Smoothness of the Beurling transform in Lipschitz domains

0.28 MB·English

by  Victor Cruz

SMOOTHNESS OF THE BEURLING TRANSFORM IN
LIPSCHITZ DOMAINS
2
1
VICTOR CRUZ AND XAVIER TOLSA
0
2

n
a
J Abstract. LetΩ⊂CbeaLipschitzdomainandconsidertheBeurlingtransform

5 of χΩ:
2 −1 1
] BχΩ(z)=εli→m0 π Zw∈Ω,|z−w|>ε (z−w)2 dm(w).
A Let 1 < p < ∞ and 0 < α < 1 with αp > 1. In this paper we show that if
C the outward unit normal N on ∂Ω belongs to the Besov space Bα−1/p(∂Ω), then
p,p
. BχΩ is in the Sobolev space Wα,p(Ω). This result is sharp. Further, together
h
with recent results by Cruz, Mateu and Orobitg, this implies that the Beurling
t
a transform is bounded in Wα,p(Ω) if N belongs to Bα−1/p(∂Ω), assuming that
m p,p
αp>2.
[

1
v 1. Introduction
5
8 In this paper we obtain sharp results on the Sobolev regularity of the Beurling
3 transform of the characteristic function of Lipschitz domains. It has been shown
5
recently in [CMO] that this plays a crucial role in the boundedness of the Beurling
.
1
transform in the Sobolev spaces on domains.
0
2 Recall that the Beurling transform of a locally integrable function f : C → C is
1
defined by the following singular integral:
:
v
Xi Bf(z) = lim −1 f(x) dm(w) z ∈ C,
ε→0 π (z −w)2
r Z|z−w|>ε
a
whenever thelimitandtheintegral makessense. Itiswell known thatforf ∈ Lp(C),
for some 1 ≤ p < ∞, the limit above exists a.e.
The Beurling transform is an operator of great importance for the study of qua-
¯
siconformal mappings in the plane, due to the fact that 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
V.C.wassupportedpartiallybygrants2009SGR-000420(Catalonia),MTM-2010-15657(Spain),
andNF-129254(Spain). X.T.wassupportedpartiallybygrants2009SGR-000420(Catalonia)and
MTM-2010-16232(Spain).
1
2 VICTOR CRUZ AND 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 constan(cid:8)ts δ and R, then we just say(cid:9)that Ω is a Lipschitz
domain.
Also, we call an open set Ω a special δ-Lipschitz domain if the exists a Lipschitz
function A : R → R with compact support such that
Ω = {(x,y) ∈ C : y > A(x)}.
As above, if we do not care about δ, then we just say that Ω is a special Lipschitz
domain.
If in the definitions of Lipschitz and special Lipschitz domains, moreover, one asks
A to be of class C1, then Ω is called a C1 or a special C1 domain, respectively.
Theresults thatwe obtaininthis paperdeal withtheSobolevsmoothness oforder
0 < α ≤ 1 of Bχ on Ω, which depends on the Besov regularity of the boundary
Ω
∂Ω. For the precise definitions of the Sobolev spaces Wα,p and the Besov spaces
Bα , see Section 2. Our first theorem concerns the Sobolev spaces W1,p(Ω):
p,q
Theorem 1.1. LetΩ ⊂ C be a either(δ,R)-Lipschitz domainora specialδ-Lipschitz
domain, and let 1 < p < ∞. Denote by N(z) the outward unit normal of Ω in
z ∈ ∂Ω. If N ∈ B˙1−1/p(∂Ω), then B(χ ) ∈ W˙ 1,p(Ω). Moreover,
p,p Ω
k∂B(χ )k ≤ ckNk ,
Ω Lp(Ω) B˙p1,−p1/p(∂Ω)
with c depending on p, δ and, in case Ω is a Lipschitz domain, on R.
Above, W˙ 1,p(Ω) stands for the homogeneous Sobolev space on Ω consisting of
the functions whose distributional derivatives belong to Lp(Ω), while B˙1−1/p(∂Ω) is
p,p
the homogeneous Besov space on ∂Ω associated to the indices p,p, with regularity
1−1/p. See Section 2 for more details.
Also, let us remark that, as B(χ ) is analytic in Ω, it turns out that
Ω
k∂B(χ )k ≈ kB(χ )k .
Ω Lp(Ω) Ω W˙ 1,p(Ω)
For the fractional Sobolev spaces Wα,p(Ω) for 0 < α < 1, we will prove the
following result, which is analogous to Theorem 1.1:
Theorem 1.2. Let Ω ⊂ C be either a (δ,R)-Lipschitz or a special δ-Lipschitz do-
main, and let 1 < p < ∞ and 0 < α < 1 such that αp > 1. Denote by N(z)
α−1/p
the outward unit normal of Ω in z ∈ ∂Ω. If N ∈ B (∂Ω), then B(χ ) ∈
p,p Ω
W˙ α,p(Ω)∩B˙α (Ω). Moreover,
p,p
kB(χ )k +kB(χ )k ≤ ckNk ,
Ω W˙ α,p(Ω) Ω B˙pα,p(Ω) B˙pα,−p1/p(∂Ω)
with c depending on p, α, δ and, in case Ω is a Lipschitz domain, on R.
THE BEURLING TRANSFORM IN LIPSCHITZ DOMAINS 3
Recall that the Beurling transform is bounded in Lp(C). Thus, saying that
B(χ ) ∈ W˙ α,p(Ω) is equivalent to saying that B(χ ) ∈ Wα,p(Ω) if Ω is bounded.
Ω Ω
Analogously, in the same situation, B(χ ) ∈ B˙α (Ω) if and only if B(χ ) ∈ Bα (Ω).
Ω p,p Ω p,p
α−1/p
The Besov spaces B appear naturally in the context of Sobolev spaces. In-
p,p
deed, it is well known that the traces of the functions from Wα,p(Ω) on ∂Ω coincide
α−1/p
with the functions from B (∂Ω), whenever Ω is a Lipschitz domain. So, by
p,p
combining this fact with Theorems 1.1 and 1.2, one deduces that B(χ ) ∈ Wα,p(Ω)
Ω
if N is the trace of some (vectorial) function from W1,p(Ω).
The results stated in Theorems 1.1 and 1.2 are sharp. In fact, it has been proved
in [To3], for 0 < α ≤ 1 with αp > 1, that if Ω is a C1 domain and B(χ ) ∈ W˙ α,p(Ω),
Ω
˙α−1/p
then N ∈ B (∂Ω). So one deduces that
p,p
B(χ ) ∈ W˙ α,p(Ω) ⇐⇒ N ∈ B˙α−1/p(∂Ω), for 0 < α ≤ 1 with αp > 1,
Ω p,p
assuming Ω to be a C1 domain. This shows that the smoothness of B(χ ) charac-
Ω
terizes the Besov regularity of the boundary ∂Ω.
The hypotehsis αp > 1 for our results is quite natural. We will prove below (see
Section 9) that if αp < 1, then B(χ ) ∈ W˙ α,p(Ω) (and in the case α < 1, B(χ ) ∈
Ω Ω
B˙α (Ω) too), without any assumption on the Besov regularity of the boundary. In
p,p
the endpoint case αp = 1 we will also obtain other partial results (see Section 9
again).
Our motivation to understand when Bχ ∈ W1,p(Ω) arises from the results of
Ω
Cruz, Mateu and Orobitg in [CMO]. In this paper one studies the smoothness
of quasiconformal mappings when the Beltrami coefficient belongs to Wα,p(Ω), for
some fixed 1 < p < ∞ and 0 < α < 1. An important step in the arguments is the
following kind of T1 theorem:
Theorem ([CMO]). Let Ω ⊂ C be a bounded C1+ε domain, for some ε > 0, and let
1 < p < ∞ and 0 < α ≤ 1 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 theorem and the results of this paper one obtains
the following.
Corollary 1.3. Let Ω ⊂ C be a bounded Lipschitz domain and 0 < α ≤ 1, 1 <
p < ∞ such that αp > 2. If the outward unit normal of Ω is in the Besov space
B˙α−1/p(∂Ω), then the Beurling transform is bounded in Wα,p(Ω).
p,p
Let us remark that, by Lemma 3.1 below, the fact that N ∈ B˙α−1/p(∂Ω) implies
p,p
that the local parameterizations of the boundary can be taken from B1+α−1/p(R) ⊂
p,p
C1+ε(R) because αp > 2, and thus theorem from Cruz-Mateu-Orobitg applies.
Let us also mention that the boundedness of the Beurling transform in the Lip-
schitz spaces Lip (Ω) for domains Ω of class C1+ε has been studied previously in
ε
[MOV], [LV], and [De], because of the applications to quasiconformal mappings and
PDE’s.
4 VICTOR CRUZ AND XAVIERTOLSA
It is well known that the Beurling transform of the characteristic function of a
ball vanishes identically inside the ball. Analogously, the Beurling transform of
the characteristic function of a half plane is constant in the half plane, and also in
the complementary half plane. So its derivative vanishes everywhere except in its
boundary. This fact will play a crucial role in the proofs of Theorems 1.1 and 1.2.
Roughly speaking, the arguments consist in comparing B(χ )(x) (or an appropriate
Ω
“α-th derivative”) to B(χ )(x) (or to the analogous “α-th derivative”), where Π is
Π
some half plane that approximates Ω near x ∈ Ω. The errors are estimated in terms
of the so called β coefficients. Given an interval I ⊂ R and a function f ∈ L1 , one
1 loc
sets
1 |f(x)−ρ(x)|
(1.1) 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. Finally, the connection with the Besov smoothness from the boundary arises
from a nice characterization of Besov spaces in terms of β ’s due to Dorronsoro [Do].
1
The plan of the paper is the following. In Section 2, some preliminary notation
andbackground is reviewed. InSection 3 we prove someauxiliary lemmas which will
be used later. In Section 4 we obtain more auxiliary results necessary for Theorem
1.1, which is proved in the subsequent section. Sections 6, 7 and 8 are devoted to
Theorem 1.2. The final Section 9 contains some results for the case αp ≤ 1.
2. Preliminaries
As usual, in the paper the letter ‘c’ stands for some constant (quite often 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
0
occurrences. 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.
For n ≥ 2 we will denote the Lebesgue measure in Rn by m or dm. On the other
hand, for n = 1, we will use the typical notation dx, dy,...
2.1. Dyadic and Whitney cubes. By a cube in Rn (in our case n = 1 or 2) 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
Q
lattice of dyadic cubes of side length 2−j is denoted by D (Rn). Also, given a > 0
j
and any cube Q, we denote by aQ the cube concentric with Q with side length
aℓ(Q).
THE BEURLING TRANSFORM IN LIPSCHITZ DOMAINS 5
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 ρ > 20 and D ≥ 1 the following holds,
0
(i) 5Q ⊂ Ω.
k
(ii) ρQ ∩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),
j j
two arcs from D(∂Ω) either are disjoint or one contains the other.
If Ω is a special Lipschitz domain, that is,SΩ = {(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
Conversely, to each Q ∈ D(∂Ω) we assign a square ψ(Q) ∈ W(Ω) such that
diam(ψ(Q)) ≈ dist(Q,ψ(Q)) ≈ ℓ(Q). One may think of ψ as a kind of inverse of φ.
As above, there exists some constant c such that for every Q ∈ D(∂Ω),
3
#{P ∈ W(∂Ω) : P = ψ(Q)} ≤ c .
3
2.2. 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
6 VICTOR CRUZ AND XAVIERTOLSA
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
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)
See [St], for example. The homogeneous Sobolev seminorm W˙ α,p(Ω) equals
kfk = kDαfk .
W˙ α,p(Ω) Lp(Ω)
2.3. Besov spaces. In this section we review some basic results concerning Besov
spaces. We pay special attention to the homogeneous Besov spaces B˙α , with 0 <
p,p
α < 1.
ConsideraradialC∞ functionη : Rn → Rn whoseFouriertransformη issupported
in the annulus A(0,1/2,3/2), such that setting η (x) = η (x) = 2kη(2kx),
k 2−k
b
(2.1) η (ξ) = 1 for all ξ 6= 0.
k
k∈Z
X
Then, for f ∈ L1 (Rn), 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˙pα,q
The homogeneous Besov space B˙α ≡ B˙α (Rn) consists of the functions such that
p,q p,q
kfk < ∞, while the functions in the Besov space Bα ≡ Bα (Rn) 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.
Given f ∈ L1 (Rn) and h > 0, denote ∆ (f)(x) = f(x+h)−f(x). For 1 ≤ p,q <
loc h
∞ and 0 < α < 1, it turns out that
k∆ (f)kp
(2.2) kfkp ≈ h q dm(h),
B˙α |h|αp+n
p,q Rn
Z
THE BEURLING TRANSFORM IN LIPSCHITZ DOMAINS 7
assuming f to be compactly supported, say. Otherwise the comparability is true
modulo polynomials, that is, above one 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
(2.3)
|∆ (f)|p |f(x)−f(y)|p
kfkp ≈ h dm(h)dm(x) = dm(x)dm(y).
B˙α |h|αp+n |x−y|αp+n
p,p ZZRn×Rn ZZRn×Rn
This fact motivates the definition of the B˙α -seminorm over domains in Rn. Given
p,p
an open set Ω ∈ Rn, one sets
|f(x)−f(y)|p
(2.4) kfkp = dm(x)dm(y),
B˙α (Ω) |x−y|αp+n
p,p ZZ(x,y)∈Ω2
and kfk = kfk + kfk . See [Di]. Analogously, if Γ is a chord arc
Bpα,p(Ω) Lp(Ω) B˙pα,p(Ω)
curve or a Lipschitz graph, one defines
|f(x)−f(y)|p
(2.5) kfkp = dH1(x)dH1(y),
B˙α (Γ) |x−y|αp+1
p,p ZZ(x,y)∈Γ2
and kfk = kfk +kfk .
Bpα,p(Γ) Lp(H1⌊Γ) B˙pα,p(Γ)
Concerning the Besov spaces of regularity 1 < α < 2, let us remark that, for
f ∈ L1 (R),
loc
(2.6) kfkp ≈ kf′kp ,
B˙pα,q B˙pα,−q1
where f′ is the distributional derivative of f. Further we will use a characterization
in terms of the coefficients β due to Dorronsoro. Recall the definition in (1.1). In
1
[Do, Theorems 1 and 2] it is shown that, for 1 ≤ 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)
(cid:0) (cid:1)
Again, this comparability should be understood modulo 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 dimensions n 6= 1 and other indices α 6∈ [1,2), there are analogous re-
sults which involve approximation by polynomials of a fixed degree instead of affine
functions, which we skip for the sake of simplicity.
Let us remark that the coefficients β (f,I) are not introduced explicitly in [Do],
1
and instead a different notation is used there.
8 VICTOR CRUZ AND XAVIERTOLSA
Finally we recall the relationship between the seminorms k·k and k·k
W˙ α,p(Ω) B˙α (Ω)
p,p
in Lipschitz domains. We have
kfk . kfk if 1 < p ≤ 2,
W˙ α,p(Ω) B˙α (Ω)
p,p
and
kfk . kfk if 2 ≤ p < ∞.
B˙α (Ω) W˙ α,p(Ω)
p,p
3. Auxiliary lemmas
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
Proof. Notice that the first estimate in (3.2) is just a restatement of (2.6), and the
second one follows from (3.1) and the characterization of B˙α in terms of differences
p,p
in (2.2). So we only have to prove (3.1).
Recall that
N (x) = (1+A′(x)2)−1/2 A′(x),−1 .
0
We will show first the inequality |∆ N(x)| . |∆ (A′)(x)|. Notice that, for arbitrary
h h(cid:0) (cid:1)
functions f,g : R → R and h > 0,
(3.3) ∆ (f g)(x) = f(x)∆ g(x)+g(x+h)∆ f(x),
h h h
and thus
(3.4) |∆ (f g)| ≤ kfk |∆ g|+kgk |∆ f|.
h ∞ h ∞ h
Also, it is easy to check that
1 −∆ f(x)
h
∆ (x) = ,
h
f f(x+h)f(x)
(cid:18) (cid:19)
and so
2
1 1
(3.5) ∆ ≤ |∆ f|.
h h
f f
(cid:12) (cid:18) (cid:19)(cid:12) (cid:13) (cid:13)∞
(cid:12) (cid:12) (cid:13) (cid:13)
(cid:12) (cid:12) (cid:13) (cid:13)
(cid:12) (cid:12) (cid:13) (cid:13)
THE BEURLING TRANSFORM IN LIPSCHITZ DOMAINS 9
On the other hand,
f(x+h)+f(x) ∆ f(x)
h
∆ 1+f2 (x) = ,
h
1+f(x+h)2 + 1+f(x)2
(cid:0) (cid:1)
(cid:16)p (cid:17)
and thus it follows that
p p
(3.6) ∆ 1+f2 (x) ≤ |∆ f(x)|.
h h
From (3.5) and (3.6) we(cid:12) inf(cid:16)erpthat (cid:17) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(3.7) |∆ N (x)| = ∆ (1+A′(x)2)−1/2 ≤ |∆ (A′)(x)|.
h 0,2 h h
Also, from (3.7) and (3.4), taking into account that kA′k ≤ c , we deduce that
(cid:12) (cid:0) (cid:1)(cid:12) ∞ 0
(cid:12) (cid:12)
|∆ N (x)| = ∆ A′(x)(1+A′(x)2)−1/2 ≤ (c +1)|∆ (A′)(x)|.
h 0,1 h 0 h
Let us see now that |∆ (A′)(x)| . |∆ N (x)|. From (3.5), we infer that
(cid:12) h (cid:0) h 0 (cid:1)(cid:12)
(cid:12) (cid:12)
∆ 1+A′(x)2 ≤ (1+c2)|∆ (N )(x)|.
h 0 h 0,2
Finally, since A′ =(cid:12)(cid:12)N (cid:16)p1+(A′)2, u(cid:17)si(cid:12)(cid:12)ng (3.4) we get
0,1
(cid:12) (cid:12)
|∆ (A′)(x)| ≤p 1+c2|∆ (N )(x)|+ ∆ 1+A′(x)2
h 0 h 0,1 h
≤ p1+c20|∆h(N0,1)(x)|+(cid:12)(cid:12)(1+(cid:16)cp20)|∆h(N0,2)((cid:17)x(cid:12)(cid:12))|,
(cid:12) (cid:12)
as wished. (cid:3)
p
Remark 3.2. From the characterization of Besov spaces in terms of differences, it
turns out that if N(z) stands for the unit normal at z ∈ Γ (with a suitable orienta-
tion), then
kN k ≈ kNk
0 B˙α B˙α (Γ)
p,p p,p
for 1 ≤ p < ∞ and 0 < α < 1.
Recall that in (1.1) we defined the coefficients β associated to a function f. Now
1
we introduce 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)
(3.8) 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 our previous results and the characteriza-
tion of homogeneous Besov spaces in terms of the β ’s from Dorronsoro, described
1
in the preceding section.
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
10 VICTOR CRUZ AND XAVIERTOLSA
with c depending on H1(∂Ω)/R.
Proof. Let δ,R > 0 be such that Ω is a (δ,R)-Lipschitz domain. Consider a finite
covering of ∂Ω by a family of balls {B(x ,R/4)} , with x ∈ ∂Ω. Notice that for
i 1≤i≤m i
any cube P ∈ D(∂Ω) with ℓ(3P) < R/4 there exists some ball B(x ,R/2) containing
i
P. Thus, to prove the lemma it is enough to see that, for each i,
p
β (∂Ω,P)
(3.9) 1 ℓ(P) . kNkp +H1(∂Ω)1−αp.
ℓ(P)α B˙α (∂Ω)
p,p
P∈D(∂Ω):XP⊂B(xi,R/2)(cid:18) (cid:19)
So fix i with 1 ≤ i ≤ m and let A : R → R be a Lipschitz functions such that,
after a suitable rotation,
Ω∩B(x ,R) = {(x,y) ∈ B(x ,R) : y > A(x)}.
i i
Moreover we may assume that A(x ) = 0 and that suppA ⊂ [−2R,2R]. Let ϕ :
i
R → R be a C∞ function which equals 1 on [−R/2,R/2] and vanishes on C \
[−3R/4,3R/4]. Consider the function A = ϕA. From (3.3) and (3.1) we deduce
that
e
|∆ (A′)| ≤ ϕ|∆ A′|+kA′k |∆ ϕ| ≤ χ ∆ N(x,A(x)) +c|∆ ϕ|.
h h ∞ h [−3R/4,3R/4] h h
Notice also that, for |h| ≤ R/4, (cid:12) (cid:12)
e (cid:12) (cid:12)
supp χ ∆ N(·,A(·)) ⊂ [−R,R].
[−3R/4,3R/4] h
(cid:16) (cid:17)
As a consequence, (x,A(x)) ∈ ∂Ω for x belonging to the support above, and so we
get
|∆ (A′)|p |N(x)−N(y)|p
h dxdh . dH1(x)dH1(y)
|h|αp+1 |x−y|αp+1
ZZ|h|≤R/4 ZZ(∂Ω)2
e
|∆ ϕ|p
h
+ dxdh
|h|αp+1
ZZ
≈ kNkp +kϕkp .
B˙α (∂Ω) B˙α
p,p p,p
It is easy to check that
kϕkp . R1−αp.
B˙α
p,p
Taking into account that k(A)′k ≤ c and that A′ vanishes out of [−R,R], we
∞
deduce
|∆ (A′)|ep e 1
h
dxdh . dxdh
|h|αp+1 |h|αp+1
ZZ|h|>R/4 Z|x|≤RZ|h|>R/4
e
1
+ dxdh
|h|αp+1
Z|x+h|≤RZ|h|>R/4
. R1−αp.