propositionnovel

Energy barrier on a fixed fiber

Energy barrier: small input, large output, and

\Delta_z\mathcal F\le V(s)\mathcal F

fixed-fiber barrier feeding the periodic cap estimate

Proposition·prop:fixed-barrier·§5 Estimating the energy on a fixed fiber

Statement (paper)

Fix an annulus

c_1\le h|\xi|\le c_2

, with

0<c_1<c_2

. There are constants

\gamma>0

K\ge1

, and

c,C>0

, depending only on

n

and the annulus, with the following property. Let

a=Kh

0<h<1/(4K)

, and let

f

be a trigonometric polynomial on the fixed torus, with

\|f\|_2=1

and frequencies in this annulus and in

|v|\le\gamma

in some orthonormal frame. Write

g_T=e^{-iT|D|}f

and let

\mathcal F(T,\tau)

be its energy from Section . Then

\begin{aligned} \mathcal F(0,\tau)&\le C a^2\|f\|_{\pc}^{\pc} && (|\tau|=4a),\\ \mathcal F(T,\tau)&\ge c a^2\|g_T\|_{\pc}^{\pc} && (|\tau|=2a),\\ \Delta_z\mathcal F(T,\tau)&\le V(s)\mathcal F(T,\tau),\qquad V(s)=C s^{-2}e^{-c s/h} && (2a\le s=|\tau|\le3). \end{aligned}

The last two inequalities hold for all

T\in\mathbb R

In plain English

For a unit-norm trigonometric polynomial

f

with frequencies in a fixed annulus and a small cone

|v|\le\gamma

, with

a=Kh

and

g_T=e^{-iT|D|}f

, the fiber energy

\mathcal F(T,\tau)

is at most

Ca^2\|f\|_{\pc}^{\pc}

T=0

|\tau|=4a

, and at least

ca^2\|g_T\|_{\pc}^{\pc}

|\tau|=2a

for all

T

. Also

\Delta_z\mathcal F\le V(s)\mathcal F

with

V(s)=Cs^{-2}e^{-cs/h}

for

2a\le s=|\tau|\le3

Why it's here

This is the main fixed-fiber estimate: it gives an input bound, an output bound, and an exponentially small potential for the Laplacian of the energy, which later propositions (the periodic cap estimate and the pencil argument) use as a barrier. It combines the wave current, fiber geometry, Dirichlet barrier, energy comparison and energy flux results. The exponentially small wall flux is divided by an interior lower bound for

\mathcal F

that loses only

e^{-C_3\gamma^2\sigma}

and a power of

\sigma

. Choosing

\gamma

small then leaves net exponential decay.

Classified novel:

This is the paper's main fixed-fiber barrier estimate. It combines the new wave-current, quaternionic Monge–Ampère Dirichlet barrier and energy-flux machinery into input/output bounds and an exponentially small potential. No prior local smoothing argument supplies such an energy subharmonicity barrier.

Uses

propWave current is positive and divergence-freelemFiber Hessian geometry and JacobianlemHessian calculus in quaternionic coordinatesthmDirichlet problem for quaternionic Monge-AmpèrelemFiber energy comparable to X_0propEnergy flux identity with full wall normal

Used by

propPeriodic cap estimate with logarithmic lossclaimWall decay and interior lower boundclaimEnergy bounds give the torus cap estimate

Inside the proof

claimSmoothed density P_* is regular and controlledclaimCurrent and density bounds with wall decayclaimHessian lower bound for rP_*claimUpper bound on boundary quotient u/rclaimEnergy and wall flux boundsclaimInterior density lower bound by Gaussian inversionclaimInterior energy lower bound via ball quadraticclaimChoosing gamma and K yields the barrier inequalities

Figures

Panel (a) draws the fiber in modulus coordinates: the horizontal axis is $|W|$ with tick $w_0$, and the vertical axis is $|Q|=|\Phi|/a$ with ticks $1/20$ and $1/10$. A quarter-ellipse arc labeled $r=0$ runs from $(0,1/10)$ to $(w_0,0)$ and bounds the shaded region $r<0$. A dotted line at height $1/20$ meets the arc, and the part of the arc below it is drawn heavy and labeled "flux wall". A small dashed box near the origin, containing a smaller solid box, is marked (b). Panel (b) enlarges that corner: a dashed rectangle $|W|\le2\gamma$, $|Q|\le1/100$, labeled "Gaussian inversion, $g(Q)=1$", contains a darker solid rectangle $|W|\le\gamma$, $|Q|\le1/400$, labeled "centers". The figure shows that the region where the wall flux lives is separated from the central region where the Gaussian multiplier is inverted. — Figure 1schematicp. 19
The fixed fiber in modulus coordinates. The wave flux through $r=0$ is supported on the heavy arc, where $|Q|\le1/20$ and hence $|W|\ge\sqrt{3}\,w_0/2$ . This separation gives Gaussian decay $e^{-cs/h}$ even for $|v|\le w_0/3$ . In the enlargement (b), once $K\gamma$ is large, the radius- $h$ balls about the marked centers project into the dashed patch. There $|W-v|\le4\gamma$ on the enlarged cap $|v|\le2\gamma$ , so Gaussian inversion costs at most $e^{C\gamma^2s/h}$ . The aperture $\gamma$ is fixed independently of $h$ ; its size relative to $w_0$ is schematic.
What it shows
Panel (a) draws the fiber in modulus coordinates: the horizontal axis is $|W|$ with tick $w_0$ , and the vertical axis is $|Q|=|\Phi|/a$ with ticks $1/20$ and $1/10$ . A quarter-ellipse arc labeled $r=0$ runs from $(0,1/10)$ to $(w_0,0)$ and bounds the shaded region $r<0$ . A dotted line at height $1/20$ meets the arc, and the part of the arc below it is drawn heavy and labeled "flux wall". A small dashed box near the origin, containing a smaller solid box, is marked (b). Panel (b) enlarges that corner: a dashed rectangle $|W|\le2\gamma$ , $|Q|\le1/100$ , labeled "Gaussian inversion, $g(Q)=1$ ", contains a darker solid rectangle $|W|\le\gamma$ , $|Q|\le1/400$ , labeled "centers". The figure shows that the region where the wall flux lives is separated from the central region where the Gaussian multiplier is inverted.
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SupportspropEnergy barrier on a fixed fiberclaimCurrent and density bounds with wall decayclaimInterior density lower bound by Gaussian inversionclaimInterior energy lower bound via ball quadraticclaimChoosing gamma and K yields the barrier inequalities

Sources

PaperOpen

Section: §5 Estimating the energy on a fixed fiber
Page: —
Label: prop:fixed-barrier
Line: sections/barrier.tex:12

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Phase 2 aligns this node to a Lean declaration.