1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

368 7. THE REDUCED DISTANCE

where a > 0 is chosen so that JB(l) rJdx = 1. This function is C^00 with

support contained in B (1) C ~n and all derivatives vanishing for lxl ~ 1
including lxl = 1. Let f: B (r)----+ ~be a continuous function. For EE (0, r)
define the mollified function

f 15 : B ( r - E) ----+ ~

by

(7.142) fz(Y)~.!_ { f(y+z)rJ(~)dz.

c;n }B(z) E

It is a standard fact that f 15 is C^00 •

We now prove a lemma about convex functions.

LEMMA 7.119 (Mollifiers and derivatives). Let f : B (r) ----+ ~ be a con-

tinuous function. Let a~i and ay~;Yj denote the standard partial derivatives

on~n.

(i) If f has first derivative DJ (x) at x EB (r), then

lim (

8

8

!~ (x))n =DJ (x).

£-+0+ Y i=l

(ii) If f has second derivative D^2 f (x) at x EB (r), then

lim (

8

(^8) :£z. (x))n = D (^2) f (x).
c-+0+ y yJ i,J-.. _ 1
(iii) If f is a convex function on B (r), then f 15 is a convex function on

B(r-s).

(iv) If f E Lfoc' where p E [1, oo), then fz converges to f in Lfoc·

PROOF. (i) D.efine j by

f ( x + y) ~ f ( x) + D f ( x) · y + j (y).

Then J (y) has first derivative D j ( 0) = 0. It suffices to prove that the

mollified function fz (y) satisfies limz-+O+ ~t~ (0) = 0 for each i. Note that

fc(y) - =---;:^1 1 f (z) - 'TJ (z-y) - dz

E B(y,z) c

and

aj~ (0) = _.!.._ r j (z) ( a'r/.) (~) dz.

8yi en } B(z) 8zi E E