corollaryroutine

Paraboloid extension and Schrödinger local smoothing

So paraboloid extension and Schrödinger smoothing hold with log loss at

p_c

payoff: carries the main result over to Schrödinger

Corollary·cor:parabolic·§12 The paraboloid and Schr\"odinger evolution

Statement (paper)

Let

d\ge1

and

p_c=2(d+1)/d

. If

1<a\le\infty

p_c<b<\infty

and

b>(d+2)a'/d

, then

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

Let

I\subset\RR

be a compact interval. For

N\ge2

and Schwartz

f

with

\supp\widehat f\subset B(0,N)\subset\RR^d

\|e^{it\Delta}f\|_{L^{p_c}(\RR^d\times I)} \lesssim_{d,I} [\log(2+N)]^{2/p_c}\|f\|_{p_c}.

For

p_c<p<\infty

and

p\le r\le\infty

, arbitrary Schwartz data satisfy

\|e^{it\Delta}f\|_{L^p(\RR^d;L^r(I))} \lesssim_{d,p,r,I}\|f\|_{B^s_{p,p}(\RR^d)} \lesssim_{d,p,s}\|(1-\Delta)^{s/2}f\|_p, \qquad s=d(1-2/p)-2/r .

Here

B^s_{p,p}

is the inhomogeneous Besov space. Taking

r=p

gives the exact spacetime Sobolev order

d-2(d+1)/p

; taking

r=\infty

gives the corresponding maximal estimate.

In plain English

The paraboloid extension operator

E

maps

L^a(B_d)

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

for

1<a\le\infty

b>p_c=2(d+1)/d

and

b>(d+2)a'/d

. Consequently

e^{it\Delta}

obeys a critical

L^{p_c}

bound with loss

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

for frequencies in

B(0,N)

, and for

p>p_c

and

p\le r\le\infty

a mixed-norm

L^p_xL^r_t

bound from the Besov space

B^s_{p,p}

and the Sobolev norm

\|(1-\Delta)^{s/2}f\|_p

, with

s=d(1-2/p)-2/r

Why it's here

This transfers the paper's main local smoothing result from the wave equation to the Schrödinger equation. A local logarithmic-loss extension bound for the sphere is rescaled to the paraboloid. Epsilon removal then gives the global extension bound and the exact-order Schrödinger estimates, including the maximal case

r=\infty

. A Fresnel-kernel localization to cubes gives the critical logarithmic estimate.

Classified routine:

Obtained from the paper's spherical extension corollary by the standard rescaling of the sphere to the paraboloid, Tao's epsilon-removal, and a Fresnel-kernel localization to cubes. These transfer the wave-equation results to Schrödinger by known techniques, so this is a direct consequence of the earlier nodes.

Uses

lemLocal smoothing implies local spherical extensionthmLocal smoothing on one frequency annuluscorGlobal spherical and cone extension estimates

Used by

Nothing in the paper

Inside the proof

claimLocal paraboloid extension with logarithmic lossclaimGlobal mixed paraboloid extension estimateclaimFar-field Schrödinger kernel decayclaimCritical Schrödinger estimate with logarithmic lossclaimMixed-norm Besov local smoothing for Schrödinger

Sources

PaperOpen

Section: §12 The paraboloid and Schr\"odinger evolution
Page: —
Label: cor:parabolic
Line: sections/parabolic.tex:20

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