2
1. PRELIMINARIES
where k = k - {k} and {k} is the entire part of k . Following customary
usage we define the space C"(Q) as the space of all functions u having
the finite norm |w|^(Q). If k is integer then we assume in addition that
the
Dau
with \a\ k have continuous extensions onto ft. Accordingly,
C (ft) denotes the space of all functions belonging to C (Q{) for all Qj
with clftt e f t . A definition of C (S) for all S is given in the book of
Stein [151], Chapter 4. The space C°°(ft) is the intersection of all
Ck(Q).
The space C0 (ft) contains all functions from C (ft) with support in ft.
The support of a continuous function u on ft (supp u) is the closure of the
set {x G ft: u(x) ^ 0} . A function u is Lipschitz on S (u e LipS) if for a
certain constant C we have \u(x) - u(y)\ C\x -y\ if x , y belong to S.
A set T in R is said to be a hypersurface of class C (Lipschitz, analytic)
if locally (and after a possible enumeration of coordinates) T is the graph of
a function x{ = y(x) of class C (Lipschitz, analytic). If Y is of class C
and admits the representation above then there is a normal n(x) to T at a
point x eT and it is
n(x) = (l + |
/
Vy|
2
r
1 / 2
(-l,
/
Vy)W .
We say T is piecewise C -smooth (analytic) if it is Tx U U Tm where
the r. are parts of C -smooth (analytic) hypersurfaces. For such surfaces
we define the spaces C (T), H
,P(T)
with the help of the mentioned local
representation of T.
THEOREM
1.1.1. i) (Extension). For any closed set S in
Rn
and a non-
negative k there is a linear operator Lext mapping C (S) into C
(Rn)
with
the finite operator norm such that Lu(x) = u(x) for x e S.
ii) (Interpolation). For any compact set S there is a constant C depending
on diam S only such that
\Dau\fi(S) C\ut^),{k+"\S)\u\l-{H+mk+"\S)
where \a\ k, P X.
A proof of part i) is given in the book of Stein [151], Chapter 4, Theorems
3 and 4.
Part ii) is a consequence of Lemma 5.1 of the paper of Agmon, Douglis,
and Nirenberg [1] and Theorem 33.4 of the book of Miranda [99].
For a function u on ft we introduce the norm
\\u\\p(£l) = (J^uf dx^j "
while the norm || H^ft) is defined as the limit of || || (Q)/measw ft for
ft of finite Lebesgue measure and in the general case as a supremum over
Previous Page Next Page