8 HABIB AMMARI AND HYEONBAE KANG

Here and throughout this paper, we define the space L5(aD) by

L~(aD)

:= {

¢

E

L

2

(aD): hn ¢da

=

0 }·

The key fact in proving Theorem 2.4 is the following: For a function u the

tangential derivative of u along aD is defined to be

au

d-l

au

EJT

:=

L

aT:

Tp,

p=l p

where T1

, ... ,

Td-l

is an orthonormal basis for the tangent plane to aD at x.

LEMMA

2.5. Let D be a bounded Lipschitz domain in

JRd,

d

~

2. Let u be a

function such that either

(i) u is Lipschitz in D and Llu

=

0

in D, or

(ii) u is Lipschitz in JRd \ D, Llu

=

0 in JRd \ D, and

lu(x)l

=

0(1/lxld-2

)

when d

~

3

and

lu(x)l

=

0(1/lxl)

when d

=

2

as

lxl

~

+oo.

Then there exists a positive constant

C

depending only on the Lipschitz character

of D such that

(2.6)

~

II;; L2(8D)

~ II~~

II£2(8D)

~

c

II;; L2(8D).

Lemma 2.5 says that the £

2

norms of the normal and tangential derivatives of

a harmonic function are comparable, and can be proved using the Rellich identity

[132];

see also

[18].

Let us briefly see how Lemma 2.5 leads us to Theorem 2.4.

Let u(x)

=

Snf(x), where f

E L~(aD).

Because of the jump formula (2.3), we

have

and by (2.6)

~~~-

£2(8D)

au

I

a ,

1/

+

£2(8D)

or equivalently

(2.7)

~ 11(~1

+ KiJ)fii£2(8D)

~ 11(~1-

KiJ)fii£2(8D)

~ c 11(~1

+ KiJ)fii£2(8D)·

Since f

=

((1/2)1

+

Ki))f

+

((1/2)1- Ki))f, (2.7) yields that

11(~1

+

Ki))fii£2(8D)

~

Cllfii£2(8D)·

We also have the following estimate from

[16].

LEMMA

2.6. There exists a constant C depending only on the Lipschitz char-

acter of D such that for any

I .AI

~

1/2

II¢11£2(8D)

~ c 1~111(.!-

Ki))

¢11£2(8D)

for all¢

E

£5(aD).