claimroutine

Kernel asymptotics and small-radius case

Rescales the ball problem and settles Nr≤2Nr\le2 using kernel asymptotics
setup and small-radius case of the ball lemma
Claim·§B.1 Observation on a ball·inside lemLocal smoothing bound on small balls

Statement (paper)

For N≥2N\ge2, 0<r≤10<r\le1 with R=Nr≤2R=Nr\le2 and every x0∈Rnx_0\in\RR^n, after translating to x0=0x_0=0 and rescaling to frequency one with T=NT=N, one has ∥U(t)Π1f∥Lp([T,2T]×B(0,R))≲T1/pmin⁡(1,Rn/p)L(R)A∥f∥p\|U(t)\Pi_1 f\|_{L^p([T,2T]\times B(0,R))}\lesssim T^{1/p}\min(1,R^{n/p})L(R)^A\|f\|_p. This rests on the kernel KK of U(t)Π1U(t)\Pi_1 satisfying, for every DD and v=∣z∣≥1v=|z|\ge1, K(t,z)=v−αk(t−v)+OD(v−α−1⟨t−v⟩−D+v−α⟨t+v⟩−D)K(t,z)=v^{-\alpha}k(t-v)+O_D\big(v^{-\alpha-1}\langle t-v\rangle^{-D}+v^{-\alpha}\langle t+v\rangle^{-D}\big), with K(t,z)=OD(⟨t⟩−D)K(t,z)=O_D(\langle t\rangle^{-D}) for v<1v<1, and on the pointwise bound ∥U(t)Π1f(x)∥Lp([T,2T])≲T1/p∥f∥p\|U(t)\Pi_1 f(x)\|_{L^p([T,2T])}\lesssim T^{1/p}\|f\|_p for every xx.

In plain English

After translating to x0=0x_0=0 and rescaling to frequency one with T=NT=N, the lemma for R=Nr≤2R=Nr\le2 reduces to the bound ∥U(t)Π1f∥Lp([T,2T]×B(0,R))≲T1/pmin⁡(1,Rn/p)L(R)A∥f∥p\|U(t)\Pi_1 f\|_{L^p([T,2T]\times B(0,R))}\lesssim T^{1/p}\min(1,R^{n/p})L(R)^A\|f\|_p. The kernel KK of U(t)Π1U(t)\Pi_1 is a profile v−αk(t−v)v^{-\alpha}k(t-v) at v=∣z∣≥1v=|z|\ge1 up to rapidly decaying errors, and is OD(⟨t⟩−D)O_D(\langle t\rangle^{-D}) for v<1v<1.

Why it's here

This sets up the rescaled problem and records the kernel asymptotics that the later large-radius claims use to compare the operator with the profile operator T\mathcal T. The small-radius case R≤2R\le2 follows from the pointwise-in-xx bound ∥U(t)Π1f(x)∥Lp([T,2T])≲T1/p∥f∥p\|U(t)\Pi_1 f(x)\|_{L^p([T,2T])}\lesssim T^{1/p}\|f\|_p, integrated over a ball of volume about min⁡(1,Rn)\min(1,R^n), which gives the factor min⁡(1,Rn/p)\min(1,R^{n/p}).
Classified routine:
Follows from the standard stationary-phase asymptotics of the frequency-localized half-wave kernel (as in Stein's Harmonic Analysis or Sogge's Fourier Integrals in Classical Analysis), together with a pointwise Hölder/Young bound integrated over a small ball. This is rescaling bookkeeping.

Uses

Nothing else in the paper

Used by

claimShell profile operator approximates propagator

Sources

PaperOpen
Section
§B.1 Observation on a ball
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