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Title On a conjecture of E.M.Stein on the Hilbert transform on vector fields English 670.8 KB 87
```                            Preface
Chapter 1. Overview of principal results
Chapter 2. Besicovitch set and Carleson's Theorem
Besicovitch set
The Kakeya maximal function
Carleson's Theorem
The weak  L 2  estimate in Theorem 1.15 is sharp
Chapter 3. The Lipschitz Kakeya maximal function
The weak  L 2  estimate
An obstacle to an  Lp estimate, for  1<p<2
Bourgain's geometric condition
Vector fields that are a function of one variable
Chapter 4. The  L2 estimate
Definitions and principal Lemmas
Truncation and an alternate model sum
Proofs of Lemmata
Chapter 5. Almost orthogonality between annuli
Application of the Fourier localization Lemma
The Fourier localization estimate
References
```
##### Document Text Contents
Page 1

MEMOIRSof the
American Mathematical Society

Number 965

On a Conjecture of E. M. Stein
on the Hilbert Transform

on Vector Fields
Michael Lacey
Xiaochun Li

May 2010 � Volume 205 � Number 965 (fourth of 5 numbers) � ISSN 0065-9266

American Mathematical Society

Page 2

May 2010 € Volume 205 € Number 965 (fourth of 5 numbers) € ISSN 0065-9266

On a Conjecture of E. M. Stein
on the Hilbert Transform

on Vector Fields

Michael Lacey
Xiaochun Li

Number 965

Page 43

34 4. THE L2 ESTIMATE

Proofs of Lemmata

Proof of Lemma 4.33. We have Φ = A− +A+ +B, so from (4.28), Lemma
4.36 and Lemma 4.37, we deduce that ‖A−‖p � 1 for all 2 < p < ∞. It remains
for us to verify that A− is of restricted weak type p0 for some choice of 1 < p0 < 2.
That is, we should verify that for all sets F,G ⊂ R2 of finite measure
(4.39) |〈A− 1F ,1G〉| � |F |1/p|G|1−1/p, p0 < p < 2.
Since A− maps L

p into itself for 2 < p < ∞, it suffices to consider the case of
|F | < |G|. Since we assume only that the vector field is Lipschitz, we can use a
dilation to assume that 1 < |G| < 2, and so this set will not explicitly enter into
our estimates.

We fix the data F ⊂ R2 of finite measure, ann, and vector field v with ‖v‖Lip ≤
κann. Take p0 = 2 − κ2. We need a set of definitions that are inspired by the
approach of Lacey and Thiele , and are also used in Lacey and Li . For
subsets S ⊂ Av := {s ∈ AT (ann) | κ−1‖v‖Lip ≤ scl(s) < κann}, set

AS =

s∈S

〈Sann 1F , αs〉as−

Set χ(x) = (1 + |x|)−1000/κ. Define

(4.40) χ
(p)
Rs

:= χ(p)s = Tc(Rs) D
p
Rs

χ, 0 ≤ p ≤ ∞.

And set χ̃
(p)
s = 1γsRsχ

(p)
s .

Remark 4.41. It is typical to define a partial order on tiles, following an
observation of C. Fefferman . In this case, there doesn’t seem to be an appropriate
partial order. Begin with this assumption on the order relation ‘<’ on tiles:

(4.42) If ωs ×Rs ∩ ωs′ ×Rs′ �= ∅, then s and s′ are comparable under ‘<’.
It follows from transitivity of a partial order that that one can have tiles s1, . . . , sJ ,
with sj+1 < sj for 1 ≤ j < J , J
log(‖v‖Lip · ann), and yet the rectangles
RsJ and Rs1 can be far apart, namely RsJ ∩ (cJ)Rs1 , for a positive constant c.
See Figure 4.3. (We thank the referee for directing us towards this conclusion.)
Therefore, one cannot make the order relation transitive, and maintain control of
the approximate localization of spatial variables, as one would wish. The partial
order is essential to the argument of , but while it is used in , it is not essential
to that argument.

We recall a fact about the eccentricity. There is an absolute constant K ′ so
that for any two tiles s, s′

(4.43) ωs ⊃ ωs′ , Rs ∩Rs′ �= ∅ implies Rs ⊂ K ′Rs′ .
Figure 3.1 illustrates this in the case where the two rectangles Rs and Rs′ have
different widths, which is not the case here.

We define an order relation on tiles by s � s′ iff ωs � ωs′ and Rs ⊂ κ−10Rs′ .
Thus, (4.42) holds for this order relation, and it is certainly not transitive.

A tree is a collection of tiles T ⊂ Av, for which there is a (non–unique) tile
ωT × RT ∈ AT (ann) with Rs ⊂ 100κ−10RT, and ωs ⊃ ωT for all s ∈ T. Here, we
deliberately use a somewhat larger constant 100κ−10 than we used in the definition
of the order relation ‘�.’

Page 44

PROOFS OF LEMMATA 35

Rs1

Rs2

RsJ

Figure 4.3. The rectangles Rs1 , . . . , RsJ of Remark 4.41.

For j = 1, 2, call T a i–tree if the tiles for all s, s′ ∈ T, if scl(s) �= scl(s′),
then ωsi ∩ ωs′i = ∅. 1–trees are especially important. A few tiles in such a tree are
depicted in Figure 4.4.

Remark 4.44. This remark about the partial order ‘�’ and trees is useful to
us below. Suppose that we have two trees T, with top s(T) and T′ with top s(T′).
Suppose in addition that s(T′) � s(T). Then, it is the case that T ∪ T′ is a tree
with top s(T). Indeed, we must necessarily have ωT � ωT, since the Rs are from
products of a central grid. Also, 100κ−1RT′ ⊂ 100κ−1RT. And so every tile in T′
could also be a tile in T.

Our proof is organized around these parameters and functions associated to
tiles and sets of tiles. Of particular note here are the first definitions of ‘density,’
which have to be formulated to accommodate the lack of transitivity in the partial
order. Note that in the first definition, the supremum is taken over tiles s′ ∈ AT of
the same annular parameter as s. We choose the collection AT as it is ‘universal,’
covering all scales in a uniform way, due to different assumptions including (4.7).

dense(S) := sup
s′∈AT

{∫
G∩v−1(ωs′ )

χ̃
(1)
s′ dx | ∃ s , s

′′ ∈ S :

ωs ⊃ ωs′ ⊃ ωs′′ , Rs ⊂ 100κ−10Rs′ ,

Rs′ ⊂ 100κ−10Rs′
}(4.45)

Δ(T)2 :=

s∈T

|〈Sann 1F , αs〉|2
|Rs|

1Rs , T is a 1–tree,(4.46)

size(S) := sup
T⊂S

T is a 1–tree

RT

Δ(T) dx.(4.47)

Observe that dense(S) only really applies to ‘tree-like’ sets of tiles, and that—and
this is important—the tile s′ that appear in (4.45) are not in S, but only assumed
to be in AT . Finally, note that

dense(s)

G∩v−1(ωs)

χ̃(1)s dx

with the implied constants only depending upon κ, χ, and other fixed quantities.

Page 86

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Page 87

ISBN 978-0-8218-4540-0

9 780821 845400

MEMO/205/965