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