claimroutineinferred_dep

Energy bounds give the torus cap estimate

Energy bounds give the torus cap estimate with [log⁡(2/h)]2[\log(2/h)]^2 loss
periodic estimate for pencil waves
Claim·§E.3 The fixed cap estimate·inside thmLocal smoothing for symmetric pencil waves

Statement (paper)

For the cap under consideration, choose γ\gamma with 2qC1γ2<c0/22qC_1\gamma^2<c_0/2 and then fix KK. Then the energy F\mathcal F satisfies cγa2σ−me−2qC1γ2σ∥gT∥pp≤F≤Ca2σm∥gT∥ppc_\gamma a^2\sigma^{-m}e^{-2qC_1\gamma^2\sigma}\|g_T\|_p^p\le\mathcal F\le Ca^2\sigma^m\|g_T\|_p^p and ΔT,τF≤Ca2s−2σme−c0σ∥gT∥pp\Delta_{T,\tau}\mathcal F\le Ca^2s^{-2}\sigma^m e^{-c_0\sigma}\|g_T\|_p^p, hence ΔF≤Cs−2e−cs/hF\Delta\mathcal F\le Cs^{-2}e^{-cs/h}\mathcal F, and consequently ∫12∥gT∥pp  dT≲[log⁡(2/h)]2∥f∥pp\int_1^2\|g_T\|_p^p\,\dd T\lesssim[\log(2/h)]^2\|f\|_p^p (after removing the normalization by homogeneity).

In plain English

For a cap with small γ\gamma (so 2qC1γ2<c0/22qC_1\gamma^2<c_0/2) and KK then fixed, the fiber energy F\mathcal F is bounded above and below by a2σ±m∥gT∥ppa^2\sigma^{\pm m}\|g_T\|_p^p, with an exponential loss e−2qC1γ2σe^{-2qC_1\gamma^2\sigma} in the lower bound. Its Laplacian satisfies ΔF≤Cs−2e−cs/hF\Delta\mathcal F\le Cs^{-2}e^{-cs/h}\mathcal F, and this yields ∫12∥gT∥pp  dT≲[log⁡(2/h)]2∥f∥pp\int_1^2\|g_T\|_p^p\,\dd T\lesssim[\log(2/h)]^2\|f\|_p^p.

Why it's here

This is where the wall decay and the interior lower bound from the previous claims are combined. The wall flux decays like e−c0σe^{-c_0\sigma} and the interior lower bound loses only e−2qC1γ2σe^{-2qC_1\gamma^2\sigma} and a power of σ\sigma. Choosing γ\gamma small makes the net decay exponential, which gives a potential V(s)=Cs−2e−cs/hV(s)=Cs^{-2}e^{-cs/h} for ΔF≤VF\Delta\mathcal F\le V\mathcal F. This is exactly what the thin-cylinder lemma and the fixed-fiber barrier need to give the periodic cap estimate with logarithmic loss for the pencil wave, which the next step then uses to remove the period.
Classified routine:
Combines the earlier wall-decay and interior lower-bound claims through a choice of parameters, giving the potential bound ΔF≤VF\Delta\mathcal F\le V\mathcal F that feeds the paper's thin-cylinder lemma and fixed-fiber barrier. The new ideas sit in those earlier components; this step is a direct assembly of them.

Uses

claimDeterminant energy and balance on pencil fibersclaimWall decay and interior lower boundpropEnergy barrier on a fixed fiberlemPropagation across a thin cylinderpropPeriodic cap estimate with logarithmic loss

Used by

claimPeriodic cap estimate gives Euclidean pencil estimate

Sources

PaperOpen
Section
§E.3 The fixed cap estimate
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sections/pencil.tex:249
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Energy bounds, potential inequality, torus cap estimate" rests on claim-thm-pencil-4
  • info
    inferred_dep
    outline: "Energy bounds, potential inequality, torus cap estimate" rests on claim-thm-pencil-5
  • info
    inferred_dep
    outline: "Energy bounds, potential inequality, torus cap estimate" rests on prop:fixed-barrier
  • info
    inferred_dep
    outline: "Energy bounds, potential inequality, torus cap estimate" rests on lem:cylinder
  • info
    inferred_dep
    outline: "Energy bounds, potential inequality, torus cap estimate" rests on prop:periodic-cap