theoremnovelstructure_uncertain

Sharp logarithm with angular regularity

Angular regularity cuts the log loss to exponent

1/p

; weak type needs no log

the angular refinement of the main estimate

Theorem·thm:angular·§B The sharp logarithm with angular regularity

Statement (paper)

Let

n\ge2

p=\pc

, and

B>1/2+1/p

. Then

\begin{aligned} \|U(T)f\|_{L^p(\RR^n\times I)} &\le C_{n,B}\big(\|L(J)^{1/p}f\|_p+\|L(\Lambda_\omega)^B f\|_p\big),\\ \|U(T)f\|_{L^{p,\infty}(\RR^n\times I)} &\le C_{n,B}\|L(\Lambda_\omega)^B f\|_p . \end{aligned}

In each inequality the angular norm on the right can be replaced by

\|\Lambda_\omega^\sigma f\|_p

for any

\sigma>0

, with a constant depending on

\sigma

In plain English

For

n\ge2

, the critical exponent

p

, and

B>1/2+1/p

, the wave propagator

U(T)f

is bounded in

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

\|L(J)^{1/p}f\|_p

plus the angular Sobolev norm

\|L(\Lambda_\omega)^B f\|_p

. It is also bounded in weak

L^{p,\infty}(\RR^n\times I)

by the angular norm alone, and the angular norm can be replaced by

\|\Lambda_\omega^\sigma f\|_p

for any

\sigma>0

Why it's here

The proof splits each Littlewood-Paley piece into a selected low angular frequency part, handled by the envelope lemma and the maximal function (strong and weak bounds), and unselected angular bands, handled by the single-annulus bound and summed using the geometric decay of angular projections when

B>1/2+1/p

. The pieces are then reassembled by the Littlewood-Paley synthesis operator and interpolation. Angular regularity thus sharpens the logarithmic loss in the estimate to the exponent

1/p

, and gives a weak-type bound with no logarithm.

Classified novel:

Sharpening the endpoint logarithmic loss at the critical exponent to

L(J)^{1/p}

, together with a log-free weak-type bound under angular regularity, goes beyond known endpoint local smoothing results. Earlier work, e.g. Heo–Nazarov–Seeger-type radial/angular results, gives no such sharp logarithm in this form, which the appendix presents as its own contribution.

Uses

Nothing else in the paper

Used by

Nothing in the paper

Inside the proof

claimLittlewood-Paley setup and small-source removalclaimAngular band projections decay geometricallyclaimSelected low-pass dominated by envelope and maximal functionclaimStrong and weak bounds for selected piecesclaimStrong bound for unselected angular bandsclaimReassembly of pieces via synthesis operator

Sources

PaperOpen

Section: §B The sharp logarithm with angular regularity
Page: —
Label: thm:angular
Line: sections/angular.tex:9

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Friction

info
structure_uncertain
No proof environment; the proof is the prose of §B.3 (sections/angular.tex 276-404), concluding 'This proves Theorem B.1'.