claimroutineinferred_dep

Wall decay and interior lower bound

Gives wall decay of AwA_w and interior lower bound μ≥mX\mu\ge m_X
inputs for the barrier potential
Claim·§E.3 The fixed cap estimate·inside thmLocal smoothing for symmetric pencil waves

Statement (paper)

Throughout 2a≤s≤32a\le s\le3, with P∗=(JDs∗∣gT∣2)1/dP_*=(J_{Ds}*|g_T|^2)^{1/d} and DD fixed large, there is mm and c0>0c_0>0 independent of γ\gamma such that μ≲s−2dσmP∗d\mu\lesssim s^{-2d}\sigma^m P_*^d for r≤0r\le0 and ∥Aw∥≲s−2dσme−c0σP∗d\|A_{\rm w}\|\lesssim s^{-2d}\sigma^m e^{-c_0\sigma}P_*^d on the wall r=0r=0. The cofactor adds O(h50na−2s−2d)O(h^{50n}a^{-2}s^{-2d}) to μ\mu, and P∗d≳hn+4P_*^d\gtrsim h^{n+4}. Moreover, for every center with ∣W∣≤γ|W|\le\gamma, ∣Q∣≤1/400|Q|\le1/400, there is C1C_1 independent of γ\gamma such that μ≥mX=cγ(hs)−dσ−me−2C1γ2σ∣gT(X)∣2\mu\ge m_X=c_\gamma(hs)^{-d}\sigma^{-m}e^{-2C_1\gamma^2\sigma}|g_T(X)|^2 on the whole spatial ball of radius hh about that center, for KK large relative to 1/γ1/\gamma.

In plain English

For 2a≤s≤32a\le s\le3, the determinant-energy density μ\mu is bounded by s−2dσmP∗ds^{-2d}\sigma^m P_*^d for r≤0r\le0, and the wall tensor AwA_{\rm w} at r=0r=0 decays like s−2dσme−c0σP∗ds^{-2d}\sigma^m e^{-c_0\sigma}P_*^d. The cofactor correction is negligible, P∗d≳hn+4P_*^d\gtrsim h^{n+4}, and on the whole spatial ball of radius hh around any center with ∣W∣≤γ|W|\le\gamma, ∣Q∣≤1/400|Q|\le1/400, one has the pointwise lower bound μ≥mX=cγ(hs)−dσ−me−2C1γ2σ∣gT(X)∣2\mu\ge m_X=c_\gamma(hs)^{-d}\sigma^{-m}e^{-2C_1\gamma^2\sigma}|g_T(X)|^2 for KK large relative to 1/γ1/\gamma.

Why it's here

This claim supplies the two estimates needed for the barrier argument: an exponentially small wall flux, with constant c0c_0 independent of γ\gamma, and an interior lower bound that loses only a power of σ\sigma and a factor e−2C1γ2σe^{-2C_1\gamma^2\sigma}. It builds on the Gaussian-multiplier lower bound ATT0≥∣S0gT∣2A^0_{TT}\ge|\mathsf S_0g_T|^2 from the conserved current. Since C1γ2C_1\gamma^2 can be made small by choosing γ\gamma small, the exponential gain at the wall wins over the interior loss, which is what the energy bounds and the torus cap estimate in the next claim use.
Classified routine:
These are technical upper and lower bounds for the determinant-energy density and the wall tensor. They follow by direct estimation from the Gaussian-multiplier conserved current (claim-thm-pencil-3) and the fixed-fiber barrier setup, as bookkeeping inside the proof of the pencil local smoothing theorem.

Uses

claimConserved current with Gaussian multiplier symbolspropEnergy barrier on a fixed fiber

Used by

claimEnergy bounds give the torus cap estimate

Sources

PaperOpen
Section
§E.3 The fixed cap estimate
Page
—
Label
—
Line
sections/pencil.tex:213
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Wall decay and interior inversion estimates" rests on claim-thm-pencil-3
  • info
    inferred_dep
    outline: "Wall decay and interior inversion estimates" rests on prop:fixed-barrier