1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. ANALYTIC SEMIGROUPS AND MAXIMAL REGULARITY THEORY 293

and such that

llTll.c(I 8 (B 0 ,B 1 ),I 8 (A 0 ,A 1 )) :S:: llTll~(t,Ao) llTll~(Bi.Ai) ·

Various formulations of interpolation spaces are found in the literature. In studying

the stability of the Ricci- DeTurck fl.ow, we use the continuous interpolation spaces

introduced in [91]. (As shown in [96], they are equivalent in norm to real inter-

polation spaces.) These continuous interpolation spaces, which one can verify are

exact, are defined as follows:

DEFINITION 35.22. Let Bo and B 1 be Banach spaces, and let () E (0, 1). The

continuous interpolation space (Bo, Bi) 11 is the set of all x E Bo such that there

exist sequences {yn} ~Bo and {zn} ~ B1 with x = Yn +Zn, where

and

as n---+ oo. The norm on (Bo, Bi) 11 is equivalent to

inf {sup ( 2n^11 llYnlls 0 , Tn(l-1:1) llznlls 1 )},

n::'.:l

where the infimum is taken over all such sequences (Yn, Zn)·

REMARK 35 .23. This is compatible with the definition given in the introduction
to this section. If A generates a strongly continuous semigroup and u E Ea, then
u E £11 if and only if etAu E h^11.

We need the following fact about continuous interpolation spaces:

LEMMA 35.24 ([419]). Lets :S:: r EN, 0 <a< p < 1, and 0 < () < l. If

() (r + p) + (1 - B) (s +a)

is not an integer, then there is a Banach space isomorphism

(35.15) ( hs+u, hr+p)

11

~ h(llr+(l-ll)s)+(llp+(l-ll)u),

and there exists C < oo such that for all 'T) E hr+p,

(35.16)

2.2. The Ricci- DeTurck operator revisited.

Let us now examine the Ricci- DeTurck operator (35.3) in the context of max-

imal regularity theory for nonlinear equations, as described in the introduction to

§2 of this chapter.

We fix a background metric [J and recall that A9 (g) is described locally by the

expression (35.6). For fixed 0 < a < p < 1, we define the following nested spaces

of metrics, where the 1ir+p are defined in Definition 35.21:

Eo ho+u

u

Xo ....:... ....,.. hO+p

(35.17) u

£1 h2+u

u

X1 ....:... ....,.. h2+P.