claimroutine

Far-field Schrödinger kernel decay

Bounds the Schrödinger kernel for

|z|\ge8N

N^d(1+N|z|)^{-M}

disposes of far-field data in the cube localization

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

Statement (paper)

Let

\chi\in C_c^\infty(B(0,2))

equal one on

B(0,1)

and

P_N=\chi(D/N)

, so

f=P_Nf

, and tile

\RR^d

by cubes

Q

of side

N

. For

t\in I_0=[1/2,1]

and

|z|\ge 8N

, the kernel of

S(t)P_N

is bounded by

C_M N^d(1+N|z|)^{-M}

. Consequently, on each cube

Q

the contribution to

S(t)f

from outside the concentric cube

30Q

is bounded by the convolution of this majorant with

|f|

, and for

M>d

this convolution has

L^p

norm

O(\|f\|_p)

uniformly over all cubes

Q

In plain English

For

t\in[1/2,1]

and

|z|\ge 8N

, the kernel of

S(t)P_N

is bounded by

C_M N^d(1+N|z|)^{-M}

. So on a cube

Q

of side

N

, the part of

S(t)f

coming from data outside

30Q

is controlled by

|f|

convolved with this majorant, which has

L^p

norm

O(\|f\|_p)

uniformly in

Q

once

M>d

Why it's here

This handles the far part of the localization of

S(t)f

to cubes: the kernel's rapid decay away from the diagonal makes this contribution harmless, with only a trivial

\|f\|_p

bound. That leaves only the near part

h=f\mathbf 1_{30Q}

, which the next claim treats with the Fresnel formula to obtain the critical estimate.

Classified routine:

A standard non-stationary phase (repeated integration by parts) bound for the frequency-localized Schrödinger kernel away from the region

|z|\lesssim N t

, followed by Young's inequality; this is the usual finite-speed-of-propagation localization step for Schrödinger evolution.

Uses

Nothing else in the paper

Used by

claimCritical Schrödinger estimate with logarithmic loss

Sources

PaperOpen

Section: §12 The paraboloid and Schr\"odinger evolution
Page: —
Label: —
Line: sections/parabolic.tex:86

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