claimroutine

Local paraboloid extension with logarithmic loss

Bounds local paraboloid extension on

B(z,R)

with loss

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

local extension feeding global and critical Schrödinger bounds

Claim·§12 The paraboloid and Schr\"odinger evolution·inside corParaboloid extension and Schrödinger local smoothing

Statement (paper)

For every

A

admissible in Lemma (in particular

A=2/p_c

), the paraboloid extension operator

E

satisfies

\|Eg\|_{L^{p_c}(B(z,R))}\lesssim_{d,A}[\log(2+R)]^A\|g\|_{p_c}

for all

z\in\RR^{d+1}

and

R\ge1

In plain English

The paraboloid extension operator

E

satisfies a local

L^{p_c}

bound on any ball

B(z,R)

\RR^{d+1}

with

R\ge1

, with loss

[\log(2+R)]^A

for each admissible

A

(in particular

A=2/p_c

\|Eg\|_{L^{p_c}(B(z,R))}\lesssim_{d,A}[\log(2+R)]^A\|g\|_{p_c}

Why it's here

The local spherical extension bound from the Kakeya-type lemma, which follows from the main local smoothing theorem, is transferred to the paraboloid by rescaling small caps of the sphere, which look like paraboloid pieces. This gives the logarithmic local estimate that the later claims use, both for the global mixed extension bound via epsilon removal and for the near-contribution critical Schrödinger estimate.

Classified routine:

Follows from the local spherical extension lemma by the standard parabolic rescaling of small spherical caps into paraboloid pieces, the classical sphere-to-paraboloid transfer used in restriction theory.

Uses

lemLocal smoothing implies local spherical extensionthmLocal smoothing on one frequency annulus

Used by

claimGlobal mixed paraboloid extension estimateclaimCritical Schrödinger estimate with logarithmic loss

Sources

PaperOpen

Section: §12 The paraboloid and Schr\"odinger evolution
Page: —
Label: —
Line: sections/parabolic.tex:45

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