claimroutine

Tube arrangement: disjoint start, compressed end

Arranges tubes: disjoint at start, outputs in a set of measure

\lesssim k^{-d}

geometric input of the counterexample

Claim·§C Disjoint packets and the weak spacetime norm·inside propWeak-type endpoint smoothing fails logarithmically

Statement (paper)

With

k=\lfloor\frac14\log_2 N\rfloor

\delta=2^{-k}

w=\alpha\delta/16

and

J=[1/4,1/2]

, the

\delta^{-d}

lines

L_\nu=(L_{j_1},\ldots,L_{j_d})

satisfy: for

X\le-1/2

the intervals of radius

w

around the

L_j(X)

are pairwise disjoint, and the output cores

S_\nu

all lie in

E=\{(X,Y):X\in J,\ Y\in (B_X)^d\}

, where

|E|\lesssim_n k^{-d}

and

\sum_\nu |S_\nu|=\delta^{-d}|J|w^d\asymp_n1

In plain English

With

\delta=2^{-k}

k=\lfloor\frac14\log_2 N\rfloor

, the

\delta^{-d}

product lines

L_\nu

start as pairwise disjoint tubes of radius

w=\alpha\delta/16

for

X\le-1/2

. Their output cores

S_\nu

all land in a set

E

(with

X\in J=[1/4,1/2]

and

Y\in(B_X)^d

) of measure

\lesssim_n k^{-d}

, while the cores' total measure is

\delta^{-d}|J|w^d\asymp_n 1

Why it's here

This is the geometric input of the counterexample: disjointness at the start lets the initial packets be summed with the

L^p

norm of

g_N

staying bounded. Compression of all the output cores into the small set

E

means the packets' mass concentrates there, which gives the

(\log N)^{d(1/2-1/p)}

gain in the weak-norm lower bound after random signs are applied in the later claims.

Classified routine:

A Kakeya/Besicovitch-type line arrangement (Perron-tree style construction in which disjoint tubes are compressed into a set of measure about 1/log), used in the standard Bourgain–Fefferman-type counterexamples; here it is a direct geometric verification for the specific product lines.

Uses

Nothing else in the paper

Used by

claimModulated packets transport along tubesclaimRandom signs give weak-norm lower bound

Sources

PaperOpen

Section: §C Disjoint packets and the weak spacetime norm
Page: —
Label: —
Line: sections/weak.tex:15

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