2.3. DIFFERENTIAL FORMS

9

2.3. Differential forms

Throughout this paper we let ^°°(A^X), 0 ^ £ ^ n = dimX, denote the space

of smooth £-forms on X. Two differential operators on forms will be of particular

interest to us. First is the exterior derivative,

(2.4) d : * f °° (A*X) - ^°° ( A m X )

Second is the formal adjoint of d, also called Hodge codifferential,

(2.5) d* = ( - l ) n £ + 1 * d* : S?°°(AmX) -• ^°°(A£X)

where * : ^^(A^X) — »

^°°(An~£X)

denotes the Hodge star duality operator. Here,

we conveniently set ^^(A^X) = 0, whenever £ 0 or £ n. Note that ** =

(-l)*(n-€) on

^°°(A£X).

The point-wise scalar product of forms a,/? e ^°°(A^X)

is given by (a, /3) dx = a A */? G

^°°(AnX),

and hence

(2.6) / (a,(3) dx= [ aA*(3

Jx Jx

The duality between d and d* is emphasized in the formula of integration by parts

(2.7) [ (dp,1)= /*,*»

for if G

^°°(A€X)

and ^ G

^oc(A€+1X).

Now a differential form p G

£?p(AeX)

is

said to be closed in the sense of distributions if J

x

((/?, d*ip) = 0 for every test form

i/j G ^ ^ ( A ^ X ) . We write it as dip = 0. Similarly, we establish what it means

for ip to be coclosed, and write it as d*ip = 0. Forms of the type da, with a G

^ ^ ( A ^ X ) , are called exact while those of type d*/3, with (3 G # ^ ( A ^

+ 1

X ) , are

called coexact. It follows from the identities dod — 0 and d* od* = 0 that the £-forms

da G

J*?P(A£X)

and d*/? G

J*fp(A£X)

are closed and coclosed, respectively. Finally,

those forms h G

Jt?p(A£X)

which are closed and coclosed will be called harmonic

fields of degree £. We denote by

H(A£X)

the space of all harmonic fields of degree £

and regard it as well known that this space is finite dimensional.

7i(AlX)

consists

of ^°°-smooth forms. Being so, all possible norms on

H(A£X)

are equivalent. For

instance, we shall benefit from the estimate

(2.8) \\h\\ , ^ \\h\\

1

,

and further,

(2.9)

ftU(A£x)

*

(JXWP)P forallP«

In relation to the imbedding _2^eak(A*X) C

5fp(AeX),

with 0 p 1, we record

the following estimate

(2.10) IMU(A,X) ^ ( J ^ ) " ^

\%\^ supy fhtdx t0\

as is easily verified by Tchebyshev inequality, see (4.6).

2.3.1. Sobolev classes of differential forms. Four spaces of differential

forms have a special place in our studies. These spaces are:

- The Sobolev space of closed forms:

(A^X)nkerd