##### 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 [20], and are also used in Lacey and Li [15]. 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 [9]. 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 [9], but while it is used in [20], 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

TITLES IN THIS SERIES

936 André Martinez and Vania Sordoni, Twisted pseudodifferential calculus and

application to the quantum evolution of molecules, 2009

935 Mihai Ciucu, The scaling limit of the correlation of holes on the triangular lattice with

periodic boundary conditions, 2009

934 Arjen Doelman, Björn Sandstede, Arnd Scheel, and Guido Schneider, The

dynamics of modulated wave trains, 2009

933 Luchezar Stoyanov, Scattering resonances for several small convex bodies and the

Lax-Phillips conjuecture, 2009

932 Jun Kigami, Volume doubling measures and heat kernel estimates of self-similar sets,

2009

931 Robert C. Dalang and Marta Sanz-Solé, Hölder-Sobolv regularity of the solution to

the stochastic wave equation in dimension three, 2009

930 Volkmar Liebscher, Random sets and invariants for (type II) continuous tensor product

systems of Hilbert spaces, 2009

929 Richard F. Bass, Xia Chen, and Jay Rosen, Moderate deviations for the range of

planar random walks, 2009

928 Ulrich Bunke, Index theory, eta forms, and Deligne cohomology, 2009

927 N. Chernov and D. Dolgopyat, Brownian Brownian motion-I, 2009

926 Riccardo Benedetti and Francesco Bonsante, Canonical wick rotations in

3-dimensional gravity, 2009

925 Sergey Zelik and Alexander Mielke, Multi-pulse evolution and space-time chaos in

dissipative systems, 2009

924 Pierre-Emmanuel Caprace, “Abstract” homomorphisms of split Kac-Moody groups,

2009

923 Michael Jöllenbeck and Volkmar Welker, Minimal resolutions via algebraic discrete

Morse theory, 2009

922 Ph. Barbe and W. P. McCormick, Asymptotic expansions for infinite weighted

convolutions of heavy tail distributions and applications, 2009

921 Thomas Lehmkuhl, Compactification of the Drinfeld modular surfaces, 2009

920 Georgia Benkart, Thomas Gregory, and Alexander Premet, The recognition

theorem for graded Lie algebras in prime characteristic, 2009

919 Roelof W. Bruggeman and Roberto J. Miatello, Sum formula for SL2 over a totally

real number field, 2009

918 Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians

and finite W -algebras, 2008

917 Salah-Eldin A. Mohammed, Tusheng Zhang, and Huaizhong Zhao, The stable

manifold theorem for semilinear stochastic evolution equations and stochastic partial

differential equations, 2008

916 Yoshikata Kida, The mapping class group from the viewpoint of measure equivalence

theory, 2008

915 Sergiu Aizicovici, Nikolaos S. Papageorgiou, and Vasile Staicu, Degree theory for

operators of monotone type and nonlinear elliptic equations with inequality constraints,

2008

914 E. Shargorodsky and J. F. Toland, Bernoulli free-boundary problems, 2008

913 Ethan Akin, Joseph Auslander, and Eli Glasner, The topological dynamics of Ellis

actions, 2008

912 Igor Chueshov and Irena Lasiecka, Long-time behavior of second order evolution

equations with nonlinear damping, 2008

For a complete list of titles in this series, visit the

AMS Bookstore at www.ams.org/bookstore/.

Page 87

ISBN 978-0-8218-4540-0

9 780821 845400

MEMO/205/965

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 [20], and are also used in Lacey and Li [15]. 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 [9]. 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 [9], but while it is used in [20], 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

TITLES IN THIS SERIES

936 André Martinez and Vania Sordoni, Twisted pseudodifferential calculus and

application to the quantum evolution of molecules, 2009

935 Mihai Ciucu, The scaling limit of the correlation of holes on the triangular lattice with

periodic boundary conditions, 2009

934 Arjen Doelman, Björn Sandstede, Arnd Scheel, and Guido Schneider, The

dynamics of modulated wave trains, 2009

933 Luchezar Stoyanov, Scattering resonances for several small convex bodies and the

Lax-Phillips conjuecture, 2009

932 Jun Kigami, Volume doubling measures and heat kernel estimates of self-similar sets,

2009

931 Robert C. Dalang and Marta Sanz-Solé, Hölder-Sobolv regularity of the solution to

the stochastic wave equation in dimension three, 2009

930 Volkmar Liebscher, Random sets and invariants for (type II) continuous tensor product

systems of Hilbert spaces, 2009

929 Richard F. Bass, Xia Chen, and Jay Rosen, Moderate deviations for the range of

planar random walks, 2009

928 Ulrich Bunke, Index theory, eta forms, and Deligne cohomology, 2009

927 N. Chernov and D. Dolgopyat, Brownian Brownian motion-I, 2009

926 Riccardo Benedetti and Francesco Bonsante, Canonical wick rotations in

3-dimensional gravity, 2009

925 Sergey Zelik and Alexander Mielke, Multi-pulse evolution and space-time chaos in

dissipative systems, 2009

924 Pierre-Emmanuel Caprace, “Abstract” homomorphisms of split Kac-Moody groups,

2009

923 Michael Jöllenbeck and Volkmar Welker, Minimal resolutions via algebraic discrete

Morse theory, 2009

922 Ph. Barbe and W. P. McCormick, Asymptotic expansions for infinite weighted

convolutions of heavy tail distributions and applications, 2009

921 Thomas Lehmkuhl, Compactification of the Drinfeld modular surfaces, 2009

920 Georgia Benkart, Thomas Gregory, and Alexander Premet, The recognition

theorem for graded Lie algebras in prime characteristic, 2009

919 Roelof W. Bruggeman and Roberto J. Miatello, Sum formula for SL2 over a totally

real number field, 2009

918 Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians

and finite W -algebras, 2008

917 Salah-Eldin A. Mohammed, Tusheng Zhang, and Huaizhong Zhao, The stable

manifold theorem for semilinear stochastic evolution equations and stochastic partial

differential equations, 2008

916 Yoshikata Kida, The mapping class group from the viewpoint of measure equivalence

theory, 2008

915 Sergiu Aizicovici, Nikolaos S. Papageorgiou, and Vasile Staicu, Degree theory for

operators of monotone type and nonlinear elliptic equations with inequality constraints,

2008

914 E. Shargorodsky and J. F. Toland, Bernoulli free-boundary problems, 2008

913 Ethan Akin, Joseph Auslander, and Eli Glasner, The topological dynamics of Ellis

actions, 2008

912 Igor Chueshov and Irena Lasiecka, Long-time behavior of second order evolution

equations with nonlinear damping, 2008

For a complete list of titles in this series, visit the

AMS Bookstore at www.ams.org/bookstore/.

Page 87

ISBN 978-0-8218-4540-0

9 780821 845400

MEMO/205/965