1. PRELIMINARIES

3

all bounded subsets of £1. For continuous functions u we have |w|0(Q) =

Hwll^Q). For an integer nonnegative k we define

\

{/P

\uhJa)

= [ ]T

\\Dau\\pp(Q)X

\a\k

The space L (Q) is the set of all functions with the finite norms || • || (Q).

The space

Hk,p(Q)

is the set of all functions having on Q (generalized)

partial derivatives of order k with the finite norm || • \\k (Q). We as-

sume 1 p +oo.

Hk,p(Q.)

is the subspace of H

,p(£l)

consisting of

the limits of sequences of functions from C™(Q) with respect to the norm

|| • \\k p(Q.). The spaces C (Q.) and H

,/7(£2)

are Banach spaces with the

norms introduced above. The space H^(Q) = H ' (Q) is even a Hilbert

space that has the scalar product

(u,v){k)(0)= ^2 [

DauD^dx.

We often write

Ck

and

Hk'p

instead of

Ck(Rn)

and

Hk'p(Rn)

and

sometimes omit the integration domain in integrals over R" .

We summarize necessary properties of H

,p

-spaces as follows.

THEOREM

1.1.2. Let Q be a bounded domain with the Lipschitz boundary.

i) If 1 p, then there is a constant C such that

\\u\\p{dO) C\\u\\lp(Q) for all u e

Cl(U).

ii) If n kp, A k - n/p, then there is C such that

\u\x(0) C\\u\\kp(Q) for all u e

Hk'p(Q).

iii) If u e

Hk'p{£l)

and

Dau

= 0 on 9ft for \a\ k - 1, then u e

HS'P(Q).

iv) (Extension). There is a linear operator Lext mapping H

,P(Q)

into

H0

jP(Rn)

with the finite operator norm such that Lu(x) = u(x) for I G Q .

v) (Interpolation ) . There is a constant C = C(Q, k) such that

\\u\\(k)(Q)

C\\u\\k(^(a)\\u\\l-k/m(n)

for all u e H{m)(Q), 0 k m.

Proofs of these results are to be found in the book of Morrey [100], Chapter

3.

We note as well that an analytic function means a function on a domain

in R" that can be locally represented as the sum of its Taylor series, and a

complex analytic function is a function on a domain in

Cn

satisfying the

Cauchy-Riemann equations.

In many situations the following formula is useful:

(1.1.1) / D.udx= / un.dT (integration by parts).

JQ

J

Jon

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