Strong and weak type estimates for singular integrals with respect to measures separated by AD-regular boundaries

0.24 MB·English

by  Vasilis Chousionis

STRONG AND WEAK TYPE ESTIMATES FOR SINGULAR
INTEGRALS WITH RESPECT TO MEASURES SEPARATED BY
AD-REGULAR BOUNDARIES
3
1
0
2 V. CHOUSIONIS AND X. TOLSA

n
a Abstract. We prove weak and strong boundedness estimates for singular inte-
J grals in Rd with respect to (d−1)-dimensional measures separated by Ahlfors-

1
David regular boundaries, generalizing and extending results of Chousionis and
1
Mattila. Our proof follows a different strategy based on new Calder´on-Zygmund

] decompositions which can be also used to extend a result of David.
A
C
.
h
t 1. Introduction
a
m A Radon measure on Rd has n-growth if there exists some constant c such that
µ
[ µ(B(x,r)) ≤ c rn for all x ∈ Rd, r > 0. If there exists some constant c such that
µ µ

1
v c−µ1rn ≤ µ(B(x,r)) ≤ cµrn for all x ∈ sptµ, 0 < r ≤ diam(sptµ),
9
2 then we say that µ is n-Ahlfors-David regular, or n-AD regular. A set E ⊂ Rd is
5
n-AD regular if the n-dimensional Hausdorff measure restricted to E, denoted by
2
1. Hn⌊E, is n-AD regular.
The space of finite complex Radon measures in U ⊂ Rd is denoted by M(U). This
0
3 is a Banach space with the norm of the total variation: kνk = |ν|(U).
1 We say that k(·,·) : Rd×Rd\{(x,y) ∈ Rd×Rd : x = y} → C is an n-dimensional
:
v Caldero´n-Zygmund (CZ) kernel if there exist constants c > 0 and η, with 0 < η ≤ 1,
i
X such that the following inequalities hold for all x,y ∈ Rd, x 6= y:
r c
a |k(x,y)| ≤ , and
|x−y|n
c|x−x′|η
|k(x,y)−k(x′,y)|+|k(y,x)−k(y,x′)| ≤ if |x−x′| ≤ |x−y|/2.
|x−y|n+η
Given a positive or complex Radon measure ν on Rd and a Caldero´n-Zygmund
kernel k, we define
Tkν(x) := k(x,y)dν(y), x ∈ Rd \sptν.
Z
2010 Mathematics Subject Classification. Primary 42B20,42B25.
Key words and phrases. Caldero´n-Zygmundsingular integrals, uniform rectifiability.
V.CispartiallysupportedbythegrantMTM2010-15657(Spain)andX.Tispartiallysupported
by grants 2009SGR-000420(Generalitat de Catalunya) and MTM2010-16232(Spain).
1
2 V.CHOUSIONISAND X.TOLSA
This integral may not converge when x ∈ sptν. For this reason, we consider the
following ε-truncated operators Tk, ε > 0:
ε
Tkν(x) := k(x,y)dν(y), x ∈ Rd.
ε
Z|x−y|>ε
Given a fixed positive Radon measure µ on Rd and f ∈ L1 (µ), we write
loc
Tkf(x) := Tk(f µ)(x), x ∈ Rd \spt(f µ),
µ
and
Tk f(x) := Tk(f µ)(x).
µ,ε ε
If ν is a positive Radon measures as well we say that Tk is bounded from Lp(ν) to
ν
Lp(µ), 1 < p < ∞, if the operators Tk are bounded from Lp(ν) to Lp(µ) uniformly
ν,ε
on ε > 0. We also say that the operators Tk are bounded from Lp(ν) to Lp,∞(µ) for
ν
1 ≤ p < ∞ if for all f ∈ Lp(ν) and for all λ > 0,
c
µ{x ∈ Rd : |T (fν)(x)| > λ} ≤ kfkp ,
ε λp Lp(ν)
uniformly on ε. Analogously Tk is bounded from M(U), U ⊂ Rd, into L1,∞(µ) if
there exists some constant c such that for all ν ∈ M(U) and all λ > 0,
ckνk
µ{x ∈ Rd : |Tkν(x)| > λ} ≤
ε λ
uniformly on ε > 0.
Our first result reads as follows.
Theorem 1.1. Let U ⊂ Rd be a domain with (d−1)-AD regular boundary Γ. Let
µ,ν two measures with (d − 1)-growth such that µ(Rd \ U¯) = ν(U) = 0. Let k be
a (d − 1)-dimensional Calder´on-Zygmund kernel such that the operator Tk :
Hd−1⌊Γ
L2(Hd−1⌊ ) → L2(Hd−1⌊ ) is bounded. Then,
Γ Γ
(i) the operators Tk : Lp(ν) → Lp(µ) and Tk : Lp(µ) → Lp(ν) are bounded for
ν µ
all 1 < p < ∞,
(ii) Tk is bounded from M(Rd \U) to L1,∞(µ) and from M(U¯) to L1,∞(ν). In
particular, the operators Tk : L1(ν) → L1,∞(µ) and Tk : L1(µ) → L1,∞(ν)
ν µ
are bounded.
An n-AD-regular set E is n-uniformly rectifiable if there exist θ,M > 0 such that
for all x ∈ E and all r > 0 there exists a Lipschitz mapping ρ from the ball B (0,r)
n
in Rn to Rd with Lip(ρ) ≤ M such that
Hn⌊ (B(x,r)∩ρ(B (0,r))) ≥ θrn.
E n
Any convolution kernel k : Rd \{0} → R such that for all x ∈ Rd \{0},
(1.1) k(−x) = −k(x) and ∇jk(x) ≤ c |x|−n−j, for j = 0,1,2,
j
(cid:12) (cid:12)
(cid:12) (cid:12)
STRONG AND WEAK ESTIMATES FOR SINGULAR INTEGRALS 3
defines a bounded operator in L2(Hn⌊ ) whenever E is an n-uniformly rectifiable
E
set. ThiswasoriginallyprovedbyDavid, seee.g. [D1]and[D2], undertheadditional
assumption |∇jk(x)| ≤ c |x|−n−j for all j ≥ 0. A proof for all kernels satisfying
j
(1.1) can be found in [T2]. It follows that all such (d−1)-dimensional kernels satisfy
the hypotheses of Theorem 1.1.
Relating the L2(µ)-boundedness of Tk with the geometric structure of µ is a hard
µ
and largely unresolved problem. After David’s result in [D2], David and Semmes
proved a result that goes in the converse direction. In [DS] they showed that the
L2(µ) boundedness of all operators associated with convolution, odd, C∞ away
from the origin CZ kernels imply that the measure µ is n-uniformly rectifiable. The
David-Semmes conjecture, dating from 1991, asks if the L2(µ)-boundedness of the
operators associated with just one of these kernels, specifically to the n-dimensional
Riesz kernel x/|x|n+1, suffices to imply n-uniform rectifiabilty. The conjecture has
beenvery recently resolved in[NToV]inthecodimension1case, thatisforn = d−1.
Mattila, Melnikov and Verdera in [MMV] had earlier proved the conjecture in the
case of 1-dimensional Riesz kernels. For all other dimensions and for other kernels
few things are known. There are several examples of kernels whose boundedness
does not imply rectifiability, see [C], [D4] and [H]. On the other hand in [CMPT]
the kernels Re(z)2n−1/|z|2n, z ∈ C,n ∈ N, were considered and it was proved that
the L2-boundedness of the operators associated with any of these kernels implies
rectifiability. By now, these are the only known examples of convolution kernels not
directly related to the Riesz kernels with this property.
With the previous discussion in mind, Theorem 1.1 elaborates that the bounded-
ness of Tk : L2(µ) → L2(ν) with µ andν being separated measures asin the theorem
µ
holds much more generally than the boundedness of Tk from L2(µ) to L2(µ). Notice
µ
that in our assumptions µ and ν can be any measures with n-growth as long as
they are separated in a reasonably nice manner. Furthermore we consider general
n-dimensional CZ-kernels requiring less smoothness than in (1.1).
In [CM] it was shown that for a smaller class of kernels and for 1 < p < ∞ the
operators Tk are bounded from Lp(ν) to Lp(µ) and from Lp(µ) to Lp(ν) whenever
µ and ν have (d − 1)-growth and they are separated by (d − 1)-Lipschitz graphs.
Theorem 1.1 extends the admissible boundaries from Lipschitz graphs to uniformly
rectifiable sets and moreover it covers the endpoint weak-(1,1) case, which did not
follow from the methods in [CM] and thus it was left untreated there.
OurprooffollowsanaltogetherdifferentapproachwhichmakesuseofnewCaldero´n-
Zygmund decompositions partially inspired by the techniques in [T1]. We should
also remark that our proof, as well as the one in [CM], makes extended use of the
following theorem of David from [D1].
Theorem 1.2. Let µ,ν two measures with compact support such that µ is n-AD
regular and ν has n-growth. Let k be an n-dimensional Calder´on-Zygmund kernel
4 V.CHOUSIONISAND X.TOLSA
such that Tk : L2(µ) → L2(µ) is bounded. Then the operators Tk : Lp(ν) → Lp(µ)
µ
and Tk : Lp(µ) → Lp(ν) are bounded for all 1 < p < ∞,
ν
Usinganexamplebasedonthefour-cornersCantorset weprovethatthereexist 1-
growth measures ν and µ, µ not AD-regular, such that the Cauchy singular integral
operator, which is associated with the Cauchy kernel 1/z, is bounded in L2(µ) but
not from L2(µ) to L2(ν). Hence we show that Theorem 1.2 fails without the AD-
regularity assumption on µ.
On the other hand the use of Caldero´n-Zygmund decompositions can be exploited
even further as only minor modifications in the proof of Theorem 1.1 allow us to
prove the following endpoint result which, as far as we know, is new.
Theorem 1.3. Under the assumptions of Theorem 1.2 the operator Tk is bounded
from M(sptµ) to L1,∞(ν). In particular Tk : L1(µ) → L1,∞(ν) is bounded.
µ
Let us remark that the boundedness of the operator Tk : M(sptν) → L1,∞(ν)
also holds. This is due to the fact that the boundedness of Tk in L2(µ) implies the
µ
boundedness from M(Rd) to L1,∞(µ) (see [T1] or [T3, Chapter 2], for example).
Thepaperisorganisedasfollows. InSection2we provetheappropriateCaldero´n-
Zygmund decompositions needed for the proof of Theorem 1.1 and in Section 3 we
prove Theorem 1.1. The proof of Theorem 1.3 is outlined in Section 4. Finally in
Section 5 we prove that the AD-regularity assumption is essential for the proof of
Theorem 1.2.
Throughout the paper the letter C stands for some constant which may change
its value at different occurrences. The notation A . B means that there is some
fixed constant C such that A ≤ CB, with C as above. Also, A ≈ B is equivalent to
A . B . A.
2. Caldero´n-Zygmund Decompositions
For any set A ⊂ Rd and ε > 0 let N(A,ε) = {x ∈ Rd : dist(x,A) ≤ ε}.
Theorem 2.1. Let U ⊂ Rd be a domain with (d−1)-AD regular boundary Γ. Let
µ,ν two measures with (d−1)-growth and compact support such that µ(Rd \U¯) =
ν(U) = 0. Suppose that sptν ⊂ N(Γ,diam(Γ)). Then for all f ∈ Lp(ν), 1 ≤ p < ∞,
and for all λ > 2d+1kfkp /kµk 1/p:
Lp(ν)
(a) There exists a family of almost disjoint balls {B } (that is, χ ≤ c)
(cid:0) (cid:1) i i i Bi
centered at sptν, with radius not exceeding 3diam(Γ), and a function h ∈
P
L1(Hd−1⌊ ) such that
Γ
λp
(2.1) |f|pdν > µ(2B ),
2d+1 i
ZBi
λp
(2.2) |f|pdν ≤ µ(2ηB ) for η > 2,
2d+1 i
ZηBi
STRONG AND WEAK ESTIMATES FOR SINGULAR INTEGRALS 5
(2.3) fν⌊ = hHd−1⌊ with |h| ≤ cλ Hd−1 a.e. in Γ .
Rd\SiBi Γ
(b) For each i, let R be a ball concentric with B , with 10r(B ) ≤ r(R ) ≤
i i i i
30diam(Γ) and denote w = χBi . Then, there exists a family of functions
i PkχBk
ϕ with spt(ϕ ) ⊂ R ∩Γ and with constant sign satisfying
i i i
(2.4) ϕ dHd−1 = w f dν,
i i
ZΓ ZBi
(2.5) |ϕ | ≤ c λ
i 1
i
X
(where c is some fixed constant), and
1
(2.6) kϕ k r(R )d−1 ≤ c |f|dν.
i L∞(Hd−1⌊Γ) i
ZBi
Proof. (a) Let
λp
F = x ∈ sptν : there exists B ,centered at x such that |f|pdν > µ(2B ) ,
x 2d+1 x
(cid:26) ZBx (cid:27)
and G = sptν \F. Notice that sptν \Γ ⊂ F and G ⊂ Γ.
For all x ∈ F let B be a maximal ball centered in x in the sense that
x
λp
|f|pdν > µ(2B ),
2d+1 x
ZBx
but for all concentric balls D with r(D ) > 2r(B )
x x x
λp
|f|pdν ≤ µ(2D ).
2d+1 x
ZDx
Notice that this maximal ball exists. Indeed, if B′ is centered at x, contains sptµ∪
x
sptν, and satisfies (2.1), we have
λp λp
|f|pdν = |f|pdν > µ(2B′) = kµk
2d+1 x 2d+1
Z ZBx′
whichcontradictstheinitialassumption. Noticealsothatsincesptν ⊂ N(Γ,diam(Γ))
all the maximal balls B satisfy r(B ) ≤ 3diam(Γ).
x x
Applying Besicovitch’s covering theorem we get an almost disjoint subfamily of
balls {B } ⊂ {B } which covers F and satisfy (2.1) and (2.2) by construction.
i i x x
Let τ = |f|pν. Recall that, given α > 1 and β > αn, a ball B(x,r) is called τ-
(α,β)-doublingifτ(αB(x,r)) ≤ βτ(B(x,r)). Denoteby D theset ofpointsz ∈ sptτ
such that there exists a sequence of τ-(2,2d+1)-doubling balls Pz centered at z such
k
that r(Pz) → 0. By standard arguments it follows that τ(G∩D) = τ(D). Therefore
k
for τ-a.e. z ∈ G, there exists a sequence of (2,2d+1)-τ-doubling balls P centered at
k
z, with r(P ) → 0, such that
k
λp
τ(P ) ≤ µ(2P ),
k 2d+1 k
6 V.CHOUSIONISAND X.TOLSA
and thus
τ(2P ) ≤ 2d+1τ(P ) ≤ λpµ(2P ).
k k k
This implies that τ⌊ is absolutely continuous with respect to µ and that τ⌊ = h µ
G G 1
with |h | ≤ λp µ-a.e., by the Lebesgue-Radon-Nikodym theorem (see [M1, p. 36-39],
1
for instance).
Notice that if A ⊂ U then h µ(A) = τ(A ∩ G) = 0 and if A ⊂ Rd \ U¯ then
1
τ(G∩A) = h µ(A) = 0. Therefore
1
(2.7) τ⌊ = h µ⌊ with 0 ≤ h ≤ λp µ−a.e. in Γ.
G 1 Γ 1
Since µ⌊ is supported on Γ and it has (d−1)-growth by standard differentiation
Γ
theory of measures, see e.g.[M1], it is absolutely continuous with respect to Hd−1⌊
Γ
with bounded Radon-Nikodym derivative. In other words, there exists a Borel
function h such that
2
(2.8) µ⌊ = h Hd−1⌊ and 0 ≤ h ≤ c Hd−1⌊ −a.e..
Γ 2 Γ 2 Γ
By (2.7) and (2.8) we deduce that
(2.9) τ⌊ = h Hd−1⌊
G 3 Γ
where h = h h and |h | ≤ cλp, Hd−1-a.e. in Γ.
3 1 2 3
Now for any ball B centered in G, using H¨older’s inequality and (2.9),
1/p
fν⌊ (B) = fdν ≤ |f|pdν ν(B)1/p′
G
ZB∩G (cid:18)ZB∩G (cid:19)
. τ(B ∩G)1/pr(B)dp−′1
1/p
. h dHd−1 Hd−1(Γ∩B)1/p′
3
(cid:18)ZB∩G (cid:19)
. λpHd−1(Γ∩B) 1/pHd−1(Γ∩B)1/p′ = λHd−1(Γ∩B).
Therefore ν⌊ is abso(cid:0)lutely continuou(cid:1)s with respect to Hd−1⌊ and
G Γ
(2.10) fν⌊ = hHd−1⌊ ,
G Γ
where 0 ≤ h ≤ cλ, µ-a.e..
(b) Assume first that the family of balls {B } is finite. Then we may suppose that
i i
this family is ordered in such a way that the sizes of the balls R are non decreasing
i
(i.e. ℓ(R ) ≥ ℓ(R )). The functions ϕ that we will construct will be of the form
i+1 i i
ϕ = α χ , with α ∈ R and A ⊂ R . We set A = R and ϕ = α χ , where the
i i Ai i i i 1 1 1 1 R1
constant α is chosen so that w f dν = ϕ dHd−1.
1 B1 1 Γ 1
Suppose that ϕ ,...,ϕ have been constructed, satisfy (2.4) and
1 k−1 R R
k−1
|ϕ | ≤ c λ,
i 1
i=1
X
STRONG AND WEAK ESTIMATES FOR SINGULAR INTEGRALS 7
where c is some constant which will be fixed below. Let R ,...,R be the sub-
1 s1 sm
family of R ,...,R such that R ∩R 6= ∅. As ℓ(R ) ≤ ℓ(R ) (because of the
1 k−1 sj k sj k
non decreasing sizes of R ), we have R ⊂ 3R .
i sj k
Since the B ’s are maximal we have that
i
1 λp
|f|pdν ≤ .
µ(6B ) 2d+1
i Z3Bi
Hence sptµ ∩ 6B 6= ∅ and thus Γ ∩ 6B 6= ∅ as well. Choosing any z ∈ Γ ∩ 6B ,
i i i i
B(z ,d ) ⊂ R for d = r(R )−6r(B ). Since r(R )− 6 r(R ) ≤ d ≤ 30diam(Γ) and
i i i i i i i 10 i i
Γ is (d−1)-AD-regular we deduce that
(2.11) Hd−1(Γ∩R ) ≥ Hd−1(Γ∩B(z ,d )) ≥ Cr(R ).
i i i i
Now taking into account that for i = 1,...,k −1, by (2.4),
|ϕ |dHd−1 ≤ |f|dν,
i
ZΓ ZBi
and using the finite ovelarpping of the balls B , H¨older’s inequality, the (d − 1)-
sj
growth of µ and ν and (2.11), it follows that
|ϕ |dHd−1 ≤ |f|dν . |f|dν
sj
j ZΓ j ZBsj Z3Rk
X X
1/p
≤ |f|pdν ν(3R )1/p′
k
(cid:18)Z3Rk (cid:19)
≤ cλµ(6R )1/pν(3R )1/p′
k k
≤ cλr(R )d−1 ≤ c λHd−1(R ∩Γ).
k 2 k
Therefore, by Chebyshev,
1 Hd−1(R ∩Γ)
Hd−1 Γ∩{ |ϕ | > 2c λ} ≤ |ϕ |dHd−1 ≤ k .
j sj 2 2c λ j sj 2
2 ZΓ
n o
P P
Setting
A = Γ∩R ∩ |ϕ | ≤ 2c λ ,
k k j sj 2
we have Hd−1(A ) ≥ Hd−1(R ∩Γ)/2. nP o
k k
The constant α is chosen so that for ϕ = α χ we have ϕ dHd−1 =
k k k Ak Γ k
w f dν. Then, using also (2.11), we obtain
Bk k R
R |f|dν 2 |f|dν
|α | ≤ Bk ≤ 21Rk
k HRd−1(Ak) HdR−1(Rk ∩Γ)
(2.12)
1/p
≤ c 21Rk |f|pdν ν(Rk)1/p′ . λ µ(Rk)1/pν(Rk)1/p′ ≤ c λ.
(cid:16)R r(Rk(cid:17))d−1 r(Rk)d−1 3
8 V.CHOUSIONISAND X.TOLSA
Thus in A ,
k
|ϕ |+ |ϕ | = |ϕ |+ |ϕ | ≤ (2c +c )λ.
k j k sj 2 3
j j
X X
Since k−1|ϕ (x)| ≤ c λ for x ∈/ A by the previous steps of the induction, if we
j=1 j 1 k
choose c = 2c +c , (2.5) follows.
1 2 3
P
Now it is easy to check that (2.6) also holds. Indeed we have
kϕ k r(R )d−1 ≤ c|α |Hd−1(Γ∩R ) ≤ c|α |Hd−1(Γ∩A )
i L∞(Hd−1⌊Γ) i i i i i
= c w f dν ≤ c |f|dν.
i
(cid:12)ZBi (cid:12) ZBi
(cid:12) (cid:12)
Suppose now that the collection o(cid:12)f balls {B(cid:12)i}i is not finite. For each fixed N
(cid:12) (cid:12)
we consider the family of balls {B } . Then, as above, we construct functions
i 1≤i≤N
ϕN,...,ϕN with spt(ϕN) ⊂ R satisfying
1 N i i
ϕN dHd−1 = w f dν,
i i
ZΓ ZBi
N
|ϕN| ≤ Bλ,
i
i=1
X
and
kϕNk r(R )d−1 ≤ c |f|dν.
i L∞(Hd−1⌊Γ) i
ZBi
Notice that the sign of ϕN equals the sign of w f dν and so it does not depend on
i i
N.
R
Then there is a subsequence {ϕk} which is convergent in the weak ∗ topol-
1 k∈I1
ogy of L∞(Hd−1⌊ ) to some function ϕ ∈ L∞(Hd−1⌊ ). Now we can consider a
Γ 1 Γ
subsequence {ϕk} with I ⊂ I which is also convergent in the weak ∗ topology
2 k∈I2 2 1
of L∞(Hd−1⌊ ) to some function ϕ ∈ L∞(Hd−1⌊ ). In general, for each j we con-
Γ 2 Γ
sider a subsequence {ϕk} with I ⊂ I that converges in the weak ∗ topology
j k∈Ij j j−1
of L∞(Hd−1⌊ ) to some function ϕ ∈ L∞(Hd−1⌊ ). It is easily checked that the
Γ j Γ
functions ϕ satisfy the required properties. (cid:3)
j
ForadomainU, ΓandµasinTheorem2.1, andacomplexmeasureν ∈ M(Rd\U)
we have the following result analogous to the preceding one.
Theorem 2.2. Let U ⊂ Rd be a domain with (d−1)-AD regular boundary Γ. Let µ
be a measure with (d−1)-growth and compact support such that µ(Rd\U¯) = 0. Then
for all ν ∈ M(Rd\U) such that sptν ⊂ N(Γ,diam(Γ)) and for all λ > 2d+1kνk/kµk:
(a) There exists a family of almost disjoint balls {B } (that is, χ ≤ c)
i i i Bi
centered at sptν, with radius not exceeding 3diam(Γ), and a function g ∈
P
L1(Hd−1⌊ ) such that
Γ
λ
(2.13) |ν|(B ) > µ(2B ),
i 2d+1 i
STRONG AND WEAK ESTIMATES FOR SINGULAR INTEGRALS 9
λ
(2.14) |ν|(ηB ) ≤ µ(2ηB ) for η > 2,
i 2d+1 i
(2.15) ν⌊ = gHd−1⌊ with |g| ≤ cλ Hd−1 a.e. in Γ .
Γ Γ\SiBi
(b) For each i, let R be a ball concentric with B , with 10r(B ) ≤ r(R ) ≤
i i i i
30diam(Γ) and denote w = χBi . Then, there exists a family of functions
i PkχBk
ϕ with spt(ϕ ) ⊂ R ∩Γ and with constant sign satisfying
i i i
(2.16) ϕ dHd−1 = w dν,
i i
ZΓ ZBi
(2.17) |ϕ | ≤ c λ
i 1
i
X
(where c is some fixed constant), and
1
(2.18) kϕ k r(R )d−1 ≤ c|ν|(B ).
i L∞(Hd−1⌊Γ) i i
This result can be derived from Theorem 2.1 by setting p = 1, taking f such that
ν = f |ν|, and replacing the measure ν there by |ν|.
3. Weak (p,p) boundedness
We will split the proof of Theorem 1.1 into two parts. We present first the proof of
the boundedness of Tk from the space of measures M(Rd\U) into L1,∞(µ). Later we
will show that Tk is bounded from Lp(ν) to Lp,∞(µ) for p > 1, by similar (although
ν
more and technical) arguments. By Theorem 1.2 and interpolation it then follows
that Tk is bounded from Lp(ν) to Lp(µ).
ν
Theorem 3.1. LetU ⊂ Rd be a domainwith (d−1)-AD regularboundaryΓ. Letµ be
a measure with (d−1)-growth such that µ(Rd\U¯) = 0. Let k be a (d−1)-dimensional
Calder´on-Zygmund kernel such that the operator Tk : L2(Hd−1⌊ ) → L2(µ) is
Hd−1⌊Γ Γ
bounded. Then the operator Tk is bounded from M(Rd \ U) into L1,∞(µ). That is
for all ν ∈ M(Rd \U) and for all λ > 0,
c
(3.1) µ({x ∈ Rd : |Tkν(x)| > λ}) ≤ kνk,
ε λ
uniformly on ε.
To simplify notation, below we will write T instead of Tk.
Proof. Supposefirstthatbothµandν havecompactsupportandsptν ⊂ N(Γ,diam(Γ)).
Clearly, we may assume that λ > 2d+1kνk/kµk.
Let {B } be the almost disjoint family of balls of Theorem 2.2. Let R = 10B
i i i i
and notice that r(R ) ≤ 30diam(Γ) recalling that r(B ) ≤ 3diam(Γ). Then we can
i i
write ν = κ+β, with
κ = ν⌊ + ϕ Hd−1⌊
Rd\SiBi i Γ
i
X
10 V.CHOUSIONISAND X.TOLSA
and
β = β := w ν⌊ −ϕ Hd−1⌊ ,
i i Bi i Γ
i i
X X(cid:0) (cid:1)
where the functions ϕ satisfy (2.16), (2.17) (2.18) and w = χBi . Moreover,
i i PkχBk
ν⌊ = gHd−1⌊ with |g| ≤ cλ Hd−1-a.e. in Γ by (2.15). Therefore κ = g˜Hd−1⌊
Rd\∪Bi Γ Γ
with g˜ = ϕ +g. In particular, |g˜| ≤ Cλ Hd−1-a.e. in Γ.
i i
By (2.1) we have
P
c c
µ 2B ≤ |ν|(B ) ≤ kνk.
i i
λ λ
(cid:16)[i (cid:17) Xi
So we have to prove that
c
(3.2) µ x ∈ Rd \ 2B : |T ν(x)| > λ ≤ kνk.
i ε
λ
n [i o
We will first show that
(3.3) |T β|dµ ≤ Ckνk.
ε
ZRd\Sk2Bi
Since β (R ) = w dν − ϕ dHd−1 = 0 and spt(β ) ⊂ R by standard estimates
i i Bi i i Γ i i i
we deduce that
R R
|T β |dµ ≤ Ckβ k ≤ C(|ν|(B )+ |ϕ |dHd−1) ≤ c|ν|(B ).
ε i i i i i
ZRd\2Ri ZΓ
We will now check that
(3.4) |T β |dµ ≤ c|ν|(B ).
ε i i
Z2Ri\2Bi
Observe that |T (w ν⌊ )(x)| ≤ c|ν|(B )/r(B ) for any x ∈ 2R \B . Therefore,
ε i Bi i i i i
|ν|(B )µ(2R )
i i
(3.5) |T (w ν⌊ )|dµ ≤ C ≤ C|ν|(B ),
ε i Bi r(B )d−1 i
Z2Ri\2Bi i
because µ has (d−1)-growth and R = 10B .
i i
On the other hand, using Cauchy-Schwarz and (2.6) we get
1/2
|T (ϕ Hd−1⌊ )|dµ ≤ |T (ϕ )|2dµ µ(2R )1/2
ε i Γ Hd−1⌊Γ,ε i i
Z2Ri (cid:18)Z (cid:19)
1/2
≤ c |ϕ |2dHd−1 µ(2R )1/2
(3.6) i i
(cid:18)ZΓ (cid:19)
1/2
≤ c kϕ k2 Hd−1(Γ∩R ) µ(2R )1/2
i L∞(Hd−1⌊Γ) i i
≤ c(cid:16)kϕ k r(R )d−1 ≤ c|ν|(cid:17)(B ).
i L∞(Hd−1⌊Γ) i i