1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. HOLDER SPACES AND SOBOLEV SPACES ON MANIFOLDS 333

where the {¢%;ii···iJ are locally defined functions on U 13. For any a E (0 , 1], we

define the Holder seminorm

(K.9) [Y'k¢Ja ~sup sup

/3 p,qEU13

p#q

Lj,i1, .. ,ik (¢%;i1· ·ie (p) - ¢%;i 1 ··ie (q))^2

d (p, q)°'

where dis the Riemannian distance with respect tog. This seminorm depends on

the choices of local coordinates and trivializations.

DEFINITION K.7. Given an integer k E [O, oo) and a E (0, 1], the Holder space

Ck,a: (M, £)is the Banach space of all sections¢ E C 1 ~c (M, £)for which the Holder

norm

k

11<1> 1 1k,a: ~ 11<1>11k + [Y'k<t>L = I )<t>lj + [Y'k<t>Ja:

j=O

is finite.

Using parallel translation, we may alternatively define a Holder seminorm as

follows. Given p, q E M, let "/p,q be a minimal geodesic from p to q in M. Let ll'Y,,,o:

(Ee)p --+ (£e)q denote the parallel translation along "/p,q defined by the connection

Y'. Define the seminorm

(K.10) [

k ]'. lll'Y,,,q ('Vk¢(p)) - 'Vk¢(q)l(t:k)

Y' ¢ =;= sup sup a: q

a: p,qEM 'Yv.o d (p, q)

p#q

When M is compact, we may restrict ourselves to finite coverings. Then, up to
uniform equivalence, it is not hard to see that the definition of the Holder norm
ll<Pllk a: is independent of the choice of connections, coordinate systems, and local

trivi~lizations of£; we may also replace the seminorm [Y'k¢Ja by [Y'k¢]~.
Although Banach space norms used in this book are primarily Holder norms,
on occasion we used Sobolev norms, such as in Subsection 3.6 of Chapter 34. Here
we recall

DEFINITION K.8. Given an integer k E [O, oo) and p E [1, oo), for any ¢ E
C 1 ~c (M, £) we define its Sobolev Wk,p_norm by

(K.11) 11</>llk,p ~ t, (JM l'Vjl~t:k)q dμ(q)) l/p

We define the Sobolev space Wk·P(M, £) to be the completion of Ck (M, £) with
respect to the wk,p_norm.

In this subsection we have given just the bare definitions. To conclude, we
remind the reader that for the use of Holder spaces and Sobolev spaces in geo-
metric analysis, compactness theorems, interpolation inequalities, and embedding
theorems play crucial roles.

2.2. Jets of maps and distances between maps.
In the proof of Proposition 33.16, on harmonic parametrizations of almost
hyperbolic pieces, we used the notion of Ck-distance between maps of manifolds.
To define this, we need to consider jets of maps.