Regularity of C^1 and Lipschitz domains in terms of the Beurling transform

0.34 MB·English

by  Xavier Tolsa

REGULARITY OF C1 AND LIPSCHITZ DOMAINS IN TERMS OF
THE BEURLING TRANSFORM
2
1
XAVIER TOLSA
0
2

n
a
J Abstract. Let Ω ⊂ C be a bounded C1 domain, or a Lipschitz domain “flat

5 enough”, and consider the Beurling transform of χΩ:
2
−1 1
A] BχΩ(z)=εli→m0 π Zw∈Ω,|z−w|>ε (z−w)2 dm(w).
Using a priori estimates, in this paper we solve the following free boundary prob-
C
lem: if BχΩ belongs to the Sobolev space Wα,p(Ω) for 0 < α ≤ 1, 1 < p < ∞
.
h such that αp > 1, then the outward unit normal N on ∂Ω is in the Besov space
at Bpα,−p1/p(∂Ω). The converse statement, proved previously by Cruz and Tolsa, also
m holds. So we have
[ B(χΩ)∈Wα,p(Ω) ⇐⇒ N ∈Bpα,−p1/p(∂Ω).

1 Together with recent results by Cruz, Mateu and Orobitg, from the preceding
v equivalence one infers that the Beurling transform is bounded in Wα,p(Ω) if and
3 onlyifthe outwardunitnormalN belongstoBα−1/p(∂Ω), assumingthatαp>2.
0 p,p
4
5
1. Introduction
.
1
0 In this paper we show that the boundedness of the Beurling transform in the
2 Sobolev spaces Wα,p(Ω), with 0 < α ≤ 1 and 1 < p < ∞ such that αp > 1,
1
: characterizes the Besov smoothness of the boundary ∂Ω, whenever Ω is a C1 domain,
v
or a Lipschitz domain “flat enough”. This can be considered as a free boundary
i
X
problem.
ar TheBeurling transformofafunctionf : C → C, withf ∈ Lp forsome1 ≤ p < ∞,
is defined by
−1 f(w)
Bf(z) = lim dm(w).
ε→0 π (z −w)2
Z|z−w|>ε
It is known that this limit exist a.e. The Beurling transform plays an essential role
in the theory of quasiconformal mappings in the plane, because it intertwines the ∂
¯
and ∂ derivatives. Indeed, in the sense of distributions, one has
¯
B(∂f) = ∂f.
Let Ω ⊂ C be a bounded domain (open and connected). We say that Ω ⊂ C
is a (δ,R)-Lipschitz domain if for each z ∈ ∂Ω there exists a Lipschitz function
Partially supported by grants 2009SGR-000420 (Generalitat de Catalunya) and MTM-2010-
16232 (Spain).
1
2 XAVIERTOLSA
A : R → R with slope kA′k ≤ δ such that, after a suitable rotation,
∞
Ω∩B(z,R) = (x,y) ∈ B(z,R) : y > A(x) .
If we do not care about the constants δ and R, then we just say that Ω is a Lipschitz
(cid:8) (cid:9)
domain. If in this definition we assume the function A to be of class C1, then we
say that Ω is a C1 domain.
In [CT] it has been shown that for any Lipschitz domain Ω and 0 < α ≤ 1 and
1 < p < ∞ such that αp > 1, if the outward unit normal is in the Besov space
Bα−1/p(∂Ω), then B(χ ) belongs to the Sobolev space Wα,p(Ω). More precisely, the
p,p Ω
following estimate has been proved:
(1.1) kB(χ )k ≤ ckNk ,
Ω W˙ α,p(Ω) B˙pα,−p1/p(∂Ω)
where N stands for the outward normal unitary vector on ∂Ω, W˙ α,p(Ω) is a homo-
geneous sobolev space on Ω, and B˙α−1/p(∂Ω) is a homogeneous Besov space on ∂Ω.
p,p
See the next section for the precise definition of Sobolev and Besov spaces, as well
as their homogeneous versions. The constant c in (1.1) may depend on p and on the
Lipschitz character of Ω, i.e. on δ and on H1(∂Ω)/R (here H1 stands for the length
or 1-dimensional Hausdorff measure). Observe that, by the Lp boundedness of the
Beurling transform,
kB(χ )k ≤ c m(Ω)1/p +kB(χ )k .
Ω Wα,p(Ω) Ω W˙ α,p(Ω)
Thus (1.1) guaranties that B(χ ) ∈ W(cid:0) α,p(Ω) whenever N ∈ B˙α−(cid:1)1/p(∂Ω).
Ω p,p
Our main result is a (partial) converse to (1.1):
Theorem 1.1. Let Ω ⊂ C be a (δ,R)-Lipschitz domain and 0 < α ≤ 1 and 1 < p <
∞ such that αp > 1. If δ = δ(p) > 0 is small enough, then
(1.2) kNk ≤ ckB(χ )k +cH1(∂Ω)−α+2/p.,
B˙pα,−p1/p(∂Ω) Ω W˙ α,p(Ω)
where c depends on δ and p.
Some remarks are in order. Notice first that C1 domains are (δ,R)-Lipschitz
domains for every δ > 0 and an appropriate R = R(δ). So the theorem applies to
all C1 domains. Then, by combining the results from [CT] with Theorem 1.1, one
infers that, for a C1 domain Ω and α,p as above,
(1.3) B(χ ) ∈ Wα,p(Ω) ⇐⇒ N ∈ Bα−1/p(∂Ω).
Ω p,p
Let us remark that the inequality
kNk ≤ ckB(χ )k
B˙pα,−p1/p(∂Ω) Ω W˙ α,p(Ω)
fails in general. Indeed, when Ω is an open ball it turns out that B(χ ) vanishes
Ω
identically on Ω. So kB(χ )k = 0, while kNk 6= 0, because N is not
Ω W˙ α,p(Ω) B˙pα,−p1/p(∂Ω)
constant.
α−1/p
Recall that the Besov spaces B appear naturally in the context of Sobolev
p,p
spaces. Indeed, a well known theorem of Gagliardo [Ga] asserts that the traces of
REGULARITY OF DOMAINS IN TERMS OF THE BEURLING TRANSFORM 3
the functions from W1,p(Ω) on ∂Ω coincide with the functions from B1−1/p(∂Ω),
p,p
whenever Ω is a Lipschitz domain. An analogous result holds for 0 < α < 1. So,
by this result theorem with (1.3), one deduces that B(χ ) ∈ Wα,p(Ω) if and only
Ω
if N is the trace of some (vectorial) function from Wα,p(Ω), which looks a rather
surprising statement at first sight.
OurmotivationforthecharacterizationofthosedomainssuchthatBχ ∈ Wα,p(Ω)
Ω
arises from the results of Cruz, Mateu and Orobitg in [CMO]. In this paper the
authors study the smoothness of quasiconformal mappings when the Beltrami co-
efficient belongs to Wα,p(Ω), for some fixed 1 < p < ∞ and 0 < α < 1. As an
important step in their arguments, they prove following kind of T1 theorem:
Theorem ([CMO]). Let Ω ⊂ C be a C1+ε domain, for some ε > 0, and let 0 < α ≤ 1
and 1 < p < ∞ be such that αp > 2. Then, the Beurling transform is bounded in
Wα,p(Ω) if and only if B(χ ) ∈ Wα,p(Ω).
Ω
As a corollary of the preceding result and Theorem 1.1 we obtain the following.
Corollary 1.2. Let Ω ⊂ C be a Lipschitz domain and let 0 < α ≤ 1 and 1 < p < ∞
be such that αp > 2. Then, the Beurling transform is bounded in Wα,p(Ω) if and
α−1/p
only if the outward unit normal of Ω is in the Besov space B (∂Ω).
p,p
Observe that the fact that N ∈ B˙α−1/p(∂Ω) implies that the local parameteriza-
p,p
tionsoftheboundarycanbetakenfromB1+α−1/p(R) ⊂ C1+ε(R)becauseαp > 2,and
p,p
thus the theorem from Cruz-Mateu-Orobitg applies. For more details, see Lemma
3.1 below.
On the other hand, it is worth mentioning that the boundedness of the Beurling
transform in the Lipschitz spaces Lip (Ω) for domains Ω of class C1+ε has been
ε
studied in [MOV], [LV], and [De], because of the applications to quasiconformal
mappings and PDE’s.
We will prove Theorem 1.1 by reducing it to the case of the so called special
Lipschitz domain, where Ω is the open set lying above a Lipschitz graph. That is,
given a Lipschitz function A : R → R, one sets
(1.4) Ω = {(x,y) ∈ C : y > A(x)}.
In this situation, we will show the following:
Theorem 1.3. Let A : R → R be a Lipschitz function with compact support and
consider the special Lipschitz domain Ω defined in (1.4). For 0 < α ≤ 1 and
1 < p < ∞ be such that αp > 1, there exists δ = δ(α,p) > 0 small enough such if
kA′k ≤ δ, then
∞
(1.5) kAk ≤ ckB(χ )k ,
B˙p1,+pα−1/p Ω W˙ α,p(Ω)
with c depending on α,p.
Above, B˙1+α−1/p and W˙ α,p(Ω) stand for homogeneous Besov and Sobolev spaces,
p,p
on R and on Ω, respectively.
4 XAVIERTOLSA
Let us remark that the converse inequality
(1.6) kB(χ )k ≤ c(δ)kAk
Ω W˙ α,p(Ω) B˙p1,+pα−1/p
also holds for special Lipschitz domains, without the smallness assumption on δ.
This has been shown in [CT]. Notice, in particular, that (1.6) shows that Bχ is
Ω
constant in Ω if this is a half plane. Of course, this can be proved without appealing
to (1.6), in a much more elementary way. This property plays a key role in the
arguments in [CT] and also in the ones of the present paper.
It is easy to check that
(1.7) kAk ≈ kNk ,
B˙p1,+pα−1/p B˙pα,−p1/p(∂Ω)
where, as above, N(z) stands for the outward unitary normal at z. So (1.5) is
analogous to (1.2). For the detailed arguments, see Lemma 3.1 below.
We will prove Theorem 1.3 by means of a Fourier type estimate. To this end, we
will need to approximate the Lipschitz graph by lines at many different scales. We
will estimate the errors in the approximation in terms of the so called β coefficients.
1
Given an interval I ⊂ R and a function f ∈ L1 , one sets
loc
1 |f(x)−ρ(x)|
(1.8) β (f,I) = inf dx,
1
ρ ℓ(I) ℓ(I)
Z3I
where the infimum is taken over all the affine functions ρ : R → R. The coefficients
β ’s (and other variants β , β ,...) appeared first in the works of Jones [Jo] and
1 p ∞
David and Semmes [DS1] on quantitative rectifiability. They have become a useful
tool in problems which involve geometric measure theory and multi-scale analysis.
See [DS2], [L´e], [MT], [To1], or [To2], for example, besides the aforementioned refer-
ences. In the present paper we will use the β ’s to measure the Besov smoothness of
1
the boundary of Lipschitz domains, by means of a characterization of Besov spaces
in terms of β ’s due to Dorronsoro [Do].
1
Theplanofthepaperisthefollowing. InSection2, somepreliminarynotationand
background is introduced. In particular, several characterizations of Besov spaces
are described. In Section 3 we prove some auxiliary lemmas which will be used later.
Theorem 1.3 is proved in Section 4, and then this is used in Section 5 to deduce
Theorem 1.1. In the final Section 6 we show that Theorems 1.1 and 1.3 also hold
replacing the W˙ α,p seminorm of Bχ by the B˙α (Ω) one, for 0 < α < 1.
Ω p,p
2. Preliminaries
2.1. Basic notation. As usual, in the paper the letter ‘c’ stands for an absolute
constant which may change its value at different occurrences. On the other hand,
constants with subscripts, such as c , retain their values at different occurrences.
0
The notation A . B means that there is a fixed positive constant c such that
A ≤ cB. So A ≈ B is equivalent to A . B . A.
The notation I(x,r) stands for an interval in R with center x and radius r.
REGULARITY OF DOMAINS IN TERMS OF THE BEURLING TRANSFORM 5
2.2. Dyadic and Whitney cubes. By a cube in Rn we mean a cube with edges
parallel to the axes. Most of the cubes in our paper will be dyadic cubes, which are
assumed to be half open-closed. The collection of all dyadic cubes is denoted by
D(Rn). They are called intervals for n = 1 and squares for n = 2. The side length
of a cube Q is written as ℓ(Q), and its center as z . The lattice of dyadic cubes of
Q
side length 2−j is denoted by D (Rn). Also, given a > 0 and any cube Q, we denote
j
by aQ the cube concentric with Q with side length aℓ(Q).
Recall that any open subset Ω ⊂ Rn can be decomposed in the so called Whitney
cubes, as follows:
∞
Ω = Q ,
k
k=1
[
where Q are disjoint dyadic cubes (the “Whitney cubes”) such that for some con-
k
stants r > 20 and D ≥ 1 the following holds,
0
(i) 5Q ⊂ Ω.
k
(ii) rQ ∩Ωc 6= ∅.
k
(iii) For each cube Q , there are at most D squares Q such that 5Q ∩5Q 6= ∅.
k 0 j k j
Moreover, for such squares Q , Q , we have 1ℓ(Q ) ≤ ℓ(Q ) ≤ 2ℓ(Q ).
k j 2 k j k
We will denote by W(Ω) the family {Q } of Whitney cubes of Ω.
k k
If Ω ⊂ C is a Lipschitz domain, then ∂Ω is a chord arc curve. Recall that a chord
arc curve is just the bilipschitz image of a circumference. Then one can define a
family D(∂Ω) of “dyadic” arcs which play the same role as the dyadic intervals in R:
for each j ∈ Z such that 2−j ≤ H1(∂Ω), D (∂Ω) is a partition of ∂Ω into pairwise
j
disjoint arcs of length ≈ 2−j, and D(∂Ω) = D (∂Ω). As in the case of D(Rn), two
j j
arcs from D(∂Ω) either are disjoint or one contains the other. The construction of
S
D(∂Ω) easy: take an arc length parameterization S1(0,r ) → ∂Ω, consider a dyadic
0
family of arcs of S1(0,r ), and then let D(∂Ω) be the image of the dyadic arcs from
0
S1(0,r ).
0
Sometimes, the arcs from D(∂Ω) will be called dyadic “cubes”, because of the
analogy with D(Rn).
If Ω is a special Lipschitz domain, that is, Ω = {(x,y) ∈ C : y > A(x)}, where
A : R → R is a Lipschitz function, there exists an analogous family D(∂Ω). In this
case, setting T(x) = (x,A(x)), one can take D(∂Ω) = T(D(R)), for instance.
If Ω is either a Lipschitz or a special Lipschitz domain, to each Q ∈ W(Ω) we
assign a cube φ(Q) ∈ D(∂Ω) such that φ(Q) ∩ ρQ 6= ∅ and diam(φ(Q)) ≈ ℓ(Q).
So there exists some big constant M depending on the parameters of the Whitney
decomposition and on the chord arc constant of ∂Ω such that
φ(Q) ⊂ M Q, and Q ⊂ B(z,Mℓ(φ(Q))) for all z ∈ φ(Q).
From this fact, it easily follows that there exists some constant c such that for every
2
Q ∈ W(Ω),
#{P ∈ D(∂Ω) : P = φ(Q)} ≤ c .
2
6 XAVIERTOLSA
2.3. Sobolev spaces. Recall that for an open domain Ω ⊂ Rn, 1 ≤ p < ∞, and a
positive integer m, the Sobolev space Wm,p(Ω) consists of the functions f ∈ L1 (Ω)
loc
such that
1/p
kfk = kDαfkp < ∞,
Wm,p(Ω) Lp(Ω)
(cid:18)0≤|α|≤m (cid:19)
X
whereDαf istheα-thderivativeoff,inthesenseofdistributions. Thehomogeneous
Sobolev seminorm W˙ m,p is defined by
1/p
kfk := kDαfkp .
W˙ m,p(Ω) Lp(Ω)
(cid:18)|α|=m (cid:19)
X
For a non integer 0 < α < 1, one sets
1
|f(x)−f(y)|2 2
(2.1) Dαf(x) = dm(y) ,
|x−y|n+2α
(cid:18)ZΩ (cid:19)
and then
1/p
kfk = kfkp +kDαfkp .
Wα,p(Ω) Lp(Ω) Lp(Ω)
(cid:18) (cid:19)
The homogeneous Sobolev seminorm W˙ α,p(Ω) equals
kfk = kDαfk .
W˙ α,p(Ω) Lp(Ω)
2.4. Besov spaces. In this section we review some basic results concerning Besov
spaces. We only consider the 1-dimensional case, and pay special attention to the
homogeneous Besov spaces B˙α , with 0 < α < 1.
p,p
Consider a radial C∞ function η : R → R whose Fourier transform η is supported
in the annulus A(0,1/2,3/2), such that setting ηk(x) = η2−k(x) = 2kη(2kx),
b
(2.2) η (ξ) = 1 for all ξ 6= 0.
k
k∈Z
X
Then, for f ∈ L1 (R), 1 ≤ p,qb< ∞, and α > 0, one defines the seminorm
loc
1/q
kfk = k2kαη ∗fkq ,
B˙α k p
p,q !
k∈Z
X
and the norm
kfk = kfk +kfk .
Bpα,q p B˙p2,−p1/p
The homogeneous Besov space B˙α ≡ B˙α (R) consists of the functions such that
p,q p,q
kfk < ∞, while the functions in the Besov space Bα ≡ Bα (R) are those such
B˙α p,q p,q
p,q
that kfk < ∞. If one chooses a function different from η which satisfies the
Bα
p,q
same properties as η above, then one obtains an equivalent seminorm and norm,
respectively. Let us remark that the seminorm k · k is a norm if one considers
B˙α
p,q
functions modulo polynomials.
REGULARITY OF DOMAINS IN TERMS OF THE BEURLING TRANSFORM 7
For any function f and any h > 0, denote ∆ (f)(x) = f(x + h) − f(x). For
h
1 ≤ p,q < ∞ and 0 < α < 1, it turns out that
∞ k∆ (f)kp
(2.3) kfkp ≈ h q dh,
B˙α hαp+1
p,q Z0
assuming f to be compactly supported, say. Otherwise the comparability is true
modulo polynomials, that is, above we should replace k∆ (f)k by
h q
inf k∆ (f +p)k .
h q
ppolynomial
See [Tr, p. 242], for instance. Analogous characterizations hold for Besov spaces
with regularity α ≥ 1. In this case it is necessary to use differences of higher order.
Observe that, for p = q and 0 < α < 1, one has
∞ |∆ (f)|p |f(x)−f(y)|p
(2.4) kfkp ≈ h dhdx ≈ dxdy,
B˙α hαp+1 |x−y|αp+1
p,p ZZ0 ZZ
for f with compact support. These results motivate the definition of the B˙α -
p,p
seminorm over a chord arc curve. If Γ is such a curve and f ∈ L1(H1⌊Γ), then one
defines
|f(x)−f(y)|p
kfkp = dH1(x)dH1(y).
B˙α (Γ) |x−y|αp+1
p,p ZZ(x,y)∈Γ2
The same definition applies if Γ is a Lipschitz graph, say. If γ : S1(0,r) → Γ or
γ : R → Γ is a bilipschitz parameterization of Γ (such as the arc length parameteri-
zation), clearly we have
kfk ≈ kf ◦γk ,
B˙α (Γ) B˙α (S1(0,r))
p,q p,q
for f compactly supported.
Concerning the Besov spaces of regularity 1 < α < 2, let us remark that, for
f ∈ L1 (R),
loc
(2.5) kfkp ≈ kf′kp ,
B˙pα,q B˙pα,−q1
where f′ is the distributional derivative of f. Further we will use the following
characterization in terms of the coefficients β defined in (1.8), due to Dorronsoro
1
[Do, Theorems 1 and 2]. For 0 < α < 2 and 1 ≤ p,q < ∞, one has:
∞ dh 1/q
kfk ≈ h−α+1kβ (f,I(·,h))k q .
B˙pα,q 1 p h
(cid:18)Z0 (cid:19)
Again, this comparability should(cid:0)be understood modu(cid:1)lo polynomials, unless f is
compactlysupported, say. Inthecasep = q,anequivalentstatementisthefollowing:
p 1/p
β (f,I)
kfkp ≈ 1 ℓ(I) .
B˙α ℓ(I)α−1
p,p
(cid:18)I∈D(R)(cid:18) (cid:19) (cid:19)
X
For other indices α ≥ 2, there are analogous results which involve approximation
by polynomials of a fixed degree instead of affine functions. Let us remark that the
8 XAVIERTOLSA
coefficients β (f,I) are not introduced in [Do], and instead a different notation is
1
used.
3. Auxiliary lemmas
3.1. About Besov spaces.
Lemma 3.1. Let A : R → R be a Lipschitz function with kA′k ≤ c and Γ ⊂ C its
∞ 0
graph. Denote by N (x) the unit normal of Γ at (x,A(x)) (whose vertical component
0
is negative, say), which is defined a.e. Then,
(3.1) |∆ (A′)(x)| ≈ |∆ N (x)|,
h h 0
with constants depending on c . Thus, for 1 ≤ p < ∞ and 0 < α < 1,
0
(3.2) kAk ≈ kA′k ≈ kN k ,
B˙pα,+p1 B˙pα,p 0 B˙pα,p
with constants depending on α and p, and also on c in the second estimate.
0
Above, we set
kN k := kN k +kN k ,
0 B˙α 0,1 B˙α 0,2 B˙α
p,p p,p p,p
where N , i = 1,2, are the components of N .
0,i 0
For the proof, see [CT].
Remark 3.2. As mentioned in Subsection 2.4, from the characterization of Besov
spaces in terms of differences, it turns out that if γ : R → Γ is an arc length
parameterization of the Lipschitz graph Γ, and N(z) stands for the unit normal at
z ∈ Γ (with a suitable orientation), then
kN ◦γk ≈ kN k ≈ kNk ,
B˙pα,−p1/p 0 B˙pα,−p1/p B˙pα,−p1/p(Γ)
for 0 < α ≤ 1 and 1 < p < ∞ such that αp > 1.
Recall that for a Lipschitz domain Ω, whose boundary has an arc length parame-
terization γ : S1(0,r ) → ∂Ω (with 2πr = H1(∂Ω)), if N(z) stands for the outward
0 0
unit normal at z ∈ ∂Ω, we also have
kNk ≈ kN ◦γk .
B˙pα,−p1/p(∂Ω) B˙pα,−p1/p(S1(0,r0))
In(1.8)wedefinedthecoefficients β associatedtoafunctionf. Nowweintroduce
1
an analogous notion replacing f by a chord arc curve Γ (which may be the boundary
of a Lipschitz domain). Given P ∈ D(Γ), we set
1 dist(x,L)
(3.3) β (Γ,P) = inf dH1(x),
1
L ℓ(P) ℓ(P)
Z3P
where the infimum is taken over all the lines L ⊂ C.
Next lemma is a direct consequence of the previous results and the characteriza-
tion of homogeneous Besov spaces in terms of the β ’s from Dorronsoro. For the
1
detailed proof, see [CT].
REGULARITY OF DOMAINS IN TERMS OF THE BEURLING TRANSFORM 9
Lemma 3.3. Let Ω be a Lipschitz domain. Suppose that the outward unit normal
satisfies N ∈ B˙α (∂Ω), for some 1 ≤ p < ∞, 0 < α < 1. Then,
p,p
p
β (∂Ω,P)
1 ℓ(P) . kNkp +cH1(∂Ω)1−αp.
ℓ(P)α B˙α (∂Ω)
p,p
P∈D(∂Ω)(cid:18) (cid:19)
X
with c depending on H1(∂Ω)/R.
Lemma 3.4. Consider functions ϕ,f : R → R and let 1 ≤ p < ∞ and 0 < α < 1.
We have
∞ |ϕ∆ (f)|p
kfkp . h dhdx+kϕkp kfkp .
B˙α hαp+1 B˙α ∞
p,p ZZ0 p,p
Proof. Recall that
∞ |∆ (ϕf)|p
(3.4) kϕfkp ≈ h dhdx.
B˙α hαp+1
p,p ZZ0
To prove the lemma, we just use that
∆ (ϕf)(x) = ϕ(x)∆ (f)(x)+f(x+h)∆ (ϕ)(x).
h h h
Then, plugging this identity into (3.4) the lemma follows easily. (cid:3)
Lemma 3.5. Let A : R → R be a Lipscthitz function supported on an interval I.
For 1 ≤ p < ∞ and 0 < α < 1, we have
kAk . kA′k ℓ(I)1−α+1/p,
B˙α ∞
p,p
with a constant depending on α and p.
Proof. Denote ℓ = ℓ(I). We set
(3.5)
∞ |∆ (A)|p dh
kAkp ≈ h dx
B˙α hαp h
p,p ZZ0
|A(x+h)−A(x)|p |A(x+h)−A(x)|p
= dhdx+ dhdx.
hαp+1 hαp+1
ZZ0<h≤ℓ ZZh>ℓ
For the first integral on the right side we use fact that the integrand vanishes unless
x ∈ 3I and the Lipschitz condition |A(x+h)−A(x)| ≤ kA′k h:
∞
|A(x+h)−A(x)|p hp
dhdx ≤ kA′kp dhdx
hαp+1 ∞ hαp+1
ZZ0<h≤ℓ Zx∈3I Zh≤ℓ
= 3kA′kp ℓ hp(1−α)−1dh ≈ kA′kp ℓp+1−αp.
∞ ∞
Z0<h≤ℓ
10 XAVIERTOLSA
Concerning the last integral in (3.5), we use again the fact that A is supported
on I and the estimate |A(x+h)−A(x)| ≤ kA′k ℓ:
∞
|A(x+h)−A(x)|p
dhdx
hαp+1
ZZh>ℓ
1 1
≤ kA′kp ℓp dhdx+ dhdx ≈ kA′kp ℓp+1−αp.
∞ hαp+1 hαp+1 ∞
(cid:18)Zx∈I Zh>ℓ Zx∈I−hZh>ℓ (cid:19)
(cid:3)
3.2. About the Beurling transform of χ . Let Ω ⊂ C be an open set. If Ω has
Ω
finite Lebesgue measure, then
−1 1
(3.6) Bχ (z) = lim χ (w)dm(w).
Ω ε→0 π (z −w)2 Ω
Z|z−w|>ε
Otherwise, B(χ ) is a BMO function and, thus, it is defined modulo constants.
Ω
Actually, a possible way to assign a precise value to B(χ )(z) is the following:
Ω
−1 1 1
(3.7) Bχ (z) = lim − χ (w)dm(w),
Ω ε→0 π Z|z−w|>ε(cid:18)(z −w)2 (z0 −w)2(cid:19) Ω
where z is some fixed point, with z 6∈ Ω, for example. It is easy to check that the
0 0
preceding principal value integral exists for all z ∈ C.
Moreover the following results hold:
Lemma 3.6. Let Ω ⊂ C be an open set. The function B(χ ) is analytic in C\∂Ω
Ω
and moreover, for every z ∈ C\∂Ω,
2 1
(3.8) ∂B(χ )(z) = χ (w)dm(w),
Ω π (z −w)3 Ω
Z|z−w|>ε
for 0 < ε < dist(z,∂Ω).
When Ω has infinite measure, saying that B(χ ) is analytic in C\∂Ω means that
Ω
the function defined in (3.7) is analytic for each choice of z . Notice that, in any
0
case, the derivative ∂B(χ ) is independent of z .
Ω 0
Lemma 3.7. Let Π ⊂ C be a half plane. Then ∂B(χ ) = 0 in C\∂Π. Equivalently,
Π
for all z 6∈ ∂Π and 0 < ε < dist(z,∂Π), we have
1
χ (w)dm(w) = 0.
(z −w)3 Π
Z|z−w|>ε
For the proofs of the preceding two lemmas, see [CT], for example. By using very
similar arguments, for any z ∈ C\∂Ω, one gets
−6 1
(3.9) ∂2B(χ )(z) = χ (w)dm(w),
Ω π (z −w)4 Ω
Z|z−w|>ε
for 0 < ε < dist(z,∂Ω). The details are left for the reader.