L(R)^A

bridge from wave smoothing to extension and Kakeya bounds

Lemma·lem:kakeya-local-extension·§9 Spherical extension and Kakeya tubes

Statement (paper)

Suppose holds with

L(N)^A

for some

A\ge0

, for a smooth radial cutoff

\chi(|\xi|/N)

with

\chi

supported in

(1/2,2)

and equal to one near

1

. For

R\ge2

z\in\RR^n

, and

g\in L^p(S^{n-1})

, the spherical extension

Eg(X)=\int_{S^{n-1}}e^{\mathrm iX\cdot\omega}g(\omega)\dd\sigma(\omega)

satisfies

\|Eg\|_{L^p(B(z,R))} \lesssim L(R)^A\|g\|_{L^p(S^{n-1})}.

In plain English

If the main local smoothing estimate holds with loss

L(N)^A

for a radial frequency cutoff

\chi(|\xi|/N)

at scale

N

, then the spherical extension operator

E

satisfies

\|Eg\|_{L^p(B(z,R))}\lesssim L(R)^A\|g\|_{L^p(S^{n-1})}

for every

R\ge2

, center

z

, and

g\in L^p(S^{n-1})

Why it's here

This lemma turns the wave-equation local smoothing hypothesis into a local extension bound on the sphere with the same loss

L(R)^A

. That bound is the input for the Kakeya-type estimates, the global extension and paraboloid corollaries, and the overlap lemma that use it.

Classified routine:

Standard implication that local smoothing implies Fourier restriction/extension estimates (as in Mockenhaupt–Seeger–Sogge and Tao's observation that local smoothing implies restriction and Kakeya); here it is obtained by rescaling and testing the wave estimate on data concentrated near the sphere.

Uses

Nothing else in the paper

Used by

corQuantitative extension and Kakeya boundscorGlobal spherical and cone extension estimateslemKakeya overlap bound for tubescorParaboloid extension and Schrödinger local smoothingclaimKakeya maximal estimate by dualityclaimLocal paraboloid extension with logarithmic loss

Sources

PaperOpen

Section: §9 Spherical extension and Kakeya tubes
Page: —
Label: lem:kakeya-local-extension
Line: sections/kakeya.tex:49

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