propositionroutineinferred_dep

Angular polynomial local smoothing bound

Angular polynomials of degree

M

obey a bound

L(N)^{1/p}+L(M)^A

bounds the selected pieces in the angular proof

Proposition·prop:angular-band·§B The sharp logarithm with angular regularity

Statement (paper)

For

N\ge2

and an angular polynomial

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

of degree at most

M

\begin{aligned} \|U(t)\Pi_N f\|_{L^p(I\times\RR^n)} &\lesssim [L(N)^{1/p}+L(M)^A]\|f\|_p,\\ \|U(t)\Pi_N f\|_{L^{p,\infty}(I\times\RR^n)} &\lesssim L(M)^A\|f\|_p . \end{aligned}

The constants are independent of

M

and

N

In plain English

For

N\ge2

and a function

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

that is an angular polynomial of degree at most

M

, the norm

\|U(t)\Pi_N f\|_{L^p(I\times\RR^n)}

is at most a constant times

[L(N)^{1/p}+L(M)^A]\|f\|_p

. The weak-type

L^{p,\infty}

norm is bounded by

L(M)^A\|f\|_p

, with constants independent of

M

and

N

Why it's here

This proposition says that angular regularity of the data removes the

N

-dependence of the logarithmic loss in the weak-type bound and leaves it only as

L(N)^{1/p}

in the strong bound, with the rest depending on

M

alone. It is proved by combining the small-ball estimate (Lemma B.3) near the origin with the pointwise angular envelope (Lemma B.4) away from it, and the annulus theorem supplies the base bound. Later steps use it for the strong and weak bounds on the selected pieces in the angular argument.

Classified routine:

Obtained by combining the paper's earlier small-ball estimate (Lemma B.3) near the origin with the angular envelope bound (Lemma B.4) away from it, using the annulus theorem as the base bound. This is a decomposition-and-assembly argument in an appendix refinement.

Uses

lemLocal smoothing bound on small ballslemAngular envelope bound and small-source estimatethmLocal smoothing on one frequency annulus

Used by

claimStrong and weak bounds for selected pieces

Sources

PaperOpen

Section: §B The sharp logarithm with angular regularity
Page: —
Label: prop:angular-band
Line: sections/angular.tex:33

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
"the unrestricted annular estimate gives both conclusions" when H>N^eta refers to the one-frequency-annulus estimate of Theorem 1.2