Kenneth Kuttler

11.9. CHANGE OF VARIABLES, NONLINEAR MAPS 335

It remains to identify g.

Lemma 11.9.5 For a.e. x, satisfying |detDh(x)|> 0, and r small enough,

Dh(x)B(0,(1− ε)r) ⊆ h(B(x,r))⊆ h(

B(x,r))⊆ Dh(x)B(0,(1+ ε)r),

mp (h(B(x,r)))mp (B(x,r))

∈ [|detDh(x)|(1− ε)p , |detDh(x)|(1+ ε)p]

limr→0

mp (h(B(x,r)))mp (B(x,r))

= |detDh(x)|

Proof: For r small enough,

h(B(x,r)) ⊆ h(x)+Dh(x)B(0,r)+Dh(x)Dh(x)−1 B(0,εr)

⊆ h(x)+Dh(x)B(0,r)+Dh(x)B(0,εr)

⊆ h(x)+Dh(x)(B(0,(1+ ε)r))

and so mp (h(B(x,r)))≤ |det(Dh(x))|mp (B(0,(1+ ε)r)) . Also,

h(x+v) = h(x)+Dh(x)v+Dh(x)Dh(x)−1o(v)

and so∥∥∥Dh(x)−1 (h(x+v)−h(x))−v

∥∥∥ = ∥∥∥Dh(x)−1o(v)∥∥∥ = ∥o(v)∥ . Thus if r is

chosen sufficiently small, it follows that for v ∈ B(0,r)∥∥∥Dh(x)−1 (h(x+v)−h(x))−v∥∥∥< εr

and so, from Lemma 8.10.1, B(0,(1− ε)r)⊆ Dh(x)−1(h(x+B(0,r)

)−h(x)

B(x,r))= h

(x+B(0,r)

)−h(x)⊇ Dh(x)B(0,(1− ε)r)

Therefore, since mp (B(x,r)) = mp

(B(x,r)

|det(Dh(x))|mp (B(0,(1− ε)r)) = |det(Dh(x))|(1− ε)p rpα p ≤ mp (h(B(x,r)))

so for r small enough,

mp (h(B(x,r)))mp (B(0,(1+ ε)r))

≤ |det(Dh(x))| ≤mp (h(B(x,r)))

mp (B(0,(1− ε)r))

The claim follows from this since ε > 0 is arbitrary. ■

Lemma 11.9.6 For a.e. x with |detDh(x)|> 0, limr→0mp(h(B(x,r)∩H))

mp(h(B(x,r)))= g(x)|detDh(x)| .

Proof: Using the result of Lemma 11.9.5, for a.e. x satisfying |detDh(x)| > 0, if rsmall enough, then

mp (h(B(x,r))) ∈ [|detDh(x)|mp (B(x,r))(1− ε)p , |detDh(x)|mp (B(x,r))(1+ ε)p]

11.9. CHANGE OF VARIABLES, NONLINEAR MAPS 335It remains to identify g.Lemma 11.9.5 For a.e. x, satisfying |\det Dh (x)| > 0, and r small enough,Dh(e)B(0.(1—e)r) <_h(B(e.r)) Ch (Bw.r) C Dh (a) BO. Fe)mp (h(B (wr) oa, . € ||detDh (w)|(1—e)”, det Dh (a)| (1 + €)”|Li Rea — |detDn(a)|Proof: For 7 small enough,h(B(a,r)) C h(a)+Dh(a) B(0,r)+Dh (x) Dh (x) | B(0,er)C h(«#)+Dh (ax) B(0,r)+Dh (ax) B(0,er)C h(#)+Dh (ax) (B(0,(1+€)r))and so m, (h(B(a,r))) < |det (Dh (a))|m, (B(0,(1+€)r)). Also,h(a+v) =h(x)+Dh(«)v+Dh(x)Dh(x) | 0(v)and so ph (x)! (h(a +) —h(2)) -»| = pn (@) | o(v)| = |lo(v)||. Thus if r ischosen sufficiently small, it follows that for v € B(0,r)| Dh (x) | (h(a+v) —h(x)) —v|| <erand so, from Lemma 8.10.1, B(0,(1—€)r) € Dh(a) (n (#+8 (0,7) —h (2) ;h (B(@,r)) =h (+B (0,7) —h (ae) 2 Dh (a) B(O, (1 -e)r)Therefore, since mp (B(a,r)) = mp (B(@.7)) ,det (Dh (x) )| mp (B(O, (1 — €) r)) = |det (Dh (a))|(1—€)? r? ap < mp (h(B(a,r)))so for r small enough,mp (h(B(x,r)))mp (B(O,(1+é)r))The claim follows from this since € > 0 is arbitrary.Mp (h(B(w,r)))mp (B(O,(1—€)r))< |det (Dh (a))| <. ‘ mp(h(B(w,r)NH)) __g(@)Lemma 11.9.6 For a.e. x with |detDh (a)| > 0, lim,—o m(F(B(@.r))) ~ |det Dh(a)| °Proof: Using the result of Lemma 11.9.5, for a.e. x satisfying |detDh (x)| > 0, if rsmall enough, thenmp (It (B(,r))) € [|detDh (a)| mp (B(a,r)) (1—e)? det Dh (a)| mp (B(a,r)) (1+ €)"]