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