claimroutineinferred_dep

Global mixed paraboloid extension estimate

Paraboloid extension

L^a\to L^b

holds globally for

b>p_c

b>(d+2)a'/d

lossless global extension for the Besov smoothing

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

Statement (paper)

For

d\ge1

p_c=2(d+1)/d

1<a\le\infty

p_c<b<\infty

and

b>(d+2)a'/d

, the extension operator satisfies

\|Eg\|_{L^b(\RR^{d+1})}\lesssim_{d,a,b}\|g\|_{L^a(B_d)}

. In particular,

E:L^{p_c}(B_d)\to L^s(\RR^{d+1})

for every

s>p_c

In plain English

For

1<a\le\infty

p_c<b<\infty

and

b>(d+2)a'/d

, the paraboloid extension operator is bounded

L^a(B_d)\to L^b(\RR^{d+1})

. In particular

E

maps

L^{p_c}(B_d)

into

L^s(\RR^{d+1})

for every

s>p_c

Why it's here

The local logarithmic-loss bound at

p_c

from the previous claim is upgraded to a global, lossless bound by epsilon removal, using that the log loss can be traded for the room

b>p_c

and the range condition on

(a,b)

. The analogous sphere argument in the spherical restriction corollary guides this, and the resulting global bound feeds into the mixed-norm Besov local smoothing estimate for Schrödinger evolution.

Classified routine:

Follows from the preceding local log-loss bound via Tao's epsilon-removal lemma (Tao 1999), mirroring the paper's own argument for the sphere in cor-spherical-restriction; the resulting range is the classical Stein–Tomas/restriction-conjecture range for the paraboloid.

Uses

corGlobal spherical and cone extension estimatesclaimLocal paraboloid extension with logarithmic loss

Used by

claimMixed-norm Besov local smoothing for Schrödinger

Sources

PaperOpen

Section: §12 The paraboloid and Schr\"odinger evolution
Page: —
Label: —
Line: sections/parabolic.tex:72

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Friction

info
inferred_dep
outline: "Global mixed paraboloid extension by epsilon removal" rests on claim-cor-parabolic-1