lemmaroutineinferred_dep

Angular envelope bound and small-source estimate

Bounds

U(t)\Pi_Ng

pointwise by profile envelopes; sources near origin negligible

reduces the solution to a profile and a maximal function

Lemma·lem:angular-envelope·§B.2 The polar kernel and the two endpoint norms

Statement (paper)

There is an integer

K

, independent of

M,N

, with the following property. Suppose

g

has angular degree at most

M

, is supported on

s=|y|\ge s_0

, and

N\min(r,s_0)\ge (2+M)^K

. Then for

t\in I

|U(t)\Pi_N g(r\theta)|\lesssim \sum_{\pm}G_g(t\pm r,\theta) \begin{cases} r^{-\alpha},&r\le4,\\r^{-d/p},&r>4.\end{cases}

For arbitrary

g\in L^p(\RR^n)

, the part supported in

|y|<N^{\eta-1}

, where

\eta=\alpha/(2n)

, satisfies

\|U(t)\Pi_N(\mathbf1_{|y|<N^{\eta-1}}g)\|_{L^p(I\times\RR^n)} \lesssim N^{-\delta}\|g\|_p,\qquad \delta=(\alpha/2)(1-2/p)>0 .

In plain English

For

g

of angular degree at most

M

, supported in

|y|\ge s_0

, with

N\min(r,s_0)\ge(2+M)^K

, the function

|U(t)\Pi_N g(r\theta)|

is bounded by

\sum_\pm G_g(t\pm r,\theta)

times

r^{-\alpha}

for

r\le4

and

r^{-d/p}

for

r>4

. For arbitrary

g\in L^p

, the part supported in

|y|<N^{\eta-1}

with

\eta=\alpha/(2n)

has

L^p(I\times\RR^n)

norm of

U(t)\Pi_N

at most

N^{-\delta}\|g\|_p

up to a constant, where

\delta=(\alpha/2)(1-2/p)>0

Why it's here

The first part gives a pointwise envelope in terms of the profile

G_g

evaluated at

t\pm r

, which lets later steps control the solution by a one-dimensional-in-time profile and a vector maximal function. The second part uses the small-ball estimate (Lemma B.3) to show that sources very close to the origin contribute a negligible

N^{-\delta}

amount, so they can be removed from the Littlewood–Paley argument.

Classified routine:

A technical step in the appendix's polar-kernel analysis: a pointwise envelope from the asymptotics of the radial/Bessel kernel for low angular degree, plus a direct consequence of the small-ball estimate (Lemma B.3) for sources near the origin. It is bookkeeping rather than a new idea.

Uses

lemLocal smoothing bound on small balls

Used by

propAngular polynomial local smoothing boundclaimLittlewood-Paley setup and small-source removalclaimSelected low-pass dominated by envelope and maximal function

Sources

PaperOpen

Section: §B.2 The polar kernel and the two endpoint norms
Page: —
Label: lem:angular-envelope
Line: sections/angular.tex:167

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
critique: Angular.tex:235-236 reads "Finally and scaling give \|U(t)\Pi_N\|_{1\to\infty}\lesssim N^{\alpha+1}". That kernel expansion is established in the proof of Lemma (angular.tex:81-90).