1 Introduction
The Calderón-Zygmund theory emerged in the late 1950’s from efforts to treat the Hilbert transform on the real line, as well as its higher dimensional version, the Riesz transforms in
R
n
. In particular, a lot of the focus has been on understanding their mapping properties when acting from the Lebesgue space
L
p
(
R
n
)
into itself, for p ∈ (1, ∞). Main protagonists include A.P. Calderón, A. Zygmund, M. Cotlar, S.G. Mihlin, B. Dahlberg, G. David, J.L. Journé, S. Semmes, E. Stein, R. Coifman, A. McIntosh, Y. Meyer, among others.
Denote by
L
n
the Lebesgue measure in
R
n
. We let L
p
denoted the standard Lebesgue scale, with respect to the measure
L
n
. The subscript “loc” indicates local absolute integrability, while the subscript “c” indicates compact support. As is customary, we also denote by
S
(
R
n
)
the class of Schwartz functions in
R
n
, and by
S
′
(
R
n
)
the class of tempered distributions in
R
n
(cf., e.g., [1]).
Definition 1.1.
Let
T
:
S
(
R
n
)
→
S
′
(
R
n
)
be a linear and continuous operator with the property that there exists a function
K
(
⋅
,
⋅
)
∈
L
loc
1
R
n
×
R
n
\
diag
such that for every
f
∈
S
(
R
n
)
with compact support the distribution Tf can be identified with the function
(1.1)
(
T
f
)
(
x
)
=
∫
R
n
K
(
x
,
y
)
f
(
y
)
d
L
n
(
y
)
for
L
n
-a.e.
x
∈
R
n
\
s
u
p
p
f
.
In such a case, we say that T is associated with the kernel K.
Definition 1.2.
Call a function
K
(
⋅
,
⋅
)
∈
L
loc
1
R
n
×
R
n
\
diag
standard in the first variable if there exit C
0 ∈ (1, ∞) and γ ∈ (0, 1] with the property that
(1.2)
|
K
(
x
,
y
)
|
≤
C
0
|
x
−
y
|
n
,
for
all
x
,
y
∈
R
n
such that
x
≠
y
,
(1.3)
|
K
(
x
,
y
)
−
K
(
x
′
,
y
)
|
≤
C
0
|
x
−
x
′
|
γ
|
x
−
y
|
n
+
γ
,
for
all
x
,
x
′
,
y
∈
R
n
such that
x
≠
y
,
x
′
≠
y
,
and
|
x
−
y
|
>
2
|
x
−
x
′
|
.
Call a function
K
(
⋅
,
⋅
)
∈
L
loc
1
R
n
×
R
n
\
diag
standard in the second variable provided its transpose K
t
(x, y)≔K(y, x), for all
x
,
y
∈
R
n
with x ≠ y, is standard in the first variable.
A singular integral operator in
R
n
is an operator
T
:
S
(
R
n
)
→
S
′
(
R
n
)
associated with a kernel K that is standard in both variables.
For a given linear and continuous map
T
:
S
(
R
n
)
→
S
′
(
R
n
)
, we define its transpose
T
t
:
S
(
R
n
)
→
S
′
(
R
n
)
via the requirement that
(1.4)
T
t
φ
,
ψ
S
S
′
≔
φ
,
T
ψ
S
′
S
for
all
φ
,
ψ
∈
S
(
R
n
)
.
A fundamental issue in the Calderón-Zygmund theory is determining additional conditions a singular integral operator should satisfy that ensure its extension as a bounded map from
L
2
(
R
n
)
into
L
2
(
R
n
)
.
For each
N
∈
N
introduce the set of normalized bump functions
(1.5)
B
N
≔
ϕ
∈
C
∞
(
R
n
)
,
s
u
p
p
ϕ
⊂
B
(
0,1
)
,
‖
ϕ
‖
C
N
(
R
n
)
≤
1
,
where
‖
ϕ
‖
C
N
(
R
n
)
≔
max
‖
α
|
≤
N
‖
∂
α
ϕ
‖
L
∞
(
R
n
)
. Also, for an arbitrary ϕ ∈ B
N
consider its translates and dilates defined as
(1.6)
ϕ
R
,
x
0
(
x
)
≔
ϕ
(
x
−
x
0
)
/
R
for
all
x
∈
R
n
,
for each
x
0
∈
R
n
and R ∈ (0, ∞).
In their celebrated result known as the T(1) Theorem, G. David and J.L. Journé gave necessary and sufficient conditions under which a singular integral operator T extends boundedly from
L
2
(
R
n
)
into itself. Let
B
M
O
(
R
n
)
denote the John-Nirenberg space of functions of bounded mean oscillations in
R
n
.
Theorem 1.3
([2]). Let T be a singular integral operator in
R
n
. Then the operator T extends as a bounded mapping
T
:
L
2
(
R
n
)
→
L
2
(
R
n
)
if and only if all of the following conditions hold:
T is weakly bounded, i.e., there exist
N
∈
N
and C ∈ (0, ∞) such that
(1.7)
⟨
T
ϕ
R
,
x
,
ψ
R
,
x
⟩
≤
C
R
n
,
∀
ϕ
,
ψ
∈
B
N
,
∀
x
∈
R
n
,
R
∈
(
0
,
∞
)
;
T
(
1
)
∈
B
M
O
(
R
n
)
;
T
t
(
1
)
∈
B
M
O
(
R
n
)
.
This is a powerful result but the main drawback is that it is hard to check the memberships in (2) and (3). There exists a version of the T(1) Theorem due to E. Stein [[3], Theorem 3, pp. 294–300] which repackages the conditions (1)–(3) in the T(1) Theorem and avoids references to the membership of T(1) and T
t
(1) to
B
M
O
(
R
n
)
.
Theorem 1.4.
Let T be a singular integral operator in
R
n
. Then the operator T extends as a bounded mapping
T
:
L
2
(
R
n
)
→
L
2
(
R
n
)
if and only if both T and T
t
are restrictedly bounded in the sense that there exist
N
∈
N
and C ∈ (0, ∞) with the property that for every function ϕ ∈ B
N
one has
T
ϕ
R
,
x
,
T
t
ϕ
R
,
x
∈
L
2
(
R
n
)
and
(1.8)
‖
T
ϕ
R
,
x
‖
L
2
(
R
n
)
≤
C
R
n
/
2
,
‖
T
t
ϕ
R
,
x
‖
L
2
(
R
n
)
≤
C
R
n
/
2
for each
x
∈
R
n
and each R ∈ (0, ∞).
Note that if T is a restrictedly bounded singular integral operator in
R
n
then T is weakly bounded (cf. (1.7)) since by the Cauchy-Schwarz inequality we may estimate
(1.9)
⟨
T
ϕ
R
,
x
,
ψ
R
,
x
⟩
≤
‖
T
ϕ
R
,
x
‖
L
2
(
R
n
)
‖
ψ
R
,
x
‖
L
2
(
R
n
)
≤
C
R
n
/
2
R
n
/
2
=
C
R
n
.
Let us now consider the Cauchy operator on a planar Lipschitz curve which plays a central role in Calderón-Zygmund theory. Specifically, if
φ
:
R
→
R
is a Lipschitz function and
Σ
≔
{
(
x
,
φ
(
x
)
)
:
x
∈
R
}
, after a suitable change of variables, one is led to defining the truncated Cauchy operator acting on functions
f
∈
L
1
R
,
d
x
1
+
|
x
|
via
(1.10)
C
φ
,
ε
f
(
x
)
≔
∫
y
∈
R
,
|
y
−
x
|
>
ε
f
(
y
)
y
−
x
+
i
(
φ
(
y
)
−
φ
(
x
)
)
d
y
,
for
L
1
−
a.e.
x
∈
R
,
where ɛ > 0 is arbitrary.
In this regard, it is natural to ask if one can apply Theorem 1.4 to conclude that C
φ,ɛ
extends as a bounded operator from
L
2
(
R
)
into
L
2
(
R
)
uniformly with respect to ɛ ∈ (0, ∞). The first hurdle comes from the fact that the kernel of C
φ,ɛ
, which is
(1.11)
k
ε
(
x
,
y
)
≔
1
|
x
−
y
|
>
ε
y
−
x
+
i
(
φ
(
y
)
−
φ
(
x
)
)
for
x
,
y
∈
R
with
x
≠
y
,
is not standard. To remedy this, we fix
θ
∈
C
∞
(
R
)
, θ ≡ 0 on (−1, 1) and θ ≡ 1 on
R
\
(
−
2,2
)
, and consider the smoothly truncated kernel
(1.12)
k
ε
̃
(
x
,
y
)
≔
θ
(
(
x
−
y
)
/
ε
)
y
−
x
+
i
(
φ
(
y
)
−
φ
(
x
)
)
for
x
,
y
∈
R
with
x
≠
y
.
Then it is straightforward to check that
k
ε
̃
is a standard kernel and, if C
φ,(ɛ) denotes the operator associated with this kernel, then there exists C ∈ (0, ∞) such that, in a point-wise fashion,
(1.13)
C
φ
,
ε
f
−
C
φ
,
(
ε
)
f
≤
C
M
f
for
all
f
∈
L
1
R
,
d
x
1
+
|
x
|
,
where
M
is the Hardy-Littlewood maximal operator in
R
.
Melnikov and Verdera have discovered a beautiful geometric argument in [4] which produces the following local estimate:
(1.14)
∫
I
C
φ
,
ε
f
2
d
L
1
1
/
2
≤
C
‖
f
‖
L
∞
(
R
)
L
1
(
I
)
1
/
2
for
all
intervals
I
⊂
R
of finite length and for
all
f
∈
L
∞
(
R
)
with
s
u
p
p
f
⊆
I
.
Employing the estimates for the kernel of C
φ,ɛ
and Minkowski’s inequality, one obtains a similar estimate when the integral in (1.14) is taken over
R
\
2
I
. Together with the version of (1.14) written for 2I, this yields the following stronger estimate
(1.15)
∫
R
C
φ
,
ε
f
2
d
L
1
1
/
2
≤
C
‖
f
‖
L
∞
(
R
)
L
1
(
I
)
1
/
2
for
all
intervals
I
⊂
R
of finite length and for
all
f
∈
L
∞
(
R
)
with
s
u
p
p
f
⊆
I
.
Writing now (1.15) with f≔ϕ
R,x
, for arbitrary ϕ ∈ B
N
,
x
∈
R
n
, and R ∈ (0, ∞), it follows that
(1.16)
C
φ
,
ε
ϕ
R
,
x
L
2
(
R
n
)
≤
C
‖
ϕ
R
,
x
‖
L
∞
(
R
)
L
1
s
u
p
p
ϕ
R
,
x
1
/
2
≤
C
R
1
/
2
,
hence C
φ,ɛ
is restrictedly bounded. In turn, (1.16), (1.13), and the L
2-boundedness of
M
imply that C
φ,(ɛ) is also restrictedly bounded since
(1.17)
‖
C
φ
,
(
ε
)
ϕ
R
,
x
‖
L
2
(
R
n
)
≤
‖
C
φ
,
ε
ϕ
R
,
x
‖
L
2
(
R
n
)
+
C
‖
M
ϕ
R
,
x
‖
L
2
(
R
n
)
≤
C
R
1
/
2
+
C
‖
ϕ
R
,
x
‖
L
2
(
R
)
≤
C
R
1
/
2
.
Observing that
C
φ
,
(
ε
)
t
=
−
C
φ
,
(
ε
)
, Theorem 1.4 applies to the singular integral operator C
φ,(ɛ) and allows us to conclude that C
φ,(ɛ) extends as a bounded operator from
L
2
(
R
n
)
into
L
2
(
R
n
)
. That the same is true for C
φ,ɛ
is a consequence of the latter, (1.13), and the fact that the Hardy-Littlewood maximal operator
M
is bounded on
L
2
(
R
n
)
.
The above argument shows that it is possible to employ Stein’s result recalled in Theorem 1.4 in order to convert Melnikov and Verdera’s local estimate (1.14) into a global estimate in
L
2
(
R
n
)
for C
φ,ɛ
, uniform with respect to ɛ > 0.
The goal here is to find a new proof for the L
2-boundedness of C
φ,ɛ
that makes use of the Melnikov-Verdera local estimate (1.14) without relying on Theorem 1.4 (whose proof in [3] is not only long and convoluted, but it uses techniques specific to
R
n
, like the Fourier transform). Our result, which directly yields L
p
estimates, is stated in the next section, as Theorem 2.1. A key feature is the fact that the underlying Euclidean space is now replaced by a generic Ahlfors regular set. After a number of preliminaries, the proof of Theorem 2.1 is provided in the last section of the paper.
2 Main result
Let
H
n
−
1
denote the n − 1-dimensional Hausdorff measure in
R
n
. A closed set
Σ
⊆
R
n
is Ahlfors regular provided there exist constants c
Σ, C
Σ ∈ (0, ∞) such that
(2.18)
c
Σ
r
n
−
1
≤
H
n
−
1
B
(
x
,
r
)
∩
Σ
≤
C
Σ
r
n
−
1
for each
x
∈
Σ
and
r
∈
0,2
d
i
a
m
(
Σ
)
.
Let
Σ
⊆
R
n
be a closed Ahlfors regular set. Then the surface ball centered at x ∈ Σ and of radius r ∈ (0, ∞) is Δ(x, r)≔B(x, r) ∩Σ. With
σ
≔
H
n
−
1
Σ
, for any
f
∈
L
loc
1
(
Σ
,
σ
)
and any surface ball Δ we denote
(2.19)
f
Δ
≔
⨏
Δ
f
d
σ
≔
1
σ
(
Δ
)
∫
Δ
f
d
σ
.
Here is the main result in this paper. A number of relevant comments and corollaries are included a little further below.
Theorem 2.1.
Fix a dimension
n
∈
N
with n ≥ 2, and suppose
Σ
⊆
R
n
is a closed, unbounded, Ahlfors regular set. Consider the “surface” measure
σ
≔
H
n
−
1
Σ
. Assume
K
:
Σ
×
Σ
\
d
i
a
g
→
C
is a σ ⊗ σ-measurable function with the property that there exists C
0 ∈ (0, ∞) such that
(2.20)
|
K
(
x
,
y
)
|
≤
C
0
|
x
−
y
|
n
−
1
for all
x
,
y
∈
Σ
satisfying
x
≠
y
.
For each ɛ ∈ (0, ∞) consider the truncated singular integral operator whose action on each function
f
∈
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
is given by
(2.21)
(
T
ε
f
)
(
x
)
≔
∫
y
∈
Σ
,
|
y
−
x
|
>
ε
K
(
x
,
y
)
f
(
y
)
d
σ
(
y
)
for
all
x
∈
Σ
.
[Local Estimate for T
ɛ
] Suppose K actually is standard in the first variable, hence there exits γ ∈ (0, 1) with the property that
(2.22)
|
K
(
x
,
y
)
−
K
(
x
′
,
y
)
|
≤
C
0
|
x
−
x
′
|
γ
|
x
−
y
|
n
−
1
+
γ
for
all
x
,
x
′
,
y
∈
Σ
with
x
≠
y
,
x
′
≠
y
,
and
|
x
−
y
|
>
2
|
x
−
x
′
|
.
In addition, assume that there exists C
1 ∈ (0, ∞) such that for every surface ball Δ ⊂ Σ, every Lipschitz function f on Σ with supp f ⊂ Δ, and every threshold ɛ ∈ (0, ∞) one has
(2.23)
⨏
Δ
|
T
ε
f
|
d
σ
≤
C
1
‖
f
‖
L
∞
(
Σ
,
σ
)
.
Fix a reference point x
* ∈ Σ and for each ɛ ∈ (0, ∞) define the modified version of T
ɛ
from (2.21) by setting, for every
f
∈
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
+
γ
and every x ∈ Σ,
(2.24)
T
ε
mod
f
(
x
)
≔
∫
Σ
K
(
x
,
y
)
1
|
x
−
y
|
>
ε
−
K
(
x
*
,
y
)
1
|
x
*
−
y
|
>
1
f
(
y
)
d
σ
(
y
)
.
Then the operator
T
ε
mod
:
L
∞
(
Σ
,
σ
)
→
B
M
O
(
Σ
,
σ
)
is well defined, linear, and bounded uniformly with respect to ɛ ∈ (0, ∞).
[Local Estimate for
T
ε
t
] Suppose K is standard in the second variable, i.e., its transpose K
t
(x, y)≔K(y, x), for all x, y ∈ Σ with x ≠ y, is standard in the first variable. Moreover, assume there exists C
2 ∈ (0, ∞) such that for every ɛ ∈ (0, ∞), every surface ball Δ ⊂ Σ, and every Lipschitz function f on Σ with supp f ⊂ Δ one has
(2.25)
⨏
Δ
|
T
ε
t
f
|
d
σ
≤
C
2
‖
f
‖
L
∞
(
Σ
,
σ
)
,
where
T
ε
t
is the truncated operator defined as in (2.21) with K replaced by K
t
, i.e., for each
f
∈
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
,
(2.26)
T
ε
t
f
(
x
)
≔
∫
y
∈
Σ
,
|
y
−
x
|
>
ε
K
(
y
,
x
)
f
(
y
)
d
σ
(
y
)
for
all
x
∈
Σ
.
Then the operator T
ɛ
: H
1(Σ, σ) → L
1(Σ, σ) is well defined, linear, and bounded uniformly with respect to ɛ ∈ (0, ∞).
If the assumptions in (1)–(2) are satisfied, then for every p ∈ (1, ∞) the operators T
ɛ
: L
p
(Σ, σ) → L
p
(Σ, σ) and
T
ε
t
:
L
p
(
Σ
,
σ
)
→
L
p
(
Σ
,
σ
)
are bounded uniformly with respect to ɛ ∈ (0, ∞). Moreover, for all exponents p, p′ ∈ (1, ∞) with
1
p
+
1
p
′
=
1
, and all thresholds ɛ ∈ (0, ∞), the functional analytic transpose of T
ɛ
: L
p
(Σ, σ) → L
p
(Σ, σ) is the operator
T
ε
t
:
L
p
′
(
Σ
,
σ
)
→
L
p
′
(
Σ
,
σ
)
.
A few comments regarding Theorem 2.1 are in order.
Remark 2.1.
(i) Conditions (2.23) and (2.25) are natural if one desires to obtain the L
p
-boundedness stated in (3). Indeed, under the assumption that for some p ∈ (1, ∞) the operator T
ɛ
: L
p
(Σ, σ) → L
p
(Σ, σ) is bounded uniformly in ɛ ∈ (0, ∞), for each f ∈ L
∞
(Σ, σ) with support contained in some surface ball Δ, we may estimate
(2.27)
⨏
Δ
|
T
ε
f
|
d
σ
≤
⨏
Δ
|
T
ε
f
|
p
d
σ
1
/
p
≤
σ
(
Δ
)
−
1
/
p
‖
T
ε
f
‖
L
p
(
Σ
,
σ
)
≤
C
σ
(
Δ
)
−
1
/
p
‖
f
‖
L
p
(
Σ
,
σ
)
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
,
for some C ∈ (0, ∞) independent of ɛ, f, and Δ. Hence, (2.23) holds. Moreover, under the assumption that there exists some p ∈ (1, ∞) such that the operator
T
ε
t
:
L
p
(
Σ
,
σ
)
→
L
p
(
Σ
,
σ
)
is bounded uniformly in ɛ ∈ (0, ∞), an estimate similar to (2.27) this time applied to
T
ε
t
implies that (2.25) also holds.
(ii) The local estimates stated in (2.23) and (2.25) are L
1-styled. By Hölder’s inequality, these are less restrictive than the L
q
-styled estimates, for some integrability exponent q ∈ (1, ∞), of the form
(2.28)
⨏
Δ
|
T
ε
f
|
q
d
σ
1
/
q
≤
C
1
‖
f
‖
L
∞
(
Σ
,
σ
)
,
⨏
Δ
|
T
ε
t
f
|
q
d
σ
1
/
q
≤
C
2
‖
f
‖
L
∞
(
Σ
,
σ
)
.
In particular, the “flat” version of our theorem (see Corollary 2.4 below) “virtually” contains Theorem 1.4.
(iii) Theorem 2.1 directly accommodates truncated singular integral operators with estimates that are uniform with respect to the truncation. This is a desirable feature given the way principal value singular integral operators are defined.
(iv) The proof we give does not use techniques specific to
R
n
(such as the Fourier transform) and, in fact, adapts to the setting of spaces of homogeneous type (the most natural measure-geometric setting for this type of result). This version brings into focus the role of the distance and measure.
The statement of Theorem 2.1 streamlines when the kernel K is standard in both variables. Since this is often the case for many of the common singular integral operators, we single out such a version in the next corollary.
Corollary 2.2.
Fix
n
∈
N
with n ≥ 2, and suppose
Σ
⊆
R
n
is a closed, unbounded, Ahlfors regular set. Abbreviate
σ
≔
H
n
−
1
Σ
and let
K
(
⋅
,
⋅
)
∈
L
loc
1
Σ
×
Σ
\
diag
,
σ
⊗
σ
be a kernel which is standard in both variables (cf. Theorem 2.1). For each ɛ ∈ (0, ∞) consider the truncated operators T
ɛ
and
T
ε
t
defined in (2.21) and (2.26).
Then the following are equivalent:
For some, or every, p, q ∈ (1, ∞) the operators T
ɛ
: L
p
(Σ, σ) → L
p
(Σ, σ) and
T
ε
t
:
L
q
(
Σ
,
σ
)
→
L
q
(
Σ
,
σ
)
are bounded uniformly with respect to ɛ.
There exists a constant C ∈ (0, ∞) such that
(2.29)
⨏
Δ
|
T
ε
f
|
d
σ
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
and
⨏
Δ
|
T
ε
t
f
|
d
σ
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
for every ɛ > 0, surface ball Δ ⊂ Σ, and Lipschitz function f on Σ with supp f ⊂ Δ.
In addition, if either (1) or rm (2) holds, then for all p, p′ ∈ (1, ∞) satisfying
1
p
+
1
p
′
=
1
, the functional analytic transpose of the operator T
ɛ
: L
p
(Σ, σ) → L
p
(Σ, σ) is the operator
T
ε
t
:
L
p
′
(
Σ
,
σ
)
→
L
p
′
(
Σ
,
σ
)
.
Also, corresponding to the end-point cases p = 1 and p = ∞, if (2) holds then
(2.30)
T
ε
:
H
1
(
Σ
,
σ
)
→
L
1
(
Σ
,
σ
)
,
T
ε
mod
:
L
∞
(
Σ
,
σ
)
→
B
M
O
(
Σ
,
σ
)
,
(2.31)
T
ε
t
:
H
1
(
Σ
,
σ
)
→
L
1
(
Σ
,
σ
)
,
T
ε
t
mod
:
L
∞
(
Σ
,
σ
)
→
B
M
O
(
Σ
,
σ
)
are bounded uniformly with respect to ɛ ∈ (0, ∞), where
T
ε
mod
is the operator defined in (2.24) and
(2.32)
(
T
ε
t
mod
f
)
(
x
)
≔
∫
Σ
K
(
y
,
x
)
1
|
x
−
y
|
>
ε
−
K
(
y
,
x
*
)
1
|
x
*
−
y
|
>
1
f
(
y
)
d
σ
(
y
)
for every
f
∈
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
+
γ
and every x ∈ Σ.
As an application of Corollary 2.2, we note that by combining this result with the work of Nazarov, Tolsa, and Volberg from [5] (in the format recorded in [[6], Theorem 5.10.3, pp. 458–460]) we see that whenever Σ is the boundary of an Ahlfors regular domain
Ω
⊆
C
for which local estimates like (2.29) hold for the truncated Cauchy operator on Σ = ∂Ω then necessarily Ω is a uniformly rectifiable domain (see [6] for all relevant terminology). Moreover, a similar result in higher dimensions is true, provided the truncated Cauchy operator is replaced by the family of truncated Riesz transforms.
Here is another application of Corollary 2.2 to the truncated Cauchy operator (1.10). As mentioned earlier, the original motivation for the work in this paper stemmed from the desire to obtain a self-contained proof of the L
p
-boundedness of the Cauchy operator on Lipschitz curves starting from (1.14).
Theorem 2.3.
Let
φ
:
R
→
R
be a Lipschitz function and for each ɛ ∈ (0, ∞) recall the truncated Cauchy operator acting on functions
f
∈
L
1
R
,
d
x
1
+
|
x
|
according to (1.10). Then the following are true.
The mapping
C
φ
,
ε
:
H
1
(
R
)
→
L
1
(
R
)
is well defined, linear and bounded with operator norm controlled uniformly with respect to ɛ ∈ (0, ∞).
Consider the modified version of C
φ,ɛ
acting on every
f
∈
L
1
R
,
d
x
1
+
|
x
|
2
according to
(2.33)
C
φ
,
ε
mod
f
(
x
)
≔
∫
y
∈
R
1
|
y
−
x
|
>
ε
y
−
x
+
i
(
φ
(
y
)
−
φ
(
x
)
)
−
1
|
y
|
>
1
y
+
i
(
φ
(
y
)
−
φ
(
0
)
)
d
y
at every
x
∈
R
. Then
C
φ
,
ε
mod
:
L
∞
(
R
)
→
B
M
O
(
R
)
is a well-defined linear and bounded map whose operator norm is controlled uniformly with respect to ɛ ∈ (0, ∞).
For every p ∈ (1, ∞) the operators
C
φ
,
ε
:
L
p
(
R
)
→
L
p
(
R
)
are bounded and their operator norms are controlled uniformly with respect to the truncation parameter ɛ ∈ (0, ∞).
Proof.
This is a direct consequence of Corollary 2.2 whose applicability is ensured by the fact that the L
2-styled local estimate (1.14) obtained by Melnikov and Verdera implies (via Hölder’s inequality) a corresponding L
1-styled local estimates and the fact that
C
φ
,
ε
t
=
−
C
φ
,
ε
.□
Since this has intrinsic value, we end this section by recording the “flat” Euclidean version of Theorem 2.1, corresponding to the case when n is replaced by n + 1 and when we choose
Σ
≔
R
n
×
{
0
}
⊆
R
n
+
1
, naturally identified with
R
n
.
Corollary 2.4.
Let
K
:
R
n
×
R
n
\
diag
→
C
be an
L
n
⊗
L
n
measurable function with the property that there exists a constant C
0 ∈ (0, ∞) such that
(2.34)
|
K
(
x
,
y
)
|
≤
C
0
|
x
−
y
|
n
for
all
x
,
y
∈
R
n
satisfying
x
≠
y
.
For each ɛ ∈ (0, ∞) consider the truncated singular integral operator whose action on each
f
∈
L
1
R
n
,
d
x
1
+
|
x
|
n
is given by
(2.35)
(
T
ε
f
)
(
x
)
≔
∫
y
∈
R
n
,
|
y
−
x
|
>
ε
K
(
x
,
y
)
f
(
y
)
d
L
n
(
y
)
for
all
x
∈
R
n
.
[Local Estimate for T
ɛ
] Suppose K is standard in the first variable (cf. Definition 1.2) and there exists a constant C
1 ∈ (0, ∞) such that for every ball
B
⊂
R
n
, and every
f
∈
C
c
∞
(
R
n
)
with supp f ⊂ B, and every ɛ ∈ (0, ∞) one has
(2.36)
⨏
B
|
T
ε
f
|
d
L
n
≤
C
1
‖
f
‖
L
∞
(
R
n
)
.
For each ɛ ∈ (0, ∞) define the modified version of T
ɛ
from (2.21) by setting
(2.37)
T
ε
mod
f
(
x
)
≔
∫
R
n
K
(
x
,
y
)
1
|
x
−
y
|
>
ε
−
K
(
0
,
y
)
1
|
y
|
>
1
f
(
y
)
d
L
n
(
y
)
for every
f
∈
L
1
R
n
,
d
x
1
+
|
x
|
n
+
γ
and every
x
∈
R
n
.
Then
T
ε
mod
:
L
∞
(
R
n
)
→
B
M
O
(
R
n
)
is a well-defined linear and bounded map whose operator norm is controlled uniformly with respect to the truncation parameter ɛ ∈ (0, ∞).
[Local Estimate for
T
ε
t
] Suppose K is standard in the second variable (cf. Definition 1.2) and there exists a constant C
2 ∈ (0, ∞) such that for every ɛ ∈ (0, ∞), every ball
B
⊂
R
n
, and every function
f
∈
C
c
∞
(
R
n
)
with supp f ⊂ B one has
(2.38)
⨏
B
|
T
ε
t
f
|
d
L
n
≤
C
2
‖
f
‖
L
∞
(
R
n
)
,
where
T
ε
t
is the truncated operator defined as in (2.35) with K replaced by K
t
, i.e., for each
f
∈
L
1
R
n
,
d
x
1
+
|
x
|
n
,
(2.39)
T
ε
t
f
(
x
)
≔
∫
y
∈
R
n
,
|
y
−
x
|
>
ε
K
(
y
,
x
)
f
(
y
)
d
L
n
(
y
)
for
all
x
∈
R
n
.
Then
T
ε
:
H
1
(
R
n
)
→
L
1
(
R
n
)
is a well-defined linear and bounded map whose operator norm is controlled uniformly with respect to ɛ ∈ (0, ∞).
If the assumptions in (1)–(2) are satisfied, then for every p ∈ (1, ∞) the maps
T
ε
:
L
p
(
R
n
)
→
L
p
(
R
n
)
and
T
ε
t
:
L
p
(
R
n
)
→
L
p
(
R
n
)
are bounded and their corresponding operator norms are controlled uniformly with respect to ɛ ∈ (0, ∞). Moreover, for all p, p′ ∈ (1, ∞) with
1
p
+
1
p
′
=
1
, and any ɛ ∈ (0, ∞), the functional analytic transpose of
T
ε
:
L
p
(
R
n
)
→
L
p
(
R
n
)
is the operator
T
ε
t
:
L
p
′
(
R
n
)
→
L
p
′
(
R
n
)
.
3 Preliminaries
Throughout this section we fix a closed, unbounded, Ahlfors regular set
Σ
⊆
R
n
and set
σ
≔
H
n
−
1
Σ
.
Given a function
f
∈
L
loc
1
(
Σ
,
σ
)
set
(3.40)
‖
f
‖
BMO
(
Σ
,
σ
)
≔
sup
Δ
⊂
Σ
⨏
Δ
f
(
x
)
−
f
Δ
d
σ
(
x
)
<
∞
where the supremum is taken over all surface balls Δ in Σ (recall the notation from (2.19)). Then the John-Nirenberg space of functions of bounded mean oscillations over Σ is defined as
(3.41)
B
M
O
(
Σ
,
σ
)
≔
f
∈
L
loc
1
(
Σ
,
σ
)
:
‖
f
‖
BMO
(
Σ
,
σ
)
<
+
∞
.
For further reference, we find it useful to record here the fact that for each function
f
∈
L
loc
1
(
Σ
,
σ
)
and each surface ball Δ ⊆Σ,
(3.42)
inf
c
∈
C
⨏
Δ
f
(
x
)
−
c
d
σ
(
x
)
≤
⨏
Δ
f
(
x
)
−
f
Δ
d
σ
(
x
)
≤
2
inf
c
∈
C
⨏
Δ
f
(
x
)
−
c
d
σ
(
x
)
.
Much as in the Euclidean setting (cf. [7], [8], [3]), Hardy spaces on Ahlfors regular sets may be defined by demanding the membership of the Fefferman-Stein grand maximal function to Lebesgue spaces (see [9], and [10]). As in the standard Euclidean setting, these may be alternatively characterized in terms of atoms. To elaborate, fix some p
* ∈ (1, ∞). Then a σ-measurable function
a
:
Σ
→
C
is called an
L
p
*
-normalized H
1-atom if there exists a surface ball Δ(x
0, r)≔B(x
0, r) ∩Σ such that
(3.43)
s
u
p
p
a
⊂
Δ
(
x
0
,
r
)
,
∫
Σ
a
d
σ
=
0
,
and
‖
a
‖
L
p
*
(
Σ
,
σ
)
≤
σ
(
Δ
(
x
0
,
r
)
)
1
/
p
*
−
1
.
The atomic characterization of the Hardy space H
1(Σ, σ) permits us to describe it as the ℓ
1-span of such atoms with convergence in L
1(Σ, σ), i.e.,
(3.44)
H
1
(
Σ
,
σ
)
=
f
∈
L
1
(
Σ
,
σ
)
:
there exist
{
λ
j
}
j
∈
N
∈
ℓ
1
(
N
)
and
L
p
*
−
normalized
H
1
−
atoms
{
a
j
}
j
∈
N
such that
f
=
∑
j
=
1
∞
λ
j
a
j
in
L
1
(
Σ
,
σ
)
equipped with the quasi-norm
(3.45)
‖
f
‖
H
1
(
Σ
,
σ
)
≔
inf
∑
j
=
1
∞
|
λ
j
|
where the infimum is taken over all writings
f
=
∑
j
=
1
∞
λ
j
a
j
of f ∈ H
1(Σ, σ) as in (3.44). For more details regarding atomic Hardy spaces on Ahlfors regular sets see [[10], Section 4.4].
The space of finite linear combinations of atoms has excellent approximation qualities vis-a-vis to the entire Hardy space. The following proposition summarizes the main properties of this space. For a proof see [[10], Proposition 4.4.4, p. 164].
Proposition 3.1.
Let
Σ
⊆
R
n
be a closed, unbounded, Ahlfors regular set and abbreviate
σ
≔
H
n
−
1
Σ
. Consider p
* ∈ (1, ∞) and let
H
fin
1
,
p
*
(
Σ
,
σ
)
stand for the vector space consisting of all finite linear combinations of
L
p
*
-normalized H
1-atoms on Σ. Also, define a norm on
H
fin
1
,
p
*
(
Σ
,
σ
)
by setting
(3.46)
‖
f
‖
H
fin
1
,
p
*
(
Σ
,
σ
)
≔
inf
∑
j
=
1
N
|
λ
j
|
p
1
/
p
:
N
∈
N
and
f
=
∑
j
=
1
N
λ
j
a
j
for some
{
λ
j
}
1
≤
j
≤
N
⊆
C
and
L
p
*
−
normalized
H
1
−
a
t
o
m
s
{
a
j
}
1
≤
j
≤
N
,
for each
f
∈
H
fin
1
,
p
*
(
Σ
,
σ
)
. Then
(3.47)
H
fin
1
,
p
*
(
Σ
,
σ
)
=
{
f
∈
L
c
p
*
(
Σ
,
σ
)
:
∫
Σ
f
d
σ
=
0
}
,
H
fin
1
,
p
*
(
Σ
,
σ
)
i
s
a
d
e
n
s
e
l
i
n
e
a
r
s
u
b
s
p
a
c
e
o
f
H
1
(
Σ
,
σ
)
,
‖
⋅
‖
H
fin
1
,
p
*
(
Σ
,
σ
)
and
‖
⋅
‖
H
1
(
Σ
,
σ
)
a
r
e
e
q
u
i
v
a
l
e
n
t
n
o
r
m
s
o
n
H
fin
1
,
p
*
(
Σ
,
σ
)
.
In the current setting, it has also been proved in [[10], Theorem 4.6.1, pp. 182–183] that
(3.48)
H
1
(
Σ
,
σ
)
*
=
B
M
O
(
Σ
,
σ
)
/
∼
where ⋅/∼ means that we are modding out constants. Henceforth we agree to denote
by
BMO
⟨
⋅
,
⋅
⟩
H
1
the canonical duality pairing. This pairing turns out to be an integral pairing when the respective H
1 function is an atom. Specifically, as a consequence of [[10], Corollary 4.8.10, p. 205] we have
(3.49)
If
g
∈
B
M
O
(
Σ
,
σ
)
and
a
is an
L
p
*
−
normalized
H
1
−
atom, then
BMO
⟨
g
,
a
⟩
H
1
=
∫
Σ
g
a
d
σ
and
∫
Σ
g
a
d
σ
≤
C
‖
g
‖
BMO
(
Σ
,
σ
)
for
some
C
∈
(
0
,
∞
)
independent of
a
.
Next, we recall a couple of useful real interpolation results. First, as proved in [[10], (4.3.1), p. 119] (in view of [ [10], (4.2.25), p. 117]), with the real interpolation method employed we have
(3.50)
L
p
(
Σ
,
σ
)
=
H
1
(
Σ
,
σ
)
,
L
∞
(
Σ
,
σ
)
1
−
1
p
,
p
for
all
p
∈
(
1
,
∞
)
.
Based on [[10], (4.3.1), p. 119], (3.48), the Duality Theorem for the real method of interpolation (cf. [[11], Theorem 3.7.1, p. 54], and Wolff’s four-space interpolation theorem (cf. [[12], Theorem 1, p. 199]) one may further show (see [13] for details) that
(3.51)
L
p
(
Σ
,
σ
)
=
L
1
(
Σ
,
σ
)
,
B
M
O
(
Σ
,
σ
)
1
−
1
p
,
p
for
all
p
∈
(
1
,
∞
)
.
Related results in the Euclidean setting are contained in [14].
This section is devoted to the proof of our main result, Theorem 2.1. Throughout, we let Lip(X) denote the space of Lipschitz functions on a set X, and use the subscript “c” to indicate compact support.
First, we observe that if
K
:
Σ
×
Σ
\
diag
→
C
is a σ ⊗ σ measurable function satisfying (2.20) and ɛ ∈ (0, ∞), then for
f
∈
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
and each x ∈ Σ relying on (2.20) we may estimate
(4.52)
∫
y
∈
Σ
,
|
y
−
x
|
>
ε
K
(
x
*
,
y
)
f
(
y
)
|
d
σ
(
y
)
≤
C
(
ε
,
x
)
∫
Σ
|
f
(
y
)
|
1
+
|
y
|
n
−
1
d
σ
(
y
)
<
∞
.
This shows that T
ɛ
f as in (2.21) is defined by an absolutely convergent integral. In addition, whenever the function
f
∈
L
1
(
Σ
,
σ
)
⊂
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
,
(4.53)
sup
x
∈
Σ
(
T
ε
f
)
(
x
)
≤
C
0
ε
n
−
1
‖
f
‖
L
1
(
Σ
,
σ
)
for
all
f
∈
L
1
(
Σ
,
σ
)
.
Hence, the operators
(4.54)
T
ε
,
T
ε
t
:
L
1
(
Σ
,
σ
)
→
L
∞
(
Σ
,
σ
)
are linear and bounded for each ɛ ∈ (0, ∞).
Step I.
We claim that the estimates in (2.23) and (2.25) self improve to similar estimates valid with the larger class of functions
L
c
∞
(
Σ
,
σ
)
replacing Lip
c
(Σ).
To see why this is true, first suppose that
(4.55)
there exists
C
1
∈
(
0
,
∞
)
independent of
ε
such that
(
2
.
23
)
holds for
every surface ball
Δ
and
every
f
∈
L
i
p
c
(
Σ
)
with
s
u
p
p
f
⊂
Δ
.
Pick an arbitrary surface ball Δ = B(x
0, r) ∩Σ for some x
0 ∈ Σ and some r ∈ (0, ∞), and let
f
∈
L
c
∞
(
Σ
,
σ
)
with supp f ⊂ Δ. Then f ∈ L
1(Σ, σ), so we may invoke [[6], Corollary 3.7.3, p. 283] with p≔1, X≔Σ, s≔n − 1, μ≔σ and obtain that there exists a sequence
(4.56)
{
g
j
}
j
∈
N
⊆
C
c
∞
(
R
n
)
with
g
j
|
Σ
→
f
in
L
1
(
Σ
,
σ
)
as
j
→
∞
,
and
g
j
|
Σ
→
f
pointwise
σ
−
a.e
.
o
n
Σ
as
j
→
∞
.
Without loss of generality, we may assume that we also have supp g
j
⊆ B(x
0, 2r) for each
j
∈
N
. Indeed, by replacing g
j
with g
j
φ where
φ
∈
C
c
∞
(
B
(
x
0
,
2
r
)
)
with φ ≡ 1 on B(x
0, r) takes care of it.
Now with
N
≔
‖
f
‖
L
∞
(
Σ
,
σ
)
define, for each
j
∈
N
,
(4.57)
g
j
,
N
:
=
min
{
max
{
g
j
,
−
N
}
,
N
}
=
max
{
min
{
g
j
,
N
}
,
−
N
}
.
Then given this definition, the properties of g
j
’s, and the fact that max and min preserve Lipschitzianity, these new functions satisfy:
(4.58)
‖
g
j
,
N
‖
L
∞
(
R
n
)
≤
‖
f
‖
L
∞
(
Σ
,
σ
)
,
g
j
,
N
∈
L
i
p
(
R
n
)
,
s
u
p
p
g
j
,
N
=
s
u
p
p
g
j
⊆
B
(
x
0
,
2
r
)
,
for each
j
∈
N
.
In view of the last line in (4.56), we also obtain σ-a.e. pointwise convergence on Σ of
{
g
j
,
N
}
j
∈
N
to f as j → ∞. The latter, (4.58), and Lebesgue’s Dominated Convergence Theorem then imply
(4.59)
g
j
,
N
|
Σ
→
f
in
L
1
(
Σ
,
σ
)
as
j
→
∞
.
If we now define
(4.60)
f
j
≔
g
j
,
N
|
Σ
for each
j
∈
N
,
then, for each
j
∈
N
,
(4.61)
f
j
∈
L
i
p
(
Σ
)
,
s
u
p
p
f
j
⊆
Δ
(
x
0
,
2
r
)
,
‖
f
j
‖
L
∞
(
Σ
,
σ
)
≤
‖
f
‖
L
∞
(
Σ
,
σ
)
,
and, after possibly passing to a subsequence, we may also arrange so that
(4.62)
f
j
→
f
in
L
1
(
Σ
,
σ
)
as
j
→
∞
and
(4.63)
f
j
(
x
)
→
f
(
x
)
for
σ
−
a.e.
x
∈
Σ
as
j
→
∞
.
Thus, using the fact that Σ is Ahlfors regular, (4.55), and (4.61), we may estimate
(4.64)
⨏
Δ
(
x
0
,
r
)
|
T
ε
f
j
|
d
σ
≤
σ
(
Δ
(
x
0
,
2
r
)
)
σ
(
Δ
(
x
0
,
r
)
)
⨏
Δ
(
x
0
,
2
r
)
|
T
ε
f
j
|
d
σ
≤
2
n
−
1
C
Σ
c
Σ
⋅
C
1
‖
f
j
‖
L
∞
(
Σ
,
σ
)
≤
2
n
−
1
C
Σ
C
1
c
Σ
‖
f
‖
L
∞
(
Σ
,
σ
)
,
uniformly with respect to
j
∈
N
. Passing to the limit as j → ∞ in (4.64), the desired conclusion in the claim will follow once we show
(4.65)
∫
Δ
(
x
0
,
r
)
|
T
ε
f
j
|
d
σ
→
∫
Δ
(
x
0
,
r
)
|
T
ε
f
|
d
σ
as
j
→
∞
.
In turn, the convergence in (4.65) is a consequence of Lebesgue’s Dominated Convergence Theorem whose applicability is guaranteed by (4.53), (4.62), and (4.61).
This completes the proof of the fact that if (2.23) holds then the same estimate will be true for f in the larger class
L
c
∞
(
Σ
,
σ
)
. That the same type of self-improvement also holds for (2.25) is seen in a similar fashion. This completes the proof of Step I.
Step II.
Proof of the statement in item (1).
Suppose K is standard in the first variable, i.e., that (2.20) and (2.22) hold. In addition, assume that the L
1-styled local estimate (2.23) is valid for every ɛ > 0, every Δ, and every
f
∈
L
c
∞
(
Σ
,
σ
)
with supp f ⊆Δ. In view of (3.41) and (3.42), to conclude that
T
ε
mod
:
L
∞
(
Σ
,
σ
)
→
B
M
O
(
Σ
,
σ
)
is bounded uniformly with respect to ɛ, it suffices to show that for every ɛ > 0,
f
∈
L
c
∞
(
Σ
,
σ
)
, and surface ball Δ ⊆Σ, there exists a constant
c
=
c
(
f
,
K
,
ε
,
Δ
)
∈
C
such that
(4.66)
⨏
Δ
T
ε
mod
f
−
c
d
σ
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
for some C ∈ (0, ∞) independent of f and ɛ.
To this end, fix ɛ > 0, a surface ball Δ = Δ(x
0, r) ⊂ Σ, and f ∈ L
∞
(Σ, σ). Decompose
(4.67)
f
=
f
1
+
f
2
,
f
1
≔
f
1
2
Δ
and
f
2
≔
f
1
Σ
\
2
Δ
,
where 2Δ≔Δ(x
0, 2r). Thus f
1 ∈ L
∞
(Σ, σ) with supp f
1 ⊆ 2Δ and we may reason as in (4.64) to write
(4.68)
⨏
Δ
T
ε
f
1
d
σ
≤
2
n
−
1
C
Σ
C
1
c
Σ
‖
f
1
‖
L
∞
(
Σ
,
σ
)
≤
2
n
−
1
C
Σ
C
1
c
Σ
‖
f
‖
L
∞
(
Σ
,
σ
)
.
In addition,
(4.69)
∫
Σ
K
(
x
*
,
y
)
1
|
x
*
−
y
|
>
1
f
1
(
y
)
d
σ
(
y
)
≤
C
0
σ
(
2
Δ
)
‖
f
‖
L
∞
(
Σ
,
σ
)
<
+
∞
.
Hence, if we set c
1≔ − (T
1
f
1)(x
*) it follows that
c
1
∈
C
. Also, from the definitions in (2.24) and (2.21) we have
T
ε
mod
f
1
(
x
)
=
(
T
ε
f
1
)
(
x
)
+
c
1
for each x ∈ Σ. When combined with (4.68) the latter implies
(4.70)
⨏
Δ
T
ε
mod
f
1
−
c
1
d
σ
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
for some C ∈ (0, ∞) independent of f and ɛ. This estimate suits our purposes.
To treat
T
ε
mod
f
2
we consider a smoothly truncated version of the operator
T
ε
mod
. Specifically, fix
θ
∈
C
∞
(
R
n
)
satisfying θ ≡ 0 on B(0, 1) and θ ≡ 1 on
R
n
\
B
(
0,2
)
. Then for each
g
∈
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
+
γ
define, at each x ∈ Σ,
(4.71)
(
T
(
ε
)
mod
g
)
(
x
)
≔
∫
Σ
K
(
x
,
y
)
θ
x
−
y
ε
−
K
(
x
0
,
y
)
θ
x
0
−
y
ε
g
(
y
)
d
σ
(
y
)
.
To determine how this smoothly truncated operator compares to the operator
T
ε
mod
, pick a function
g
∈
L
∞
(
Σ
,
σ
)
⊆
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
+
γ
. After choosing
(4.72)
R
>
max
{
1
/
3
,
ε
,
|
x
*
−
x
0
|
}
observe that since |x
0 − y|≥|x
* − y| − |x
* − x
0| we have
(4.73)
y
∈
R
n
\
B
(
x
*
,
3
R
)
⇒
|
x
0
−
y
|
>
2
R
>
max
{
2
ε
,
2
|
x
*
−
x
0
|
}
and
|
x
0
−
y
|
>
2
3
|
x
*
−
y
|
.
Hence, we may apply (2.22) to write
(4.74)
K
(
x
0
,
y
)
θ
x
0
−
y
ε
−
K
(
x
*
,
y
)
1
|
x
*
−
y
|
>
1
=
|
K
(
x
0
,
y
)
−
K
(
x
*
,
y
)
|
≤
C
0
|
x
*
−
x
0
|
γ
|
x
0
−
y
|
n
−
1
+
γ
for
all
y
∈
Σ
\
Δ
(
x
*
,
3
R
)
,
with C
0 as in (2.21) and (2.22). In light of (4.74), and recalling (2.21) and (4.73), we may estimate
(4.75)
∫
Σ
K
(
x
0
,
y
)
θ
x
0
−
y
ε
g
(
y
)
−
K
(
x
*
,
y
)
1
|
x
*
−
y
|
>
1
g
(
y
)
d
σ
(
y
)
≤
∫
Δ
(
x
*
,
3
R
)
K
(
x
0
,
y
)
θ
x
0
−
y
ε
g
(
y
)
d
σ
(
y
)
+
∫
Δ
(
x
*
,
3
R
)
K
(
x
*
,
y
)
1
|
x
*
−
y
|
>
1
g
(
y
)
d
σ
(
y
)
+
C
0
‖
g
‖
L
∞
(
Σ
,
σ
)
∫
Σ
\
Δ
(
x
*
,
3
R
)
|
x
*
−
x
0
|
γ
|
x
0
−
y
|
n
−
1
+
γ
d
σ
(
y
)
≤
C
0
‖
g
‖
L
∞
(
Σ
,
σ
)
∫
Δ
(
x
*
,
3
R
)
1
|
x
0
−
y
|
>
ε
|
x
0
−
y
|
n
−
1
+
1
|
x
*
−
y
|
>
1
|
x
*
−
y
|
n
−
1
d
σ
(
y
)
+
C
R
γ
‖
g
‖
L
∞
(
Σ
,
σ
)
∫
Σ
\
Δ
(
x
*
,
3
R
)
d
σ
(
y
)
|
x
*
−
y
|
n
−
1
+
γ
≤
C
‖
g
‖
L
∞
(
Σ
,
σ
)
R
n
−
1
ε
n
−
1
+
R
n
−
1
<
∞
.
In (4.75) we have also relied on the fact that
(4.76)
∫
Σ
\
Δ
(
x
*
,
3
R
)
d
σ
(
y
)
|
x
*
−
y
|
n
−
1
+
γ
≤
C
R
−
γ
,
which follows from [[6], (7.2.5), p. 574]. The constant C in (4.76) and (4.75) depends only on n, K, Σ.
Consequently, it is meaningful to define
(4.77)
A
2
(
g
)
≔
∫
Σ
K
(
x
0
,
y
)
θ
x
0
−
y
ε
−
K
(
x
*
,
y
)
1
|
x
*
−
y
|
>
1
g
(
y
)
d
σ
(
y
)
∈
C
.
Pick now x ∈ Σ arbitrary. Since
θ
x
−
y
ε
−
1
|
x
−
y
|
>
ε
≡
0
if
y
∈
R
n
is such that |x − y| < ɛ or |x − y| > 2ɛ, relying again on (2.21) and the Ahlfors regularity of Σ, we may also estimate
(4.78)
∫
Σ
K
(
x
,
y
)
θ
x
−
y
ε
−
1
|
x
−
y
|
>
ε
g
(
y
)
d
σ
(
y
)
≤
C
0
∫
ε
≤
|
x
−
y
|
≤
2
ε
y
∈
Σ
‖
θ
‖
L
∞
(
Σ
,
σ
)
+
1
|
x
−
y
|
n
−
1
|
g
(
y
)
|
d
σ
(
y
)
≤
C
0
‖
g
‖
L
∞
(
Σ
,
σ
)
‖
θ
‖
L
∞
(
Σ
,
σ
)
+
1
ε
n
−
1
σ
(
Δ
(
x
,
2
ε
)
)
≤
C
‖
g
‖
L
∞
(
Σ
,
σ
)
.
Thus,
(4.79)
A
1
(
g
)
(
x
)
≔
∫
Σ
K
(
x
,
y
)
1
|
x
−
y
|
>
ε
−
θ
x
−
y
ε
g
(
y
)
d
σ
(
y
)
is meaningfully defined and
(4.80)
⨏
Δ
(
x
0
,
r
)
A
1
(
g
)
d
σ
≤
C
‖
g
‖
L
∞
(
Σ
,
σ
)
for some C ∈ (0, ∞) independent of g and ɛ.
In concert, (4.77), (4.79), (4.71), and (2.24) allow us to write
(4.81)
T
ε
mod
g
(
x
)
−
(
T
(
ε
)
mod
g
)
(
x
)
=
A
1
(
g
)
(
x
)
+
A
2
(
g
)
,
for
all
g
∈
L
∞
(
Σ
,
σ
)
and
x
∈
Σ
.
If in (4.81) we now take g≔f
2, define the constant
(4.82)
c
2
=
c
2
(
r
,
ε
,
θ
)
≔
A
2
(
f
2
)
∈
C
,
then compute the integral average over Δ of the resulting identity, and finally invoke (4.80), we deduce that
(4.83)
⨏
Δ
T
ε
mod
f
2
−
T
(
ε
)
mod
f
2
−
c
2
d
σ
≤
⨏
Δ
|
A
1
(
f
2
)
|
d
σ
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
for some C ∈ (0, ∞) independent of f and ɛ. In view of the conclusion we presently seek, this allows us to focus on the operator
T
(
ε
)
mod
, with smooth truncation, in place of the original
T
ε
mod
. Specifically, first we employ (4.71) and the definition of f
2 to estimate
(4.84)
⨏
Δ
|
T
(
ε
)
mod
f
2
|
d
σ
≤
1
σ
(
Δ
)
∫
Δ
∫
Σ
\
2
Δ
|
K
(
x
,
y
)
θ
x
−
y
ε
−
K
(
x
0
,
y
)
θ
x
0
−
y
ε
‖
f
(
y
)
|
d
σ
(
y
)
d
σ
(
x
)
.
To bound the integrand, for each x ∈ Δ and y ∈ Σ _ 2Δ, we write
(4.85)
K
(
x
,
y
)
θ
x
−
y
ε
−
K
(
x
0
,
y
)
θ
x
0
−
y
ε
≤
K
(
x
0
,
y
)
θ
x
−
y
ε
−
θ
x
0
−
y
ε
+
θ
x
−
y
ε
K
(
x
,
y
)
−
K
(
x
0
,
y
)
.
To further bound the terms in (4.85), first observe that
(4.86)
y
∈
Σ
\
Δ
(
x
0
,
2
r
)
x
∈
Δ
(
x
0
,
r
)
⇒
|
y
−
x
0
|
≤
2
|
y
−
ξ
|
for
all
ξ
∈
[
x
,
x
0
]
.
Now fix y ∈ Σ _ 2Δ and x ∈ Δ. Then the properties of θ and the Mean Value Theorem imply that there exists ξ
* ∈ [x, x
0] such that
(4.87)
θ
x
−
y
ε
−
θ
x
0
−
y
ε
≤
(
∇
θ
)
ξ
*
−
y
ε
⋅
|
x
−
x
0
|
ε
1
ε
<
|
ξ
*
−
y
|
<
2
ε
≤
4
‖
∇
θ
‖
L
∞
(
Σ
,
σ
)
⋅
|
x
−
x
0
|
|
y
−
x
0
|
,
where the last inequality in (4.87) uses the fact that under the current assumptions
2
ε
≥
|
y
−
ξ
*
|
≥
1
2
|
y
−
x
0
|
(relying on (4.86)). Combined, (2.21) and (4.87) give
(4.88)
K
(
x
0
,
y
)
θ
x
−
y
ε
−
θ
x
0
−
y
ε
≤
C
|
x
−
x
0
|
|
x
0
−
y
|
n
.
In addition, we may apply (2.22) to also conclude that
(4.89)
θ
x
−
y
ε
K
(
x
,
y
)
−
K
(
x
0
,
y
)
≤
C
|
x
−
x
0
|
γ
|
x
0
−
y
|
n
−
1
+
γ
.
Together, (4.84)–(4.89) and the Ahlfors regularity of Σ yield
(4.90)
⨏
Δ
T
(
ε
)
mod
f
2
d
σ
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
σ
(
Δ
)
∫
Δ
∫
Σ
\
2
Δ
r
|
x
0
−
y
|
n
+
r
γ
|
x
0
−
y
|
n
−
1
+
γ
d
σ
(
y
)
d
σ
(
x
)
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
for some C ∈ (0, ∞) independent of ɛ, f, x
0, and r. For the last inequality in (4.90) we have used versions of (4.76) with x
* replaced by x
0, 3R replaced by 2r, and with γ also equal to 1 (based on [[6], (7.2.5), p.5̇74]).
Finally, if we set
c
≔
c
1
+
c
2
∈
C
, then (4.67), (4.70), (4.83), and (4.90) imply
(4.91)
⨏
Δ
T
ε
mod
f
−
c
d
σ
≤
⨏
Δ
T
ε
mod
f
1
−
c
1
d
σ
+
⨏
Δ
T
ε
mod
f
2
−
c
2
d
σ
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
+
⨏
Δ
T
ε
mod
f
2
−
T
(
ε
)
mod
f
2
−
c
2
d
σ
+
⨏
Δ
T
(
ε
)
mod
f
2
|
d
σ
≤
C
‖
f
‖
L
∞
(
Σ
,
σ
)
for some constant C ∈ (0, ∞) independent of ɛ, f, x
0, and r. This completes the justification of (4.66) and, with it, the treatment of Step II.
Step III.
Proof of the statement in item (2).
Suppose K is standard in the second variable, i.e., K
t
satisfies (2.21) and (2.22). In addition, assume that the L
1-styled local estimate (2.25) holds for every ɛ > 0, surface ball Δ, and function
f
∈
L
c
∞
(
Σ
,
σ
)
with supp f ⊆Δ. The goal in this step is to show that the operator T
ɛ
: H
1(Σ, σ) → L
∞
(Σ, σ) is bounded uniformly with respect to ɛ. To this end, fix an arbitrary ɛ ∈ (0, ∞) and observe that, as a consequence of (4.54),
(4.92)
T
ε
g
∈
L
∞
(
Σ
,
σ
)
and
T
ε
t
g
∈
L
∞
(
Σ
,
σ
)
whenever
g
∈
L
c
∞
(
Σ
,
σ
)
.
Also, note that
(4.93)
For
each
f
∈
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
there exist
c
f
,
c
f
̃
∈
C
such
that
T
ε
f
−
T
ε
mod
f
=
c
f
and
T
ε
t
f
−
T
ε
t
mod
f
=
c
f
̃
.
Indeed, (4.93) follows by taking c
f
≔(T
1
f)(x
*) and
c
f
̃
≔
T
1
t
f
(
x
*
)
.
To proceed, fix p* ∈ (1, ∞) and we claim that
(4.94)
∫
Σ
g
(
T
ε
a
)
d
σ
=
∫
Σ
a
T
ε
t
mod
g
d
σ
for
every
g
∈
L
c
∞
(
Σ
,
σ
)
and
every
L
p
*
−
normalized
H
1
−
atom
a
.
To prove (4.94), let us first note that since
T
ε
a
,
T
ε
t
g
∈
L
∞
(
Σ
,
σ
)
(as a consequence of (4.92) and (4.93)), the integrals in (4.94) are absolutely convergent. Also, with
c
≔
c
g
̃
∈
C
defined in relation to g as in (4.93), we have
(4.95)
∫
Σ
g
(
T
ε
a
)
d
σ
=
∫
Σ
g
(
x
)
∫
Σ
K
(
x
,
y
)
1
|
x
−
y
|
>
ε
a
(
y
)
d
y
d
σ
(
x
)
=
∫
Σ
a
(
y
)
∫
Σ
K
(
x
,
y
)
1
|
x
−
y
|
>
ε
g
(
x
)
d
x
d
σ
(
y
)
=
∫
Σ
a
T
ε
t
g
d
σ
=
∫
Σ
a
(
T
ε
t
mod
g
+
c
)
d
σ
=
∫
Σ
a
T
ε
t
mod
g
d
σ
where the second equality is an application of Fubini’s Theorem, which is permissible since
(4.96)
∫
Σ
∫
Σ
|
g
(
x
)
‖
K
(
x
,
y
)
|
1
|
x
−
y
|
>
ε
|
a
(
y
)
|
d
σ
(
y
)
d
σ
(
x
)
≤
C
0
∫
Σ
∫
Σ
|
g
(
x
)
|
1
|
x
−
y
|
>
ε
|
x
−
y
|
n
−
1
|
a
(
y
)
|
d
σ
(
y
)
d
σ
(
x
)
≤
C
0
ε
n
−
1
‖
g
‖
L
1
(
Σ
,
σ
)
‖
a
‖
L
1
(
Σ
,
σ
)
<
+
∞
.
The fourth equality in (4.95) uses (4.93), while the last equality employs the moment condition for the atom a. This proves (4.94).
Our next claim is that:
there exists C ∈ (0, ∞) independent of ɛ such that
(4.97)
‖
T
ε
a
‖
L
1
(
Σ
,
σ
)
≤
C
for
every
L
p
*
−
normalized
H
1
−
atom
a
.
To show (4.97), pick
g
∈
L
c
∞
(
Σ
,
σ
)
and invoke Step II (recall that by assumption K
t
is standard in the first variable) to conclude that the operator
(4.98)
T
ε
t
mod
:
L
∞
(
Σ
,
σ
)
→
B
M
O
(
Σ
,
σ
)
is
bounded
uniformly with respect to ɛ.
In particular,
T
ε
t
mod
g
∈
B
M
O
(
Σ
,
σ
)
. The latter, (3.48), (3.49), and (4.98), allow us to conclude that there exists C ∈ (0, ∞) such that
(4.99)
∫
Σ
a
T
ε
t
mod
g
d
σ
≤
C
‖
T
ε
t
mod
g
‖
BMO
(
Σ
,
σ
)
≤
C
‖
g
‖
L
∞
(
Σ
,
σ
)
,
for every
L
p
*
-normalized H
1-atom a. Hence, making use of (4.94) and (4.99), we obtain
(4.100)
‖
T
ε
a
‖
L
1
(
Σ
,
σ
)
=
sup
∫
Σ
g
(
T
ε
a
)
d
σ
:
g
∈
L
c
∞
(
Σ
,
σ
)
,
‖
g
‖
L
c
∞
(
Σ
,
σ
)
≤
1
=
sup
∫
Σ
a
T
ε
t
mod
g
d
σ
:
g
∈
L
c
∞
(
Σ
,
σ
)
,
‖
g
‖
L
c
∞
(
Σ
,
σ
)
≤
1
≤
C
.
This finishes the proof of (4.97).
To make our next claim, recall the space
H
fin
1
,
p
*
and its properties discussed in Proposition 3.1. In relation to these, we claim that:
there exists C ∈ (0, ∞) independent of ɛ such that
(4.101)
‖
T
ε
f
‖
L
1
(
Σ
,
σ
)
≤
C
‖
f
‖
H
1
(
Σ
,
σ
)
for
every
f
∈
H
fin
1
,
p
*
(
Σ
,
σ
)
.
Starting with
f
∈
H
fin
1
,
p
*
(
Σ
,
σ
)
so that
f
=
∑
j
=
1
N
λ
j
a
j
for some
{
λ
j
}
1
≤
j
≤
N
⊆
C
and some
L
p
*
-normalized H
1-atoms
{
a
j
}
1
≤
j
≤
N
, we have
T
ε
f
=
∑
j
=
1
N
λ
j
T
ε
a
j
since T
ɛ
is linear. In addition, making use of the estimate in (4.97), we obtain
(4.102)
‖
T
ε
f
‖
L
1
(
Σ
,
σ
)
≤
∑
j
=
1
N
|
λ
j
|
‖
T
ε
a
j
‖
L
1
(
Σ
,
σ
)
≤
C
∑
j
=
1
N
|
λ
j
|
.
Now taking the infimum in the resulting inequality in (4.102) over all finite atomic representations of f, and then invoking the equivalence of norms stated in the last line in (3.47), we arrive at
(4.103)
‖
T
ε
f
‖
L
1
(
Σ
,
σ
)
≤
C
‖
f
‖
H
fin
1
,
p
*
(
Σ
,
σ
)
≤
C
‖
f
‖
H
1
(
Σ
,
σ
)
,
which is valid uniformly with respect to ɛ. This proves (4.101).
Having established (4.101) and recalling that
H
fin
1
,
p
*
(
Σ
,
σ
)
is a dense linear subspace of H
1(Σ, σ) (see (3.47)), we may now uniquely extend the operator
(4.104)
T
ε
|
H
fin
1
,
p
*
(
Σ
,
σ
)
:
H
fin
1
,
p
*
(
Σ
,
σ
)
→
L
1
(
Σ
,
σ
)
,
to a linear and continuous operator Q
ɛ
: H
1(Σ, σ) → L
1(Σ, σ) with preservation of operator norm so that the operator norm of Q
ɛ
is also independent of ɛ. Upon observing that
H
1
(
Σ
,
σ
)
⊂
L
1
(
Σ
,
σ
)
⊂
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
, the boundedness stated in item (2) will follow as soon as we prove that
(4.105)
T
ε
|
H
1
(
Σ
,
σ
=
Q
ε
.
To justify (4.105), let f ∈ H
1(Σ, σ) be arbitrary and consider a sequence
{
f
k
}
k
∈
N
⊂
H
fin
1
,
p
*
(
Σ
,
σ
)
such that f
k
→ f as k → ∞ in H
1(Σ, σ). By the continuity of Q
ɛ
it follows that
(4.106)
T
ε
f
k
=
Q
ε
f
k
→
Q
ε
f
as
k
→
∞
in
L
1
(
Σ
,
σ
)
.
In particular, there exists a subsequence
{
f
k
j
}
j
∈
N
such that
(4.107)
T
ε
f
k
j
→
Q
ε
f
as
j
→
∞
point
−
wise
σ
−
a.e.
on
Σ
.
In addition, at each x ∈ Σ, the estimate on K and the fact that H
1(Σ, σ) is continuously embedded in L
1(Σ, σ), imply
(4.108)
(
T
ε
f
k
)
(
x
)
−
(
T
ε
f
)
(
x
)
≤
∫
y
∈
Σ
,
|
x
−
y
|
>
ε
|
K
(
x
,
y
)
|
f
k
(
y
)
−
f
(
y
)
d
σ
(
y
)
≤
C
0
ε
n
−
1
‖
f
k
−
f
‖
L
1
(
Σ
,
σ
)
≤
C
ε
n
−
1
‖
f
k
−
f
‖
H
1
(
Σ
,
σ
)
→
0
as
k
→
∞
.
Thus, the sequence
{
T
ε
f
k
}
k
∈
N
converges pointwise to T
ɛ
f. The latter, in concert with the convergence in (4.107) forces Q
ɛ
f = T
ɛ
f at σ-a.e. point on Σ and proves (4.105). The proof of Step III is therefore finished.
Step IV.
Proof of the statement in item (3).
From Step II and Step III we know that the operators
(4.109)
T
ε
:
H
1
(
Σ
,
σ
)
→
L
1
(
Σ
,
σ
)
and
T
ε
mod
:
L
∞
(
Σ
,
σ
)
→
B
M
O
(
Σ
,
σ
)
are bounded, with operator norms controlled independently of ɛ. To obtain the stated L
p
-bounds, the idea is to make use of the interpolation results from (3.50) and (3.51). This has to be done in a way that accounts for the fact that the boundedness results in (4.109) are for two different operators.
To proceed, fix p ∈ (1, ∞) and let f ∈ L
p
(Σ, σ) be arbitrary. Split f = f
0 + f
1 with f
0 ∈ H
1(Σ, σ) and with f
1 ∈ L
∞
(Σ, σ). Then
(4.110)
f
0
∈
H
1
(
Σ
,
σ
)
⊂
L
1
(
Σ
,
σ
)
⊂
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
and, while L
∞
(Σ, σ) is not contained in the space
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
, we make use of
L
p
(
Σ
,
σ
)
⊂
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
to conclude that
(4.111)
f
1
=
f
−
f
0
∈
L
1
Σ
,
σ
(
x
)
1
+
|
x
|
n
−
1
.
Hence, T
ɛ
f
0 and T
ɛ
f
1 are meaningfully defined and T
ɛ
f = T
ɛ
f
0 + T
ɛ
f
1. In addition, from Step II we know T
ɛ
f
0 ∈ L
1(Σ, σ), while (4.93) ensures the existence of some
c
∈
C
such that
T
ε
f
1
=
T
ε
mod
f
1
+
c
. At the same time, as a consequence of Step II, we have
T
ε
mod
f
1
∈
B
M
O
(
Σ
,
σ
)
, thus
T
ε
f
1
=
T
ε
mod
f
1
+
c
∈
B
M
O
(
Σ
,
σ
)
. In summary, we have proved that
if f
0 ∈ H
1(Σ, σ) and f
1 ∈ L
∞
(Σ, σ) are such that
f
=
f
0
+
f
1
,
(4.112)
then
T
ε
f
=
T
ε
f
0
+
T
ε
mod
f
1
+
c
,
with
T
ε
f
0
∈
L
1
(
Σ
,
σ
)
and
T
ε
mod
f
1
+
c
∈
B
M
O
(
Σ
,
σ
)
.
Thus, for each t ∈ (0, ∞), we have (for Petree’s K-functional)
(4.113)
K
t
,
T
ε
f
,
L
1
(
Σ
,
σ
)
,
B
M
O
(
Σ
,
σ
)
=
inf
‖
g
0
‖
L
1
(
Σ
,
σ
)
+
t
‖
g
1
‖
BMO
(
Σ
,
σ
)
:
T
ε
f
=
g
0
+
g
1
,
w
i
t
h
g
0
∈
L
1
(
Σ
,
σ
)
and
g
1
∈
B
M
O
(
Σ
,
σ
)
≤
‖
T
ε
f
0
‖
L
1
(
Σ
,
σ
)
+
t
‖
T
ε
mod
f
1
+
c
‖
BMO
(
Σ
,
σ
)
=
‖
T
ε
f
0
‖
L
1
(
Σ
,
σ
)
+
t
‖
T
ε
mod
f
1
‖
BMO
(
Σ
,
σ
)
≤
C
‖
f
0
‖
H
1
(
Σ
,
σ
)
+
t
‖
f
1
‖
L
∞
(
Σ
,
σ
)
,
where the first inequality is a consequence of (4.112), while the last inequality is based on (4.109). Hence,
(4.114)
K
t
,
T
ε
f
,
L
1
(
Σ
,
σ
)
,
B
M
O
(
Σ
,
σ
)
≤
C
inf
‖
h
0
‖
H
1
(
Σ
,
σ
)
+
t
‖
h
1
‖
L
∞
(
Σ
,
σ
)
:
f
=
h
0
+
h
1
,
with
h
0
∈
H
1
(
Σ
,
σ
)
and
h
1
∈
L
∞
(
Σ
,
σ
)
=
C
K
t
,
f
,
H
1
(
Σ
,
σ
)
,
L
∞
(
Σ
,
σ
)
.
Consequently, using (3.51), (4.114), and (3.50), we have
(4.115)
‖
T
ε
f
‖
L
p
(
Σ
,
σ
)
p
≈
∫
0
∞
t
1
p
−
1
K
t
,
T
ε
f
,
L
1
(
Σ
,
σ
)
,
B
M
O
(
Σ
,
σ
)
p
d
t
t
≤
C
∫
0
∞
t
1
p
−
1
K
t
,
f
,
H
1
(
Σ
,
σ
)
,
L
∞
(
Σ
,
σ
)
p
d
t
t
≈
‖
f
‖
L
p
(
Σ
,
σ
)
p
.
Note that also, as wanted, all constants in (4.113)–(4.115) are independent of ɛ. This proves that the operator T
ɛ
: L
p
(Σ, σ) → L
p
(Σ, σ) is bounded for every p ∈ (1, ∞) with operator norm independent of ɛ. That the same is true for the operator
T
ε
t
follows from this since K
t
is standard in both variables whenever K is. This finishes the proof of .
Finally, the last statement in the theorem is a direct consequence of and Fubini’s Theorem.
and
Marius Mitrea
