claimroutineinferred_dep

Periodic cap estimate gives Euclidean pencil estimate

Removes periodicity: gives Euclidean pencil bound with

[\log(2/h)]^{2/p_c}

completes the pencil theorem by rescaling

Claim·§E.3 The fixed cap estimate·inside thmLocal smoothing for symmetric pencil waves

Statement (paper)

The Euclidean estimate holds for every

b\in C_c^\infty(\Gamma)

0<h\le1

and Schwartz

f

: finitely many caps cover the directions in

\supp b

, and the periodic (torus) estimate applied on the same lattice to

C/M

and

b_M(\eta)=b(\eta/M)

, with

M\ge1

larger than

\sup_{\supp b}|\nabla\phi|

, yields by the cube argument and rescaling

f_M(X)=f(MX)

the bound

\|e^{-\mathrm iT\phi(D)}b(hD)f\|_{L^{p_c}(\RR^n\times[1,2])}\le C_b[\log(2/h)]^{2/p_c}\|f\|_{L^{p_c}(\RR^n)}

for small

h

, the remaining compact range of

h

follows from smooth multiplier estimates, and the assertion for a simple smallest eigenvalue with negative definite transverse Hessian follows by applying the result to

-C

with cutoff

\bar b

and conjugating with

f(X)\mapsto\overline{f(-X)}

In plain English

The Euclidean estimate

\|e^{-\mathrm iT\phi(D)}b(hD)f\|_{L^{p_c}(\RR^n\times[1,2])}\le C_b[\log(2/h)]^{2/p_c}\|f\|_{L^{p_c}}

holds for every

b\in C_c^\infty(\Gamma)

0<h\le1

and Schwartz

f

, and also for a simple smallest eigenvalue with negative definite transverse Hessian.

Why it's here

This step removes the periodicity assumption: the torus cap estimate from the previous claim is applied to rescaled data

C/M

b_M

, and then transferred to

\RR^n

by the cube argument and rescaling, with finitely many caps covering

\supp b

. Large

h

is handled by smooth multiplier bounds, and the smallest-eigenvalue case follows by passing to

-C

and conjugating. This completes the proof of the pencil theorem, which the paper uses to obtain the main annulus estimate with its logarithmic loss.

Classified routine:

Transfers the torus cap estimate of the previous claim to Euclidean space by rescaling and a standard periodization (cube) argument. It then covers

\mathrm{supp}\, b

by finitely many caps, handles the compact

h

-range with smooth multiplier bounds, and treats the other eigenvalue case by conjugation symmetry.

Uses

claimEnergy bounds give the torus cap estimatethmLocal smoothing on one frequency annulus

Used by

Nothing in the paper

Sources

PaperOpen

Section: §E.3 The fixed cap estimate
Page: —
Label: —
Line: sections/pencil.tex:283

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Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Remove period by scaling; smallest eigenvalue case" rests on claim-thm-pencil-6
info
inferred_dep
outline: "Remove period by scaling; smallest eigenvalue case" rests on thm:main