Vol. 98, No. 2 DUKE MATHEMATICAL JOURNAL © 1999
L
2
-BOUNDEDNESS OF THE CAUCHY INTEGRAL OPERATOR
FOR CONTINUOUS MEASURES
XAVIER TOLSA
1. Introduction. Let µ be a continuous (i.e., without atoms) positive Radon mea-
sure on the complex plane. The truncated Cauchy integral of a compactly supported
function f in L
p
(µ),1≤ p ≤+∞, is defined by
Ꮿ
ε
f(z)=
|ξ−z|>ε
f(ξ)
ξ −z
dµ(ξ ), z ∈ C,ε>0.
In this paper, we consider the problem of describing in geometric terms those measures
µ for which
|
Ꮿ
ε
f |
2
dµ ≤ C
|f |
2
dµ, (1)
for all (compactly supported) functions f ∈ L
2
(µ) and some constant C independent
of ε>0. If (1) holds, then we say, following David and Semmes [DS2, pp. 7–8], that
the Cauchy integral is bounded on L
2
(µ).
A special instance to which classical methods apply occurs when µ satisfies the
doubling condition
µ(2) ≤ Cµ(),
for all discs centered at some point of spt(µ), where 2 is the disc concentric with
of double radius. In this case, standard Calderón-Zygmund theory shows that (1)
is equivalent to
Ꮿ
∗
f
2
dµ ≤ C
|f |
2
dµ, (2)
where
Ꮿ
∗
f(z)= sup
ε>0
|Ꮿ
ε
f(z)|.
If, moreover, one can find a dense subset of L
2
(µ) for which
Ꮿf(z)= lim
ε→0
Ꮿ
ε
f(z) (3)
Received 13 May 1997. Revision received 3 November 1998.
1991 Mathematics Subject Classification. Primary 30E20; Secondary 42B20, 30E25, 30C85.
Author partially supported by Direccion General de Investigacion Cientifica y Tecnica grant number
PB94-0879.
269
270 XAVIER TOLSA
exists a.e. (µ) (i.e., almost everywhere with respect to µ), then (2) implies the a.e.
(µ) existence of (3), for any f ∈ L
2
(µ), and
|
Ꮿf |
2
dµ ≤ C
|f |
2
dµ,
for any function f ∈ L
2
(µ) and some constant C.
For a general µ, we do not know if the limit in (3) exists for f ∈ L
2
(µ) and almost
all (µ) z ∈ C. This is why we emphasize the role of the truncated operators
Ꮿ
ε
.
Proving (1) for particular choices of µ has been a relevant theme in classical
analysis in the last thirty years. Calderón’s paper [Ca] is devoted to the proof of (1)
when µ is the arc length on a Lipschitz graph with small Lipschitz constant. The
result for a general Lipschitz graph was obtained by Coifman, McIntosh, and Meyer
in 1982 in the celebrated paper [CMM]. The rectifiable curves , for which (1) holds
for the arc length measure µ on the curve, were characterized by David [D1] as those
satisfying
µ
(z, r)
≤ Cr, z ∈ , r > 0, (4)
where (z,r) is the closed disc centered at z of radius r. It has been shown in [MMV]
that if µ satisfies the Ahlfors-David regularity condition
C
−1
r ≤ µ
(z, r)
≤ Cr, z ∈ E, 0 <r<diam(E),
where E is the support of µ, then (1) is equivalent to E being a subset of a rectifiable
curve satisfying (4).
A necessary condition for (1) is the linear growth condition
µ
(z, r)
≤ C
0
r, z ∈ spt(µ), r > 0, (5)
as shown, for example, in [D2, p. 56]. To find another relevant necessary condition,
we need to introduce a new object. The Menger curvature of three pairwise different
points x,y,z ∈ C is
c(x,y,z) =
1
R(x,y,z)
,
where R(x,y,z) is the radius of the circumference passing through x,y,z (with
R(x,y,z) =∞and c(x,y,z) = 0, if x,y,z lie on the same line). If two among the
points x,y,z coincide, we let c(x,y,z) = 0. The relation between the Cauchy kernel
and Menger curvature was found by Melnikov in [Me2]. It turns out that a necessary
condition for (1) is (see [MV] and [MMV])
c(x,y,z)
2
dµ(x) dµ(y) dµ(z) ≤ C
1
µ(), (6)
for all discs . The main result of this paper is that, conversely, (5) and (6) are
also sufficient for (1). It is not difficult to realize that (6) can be rewritten as
Ꮿ(1) ∈
L
2
-BOUNDEDNESS OF THE CAUCHY OPERATOR 271
BMO(µ), when µ is doubling and satisfies(5). Therefore, our result can be understood
as a T(1)-theorem for the Cauchy kernel with an underlying measure not necessarily
doubling. In fact, the absence of a doubling condition is the greatest problem we
must confront. We overcome the difficulty thanks to the fact that the operators to
be estimated have positive kernels. Following an idea of Sawyer, we resort to an
appropriate “good λ-inequality” to obtain a preliminary weak form of the L
2
-estimate.
In a second step, we use an inequality of Melnikov [Me2] relating analytic capacity
to Menger curvature to prove the weak (1,1)-estimate for
Ꮿ
ε
, uniform in ε>0. It
is worthwhile to mention that this part of the argument involves complex analysis
in an essential way and no real variables proof is known to the author. From the
weak (1,1)-estimate, we get the restricted weak-type (2,2) of
Ꮿ
ε
. By interpolation,
one obtains the strong-type (p, p), for 1 <p<2, and then by duality, one obtains
the strong-type (p, p), for 2 <p<∞. One more appeal to interpolation finally gives
the strong-type (2,2). For other applications of the notion of Menger curvature, see
[L] and [Ma2].
We now proceed to introduce some notation and terminology to state a more formal
and complete version of our main result. We say that µ satisfies the local curvature
condition if there is a constant C
1
such that (6) holds for any disc centered at some
point of spt(µ).
We say that the Cauchy integral is bounded on L
p
(µ) whenever the operators Ꮿ
ε
are bounded on L
p
(µ) uniformly on ε. Let M(C) be the set of all finite complex
Radon measures on the plane. If ν ∈ M(C), then we set
Ꮿ
ε
(ν)(z) =
|ξ−z|>ε
1
ξ −z
dν(ξ ).
We say that the Cauchy integral is bounded from M(C) to L
1,∞
(µ), the usual space
of weak L
1
-functions with respect to µ, whenever the operators Ꮿ
ε
are bounded from
M(C) to L
1,∞
(µ) uniformly on ε.
We can now state our main result.
Theorem 1.1. Let µ be a continuous positive Radon measure on C. Then the
following statements are equivalent.
(1) µ has linear growth and satisfies the local curvature condition.
(2) The Cauchy integral is bounded on L
2
(µ).
(3) The Cauchy integral is bounded from M(C) to L
1,∞
(µ).
(4) The Cauchy integral is bounded from L
1
(µ) to L
1,∞
(µ).
Notice that if any of the statements (1), (2), (3), or (4) of Theorem 1.1 holds,
then the Cauchy integral is bounded on L
p
(µ), for 1 <p<∞, by interpolation
and duality. Conversely, if there exists p ∈ (1, ∞) such that the Cauchy integral is
bounded on L
p
(µ), then the Cauchy integral is bounded on L
2
(µ) by duality and
interpolation.
272 XAVIER TOLSA
Using the results of Theorem 1.1, in the final part of the paper we give a geometric
characterization of the analytic capacity γ
+
, and we show that γ
+
is semiadditive for
sets of area zero.
The paper is organized as follows. In Section 2 we define the curvature operator
K, and we prove that if µ has linear growth and satisfies the weak local curvature
condition, then K is bounded on L
p
(µ), for all p ∈ (1, ∞). As a consequence, we
get that for each µ-measurable subset A ⊂ C,
A
A
A
c(x,y,z)
2
dµ(x) dµ(y) dµ(z) ≤ Cµ(A).
In Section 3 we explore the relation between the Cauchy integral, analytic capacity,
and curvature. In Section 4 we complete the proof of Theorem 1.1. Finally, in Section
5 we study the analytic capacity γ
+
.
A constant with a subscript, such as C
0
, retains its value throughout the paper,
while constants denoted by the letter C may change in different occurrences.
Acknowledgments. I would like to thank Mark Melnikov for introducing me to
this subject and for his valuable advice. Also, I wish to express my thanks to Joan
Verdera for many helpful suggestions and comments.
2. The curvature operator. Throughout the paper, µ is a positive continuous
Radon measure on the complex plane. Also, if A ⊂ C is µ-measurable, we set
c
2
(x,y,A)=
A
c(x,y,z)
2
dµ(z), x, y ∈ C,
and, if A,B,C ⊂ C are µ-measurable, then
c
2
(x,A,B) =
A
B
c(x,y,z)
2
dµ(y)dµ(z), x ∈ C,
and
c
2
(A,B,C)=
A
B
C
c(x,y,z)
2
dµ(x) dµ(y) dµ(z).
The total curvature of A (with respect to µ) is defined as
c
2
(A) =
A
A
A
c(x,y,z)
2
dµ(x) dµ(y) dµ(z).
Also, we define the curvature operator K as
K(f )(x) =
k(x, y)f (y) dµ(y), x ∈ C,f ∈ Ł
1
loc
(µ),
where k(x, y) is the kernel
k(x, y) =
c(x,y,z)
2
dµ(z) = c
2
(x,y, C), x, y ∈ C.
L
2
-BOUNDEDNESS OF THE CAUCHY OPERATOR 273
For a µ-measurable A ⊂ C,weset
K
A
(f )(x) =
c
2
(x,y, A)f (y) dµ(y), x ∈ C,f ∈ Ł
1
loc
(µ).
Thus,
K(f) = K
A
(f )+K
C\A
(f ).
We say that µ satisfies the weak local curvature condition if there are constants
0 <α≤ 1 and C
2
such that for each disc centered at some point of spt(µ), there
exists a compact subset S ⊂ such that
µ(S) ≥ α µ() and c
2
(S) ≤ C
2
µ(S). (7)
In this section we prove the following result.
Theorem 2.1. Let µ be a positive Radon measure with linear growth that satisfies
the weak local curvature condition. Then K is bounded from L
p
(µ) to L
p
(µ), 1 <
p<∞, and from M(C) to L
1,∞
(µ).
Corollary 2.2. Let µ be a positive Radon measure with linear growth that sat-
isfies the weak local curvature condition. Then there exists a constant C such that for
all µ-measurable sets A,B ⊂ C,
c
2
(A,B,C) ≤ C
µ(A)µ(B).
In particular,
c
2
(A) ≤ Cµ(A).
Proof. Since K is of strong-type (2,2),
c
2
(A,B,C) =
A
K(χ
B
)dµ
≤χ
A
L
2
(µ)
K(χ
B
)
L
2
(µ)
≤ C
µ(A)µ(B).
From Corollary 2.2 it follows that if µ has linear growth, the local and the weak
local curvature conditions are equivalent.
Some remarks about Theorem 2.1 are in order. The proof of the L
p
-boundedness of
the curvature operator is based on a “good λ-inequality.” The fact that K is a positive
operator seems to be essential to proving the L
p
-boundedness of K without assuming
that µ is a doubling measure. Recall that in [S1], [S2], and [SW] the boundedness
of some positive operators in L
p
(µ) is studied without assuming that µ is doubling,
too. Our proof is inspired by these papers.
We consider the centered Hardy-Littlewood maximal operator
M
µ
(f )(x) = sup
r>0
1
µ
(x,r)
(x,r)
|f |dµ.