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 +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™
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