4 L. FONTANA, S. KRANTZ, AND M. PELOSO

In the discussion that follows we let Q denote either a smoothly

bounded domain fi, or the half space

R++1.

If s is a non-negative

integer then the s order Sobolev space norm on functions on Q is given

by

11/112= E

The associated inner product is

\a\s

Here dV stands for ordinary Lebesgue volume measure.

o s

For s a non-negative integer we define the Sobolev space W (fi) as

the closure of C™(Q). When s E K+ we define Ws(n) by interpolation

o

(see [LIM], for instance). Moreover, for s G K+, we denote byW8 (fi)

the closure of

CQ°(Q)

in

W8(£l).

When s 0 we define the negative

o

Sobolev space

WS(Q)

to be the dual of

Ws

(Q) with respect to the

standard L2-pairing.

On the Euclidean space RN+1 we consider the Fourier transform

defined initially for a testing function / £ CQ° as

JRN+I

We will also consider the tangential Fourier transform of functions de-

fined on the half space R£ +1 : If / € C0°°(E^+1) we set

JRN

We denote the inverse tangential Fourier transform of a function g{x$, £')

by g{xz,x').

For any s E K the Sobolev space ^(E^ 4 " 1 ) can be defined via the

Fourier transform. Indeed, we set

W'{RN+1) = {/ € L2(RN+l) : / ^ ( l + K|2)W(0|2de oo}.

We will also consider the Sobolev spaces

Ws(bQ)

defined on the

boundary of our domain, s G l . In the case Q =

R++1, Ws(bQ)

is just

the classical Sobolev space on

RN.

In the case of a smoothly bounded

domain Q, the Sobolev space can be defined by fixing a smooth atlas

{Xj} on 90, letting (j)j be a partition of unity subordinate to this atlas,

and defining the Sobolev norm of a function / on bQ as

WfWw^bQ) = £ Wfaf °

Xj~l\\ws{RNy

3

Qaf

dxa

/-,2™