6
HABIB AMMARI AND HYEONBAE KANG
and aD is a manifold of dimension d- 1, the righthand side of the above is not
absolutely integrable. On the other hand, (1/wd)(x-
y, Vz)/lx- Yld
has a structure
of the Poisson kernel. To see this, let us suppose that d
=
2,
z
=
0, aD is the x-axis
near 0, and Dis a part below the x-axis. In this case, if x
=
z-
Wz
=
(0, -E), then
for y
=
(y1, 0)
E
aD, we have
1

(vT(x- y),
vz) =-
27r IY112
+
€2'
which is exactly -1/2 times the Poisson kernel. So we can expect that
r
(Vr(x-
y), Vz)r/J(y)du(y) lav
will pick up -(1/2)¢J(z) as
x
-t
z
from inside
D.
The following theorem shows that
it is indeed the case. Proofs of the theorem can be found in
[70]
for when D has a
C2-boundary, and in [132] for D with a Lipschitz boundary.
THEOREM 2.1 (Jump formula).
Let D be a bounded Lipschitz domain in
JR.d.
For
rjJ
E
L 2
(aD)
Svr/JI+(x)
=
SvrfJI_(x) a.e.
X
E
aD,
(2.3)
:vSvr/JI± (x)
=
(±~I
+Kn )r/J(x) a.e. x
E
aD,
(2.4)
(Vvr/J)I±(x)
= (
:r=~J
+
Kv
)rjJ(x)
a.e. x
E
aD,
where K v is defined by
( )
1
i
(y-
X, Vy) ( ) ( )
Kvr/J x
=
-p.v.
I ld
rjJ
y du y
Wd 8D X-
y
and KL is the L
2-adjoint
of Kv, i.e.,
V"* ( )
1
i
(x-
y, Vx) ( ) ( )
"-'DrP
x
= -p.v.
I ld
rjJ
y du y .
Wd 8D X-
y
Here
p.v.
denotes the Cauchy principal value.
This theorem is based on the fact that the operators
Kv
is well-defined, which
is guaranteed by the following celebrated theorem of Coifman-Mclntosh-Meyer
[60].
THEOREM 2.2.
The opemtors Kv and KiJ are singular integml opemtors and
bounded on L
2(aD).
We strongly recommend the readers to read
[60]
to have feeling of power and
beauty of the classical harmonic analysis.
If
D has C2 boundary, or C1•0 (a 0)
boundary for that matter, then there is an extra orthogonality, which is absent for
the Lipschitz boundary, between
x-y (x, y
E aD) and
Vx.
Using this orthogonality,
we can show that
I
(x -
y, Vx)
I
C
lx _ yld :::; lx _ Yld-2
(:::; lx _
y~-l-a
if aD is C1•0
) ,
and hence
KL
becomes a compact operator on
L
2(aD).
See
[70]
for this.
Let us state further mapping properties of layer potentials, whose proof can be
found in [132]; see also [18].
THEOREM 2.3.
Let D be a bounded Lipschitz domain in
JR.d.
Then
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