claimroutineinferred_dep

Shell profile operator approximates propagator

Shows profile operator T\mathcal T is LpL^p bounded and approximates a shell's evolution
single-shell estimate independent of the radius
Claim·§B.1 Observation on a ball·inside lemLocal smoothing bound on small balls

Statement (paper)

Suppose 2<R≤T2<R\le T and S=100R3+n/pS=100R^{3+n/p}. For h(u,ω)h(u,\omega) vanishing for ∣u∣>5R|u|>5R and Th(b,x)=∫k(b−u+x⋅ω)h(u,ω) du dω\mathcal T h(b,x)=\int k(b-u+x\cdot\omega)h(u,\omega)\dd u\dd\omega, one has ∥Th∥Lp({∣b∣,∣x∣≤R})≲L(R)A∥h∥Lp( du dω)\|\mathcal T h\|_{L^p(\{|b|,|x|\le R\})}\lesssim L(R)^A\|h\|_{L^p(\dd u\dd\omega)}. Moreover, for Q≥SQ\ge S and fQ(sω)=s−αh(s−Q,ω)f_Q(s\omega)=s^{-\alpha}h(s-Q,\omega), ∥U(Q+b)Π1fQ−Th∥Lp({∣b∣,∣x∣≤R})≲(R2+n/p/Q)∥h∥Lp( du dω)\|U(Q+b)\Pi_1 f_Q-\mathcal T h\|_{L^p(\{|b|,|x|\le R\})}\lesssim (R^{2+n/p}/Q)\|h\|_{L^p(\dd u\dd\omega)}.

In plain English

For 2<R≤T2<R\le T and S=100R3+n/pS=100R^{3+n/p}, the profile operator Th(b,x)=∫k(b−u+x⋅ω)h(u,ω) du dω\mathcal T h(b,x)=\int k(b-u+x\cdot\omega)h(u,\omega)\,du\,d\omega (for hh supported in ∣u∣≤5R|u|\le5R) is bounded in LpL^p on {∣b∣,∣x∣≤R}\{|b|,|x|\le R\} with norm ≲L(R)A\lesssim L(R)^A. For a radial shell datum fQ(sω)=s−αh(s−Q,ω)f_Q(s\omega)=s^{-\alpha}h(s-Q,\omega) with Q≥SQ\ge S, U(Q+b)Π1fQU(Q+b)\Pi_1 f_Q differs from Th\mathcal T h there by ≲(R2+n/p/Q)∥h∥Lp\lesssim (R^{2+n/p}/Q)\|h\|_{L^p}.

Why it's here

This gives a bound independent of the radius for one shell: the kernel asymptotics from the previous claim turn U(Q+b)Π1U(Q+b)\Pi_1 on a shell at distance QQ into the fixed operator T\mathcal T, with an error that decays in QQ. The next claim sums over shells, using T\mathcal T for the main part and the 1/Q1/Q error for the remainder, with the main theorem presumably used for the LpL^p bound on T\mathcal T.
Classified routine:
A step inside the proof of the small-ball lemma: it combines the kernel asymptotics from the previous claim with a Taylor expansion of the phase for ∣x∣eeReeQ|x| e e R e e Q, and uses the main theorem for the LpL^p bound on T\mathcal T. This is standard stationary-phase bookkeeping for radial shell data, not a new idea of the paper.

Uses

thmLocal smoothing on one frequency annulusclaimKernel asymptotics and small-radius case

Used by

claimSumming shell estimates over radii

Sources

PaperOpen
Section
§B.1 Observation on a ball
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sections/angular.tex:103
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Shell profile estimate independent of radius" rests on claim-lem-angular-ball-1