claimroutine

Smoothed density P_* is regular and controlled

Smoothed density

P_*

is positive, slowly varying, with controlled

L^{d+1}

mass

comparison weight for the barrier bounds

Claim·§5 Estimating the energy on a fixed fiber·inside propEnergy barrier on a fixed fiber

Statement (paper)

Let

d=n-1

s\in[2a,3]

D\ge1

a dimensional constant,

M=2n+4

, and

P_*(X)=\big((J_{Ds}*|g_T|^2)(X)\big)^{1/d}

, where convolution is in the real spatial variables

X

. Then

P_*>0

|\nabla_X^jP_*|\le C(Ds)^{-j}P_*

for

j=1,2

, and

\int P_*^{d+1}\,dX\le\|g_T\|_{\pc}^{\pc}

In plain English

Smoothing

|g_T|^2

in the spatial variables

X

with the kernel

J_{Ds}

and taking the

1/d

power (with

d=n-1

) gives a function

P_*

that is strictly positive, has

|\nabla_X^j P_*|\le C(Ds)^{-j}P_*

for

j=1,2

, and satisfies

\int P_*^{d+1}\,dX\le\|g_T\|_{\pc}^{\pc}

Why it's here

This supplies the comparison function

P_*

used in the later claims: positivity and the logarithmic-derivative bounds at scale

Ds

let it serve as a slowly varying weight, for example in the Hessian bound for

rP_*

. The integral bound lets the later energy estimates be converted into bounds by

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

. The integral bound should follow from Hölder/Jensen-type control of the smoothed density, since the

\pc

norm is the relevant

L^{d+1}

-type exponent.}

Classified routine:

Standard properties of convolution with a positive smooth bump kernel with tails (e.g. Poisson-like J_s): positivity, log-derivative bounds from |∇^j J_s| ≲ s^{-j} J_s, and the integral bound via Jensen/Young's inequality, with p_c = (d+1)·... matching exponents.

Uses

Nothing else in the paper

Used by

claimCurrent and density bounds with wall decayclaimHessian lower bound for rP_*claimEnergy and wall flux bounds

Sources

PaperOpen

Section: §5 Estimating the energy on a fixed fiber
Page: —
Label: —
Line: sections/barrier.tex:34

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