AND IMPLICIT FUNCTION THEOREMS
MIHAI CRISTEA
We establish some mean value theorems concerning the generalized derivative on a direction in the sense of Clarke, in connection with a mean value theorem of Lebourg [14] and Pourciau [18] for locally lipschitzian maps. We use the results to generalize the lipschitzian local inversion theorem of Clarke [2] and give global univalence results of Hadamard-Levy-John type, extending earlier results from [4] and [9]. We prove some extensions of some known univalence theorems of Warschawski and Reade from complex univalence theory. Our extensions hold for a class of mappings defined by a generalized ACL property, containing the locally lipschitzian mappings, the quasiregular mappings, and the space of Sobolev mappings Wloc1,1(D,Rn)∩C(D,Rn). We also give in this class some implicit function theorems.
AMS 2000 Subject Classification: 30C45, 26B10, 30C65.
Key words: mean value, local and global univalence, implicit function theorem.
1. INTRODUCTION
An extensive literature has been devoted in the last 30 years to the so- called generalized derivative of Clarke, whose natural setting is in the class of locally lipschitzian mappings f : D → Rm, with D ⊂ Rn open. Such mappings are a.e. differentiable and if E ⊂ D is such that mn(E) = 0 and f is differentiable on D\E, the generalized derivative ∂Ef(x) of f at x is defined as
co{A∈ L(Rn,Rn)|there existsxp →x, xp∈D\E such thatf‘(xp)→A}.
Here, mq is the q-Hausdorff measure in Rn. A set A ⊂ Rn has q- dimensional measure if A =
∞
S
p=1
Ap with mq(Ap) <∞ for every p ∈N. The generalized derivative∂Ef(x) is defined at all pointsx∈D, althoughf is only a.e. differentiable. However, ∂Ef(x) does not usually reduce to the ordinary derivative f‘(x) becausef‘ may be discontinuous atx.
REV. ROUMAINE MATH. PURES APPL.,54(2009),2, 131–145
If v ∈ Sn = {x ∈ Rn | kxk = 1}, D ⊂ Rn is open, x ∈ D and f :D→Rm is a map, we define
Df,v(x) =
w∈Rm|there existstp →0 so that f(x+tpv)−f(x) tp
→w
, the derivative set of the mapf at the pointxon the directionv, and ifA⊂D, we set Df,v(A) = S
x∈A
Df,v(x). If Df,v(x) ⊂ Rm and cardDf,v(x) = 1, then there exists lim
t→0
f(x+tv)−f(x)
t = ∂f∂v(x), the directional derivative of f in x on the direction v. We put Df,v+ (x) = lim sup
t→0
f(x+tv)−f(x) t
. If E ⊂ D is such that mn(E) = 0 and ∂f∂v exists on D\E, we define the generalized derivative of f atx on the directionv as
∂E ∂f
∂v
(x) = con
w∈Rm |there existsxp ∈D\E, xp →x such that ∂f
∂v(xp)→w o
, and if A ⊂ D, we put ∂E(∂f∂v)(A) = co S
x∈A
∂E(∂f∂v)(x). We see that if ∂f∂v is bounded near x, then∂E(∂f∂v)(x) is a compact convex subset ofRm.
For maps f : D ⊂ Rn → Rm with D ⊂ Rn open and E ⊂ D with mn(E) = 0 such that f is differentiable on D\ E, we can also define the generalized derivative of f atx in the sense of Clarke as
∂Ef(x) = co{A∈ L(Rn,Rm)|there existsxp∈D\E, xp →x such thatf‘(xp)→A},
since the definition is consistent even if f‘ is not bounded near x. But if f‘ is bounded near x, then ∂Ef(x) is a compact convex subset of L(Rn,Rm).
If A ⊂ D, we put ∂Ef(A) = co S
x∈A
∂Ef(x). If D ⊂ Rn is open, x ∈ D and f :D→Rm is a map, we put
D+f(x) = lim sup
y→x
kf(y)−f(x)k
ky−xk , D−f(x) = lim inf
y→x
kf(y)−f(x)k ky−xk . If D ⊂ Rn is open, v ∈ Sn and f : D → Rm is continuous, we say that f is v-ACL (absolutely continuous on the direction v) if there exists B ⊂Hv ={x∈Rn| hx, vi= 0} withmn−1(B) = 0 such that f|Ix:Ix→Rm is absolutely continuous for every compact intervalIx ⊂P−1(x)∩Dand every x ∈Hv\B, where P :Rn →Hv is the projection on Hv. If e1, . . . , en is the canonical base inRnandfisei-ACL fori= 1, . . . , n, we say as in [22, page 88]
that f is ACL, and if f is v-ACL for every v ∈ Sn, we say as in [9] that f is a GACL map. Using [20, page 6], we see that a continuous map from the
Sobolev space Wloc1,1(D,Rm) is a GACL map. We can also easily see that a locally lipschitzian map is GACL. If A∈ L(Rn,Rm), we put
kAk= sup
kxk=1
kA(x)k, l(A) = inf
kxk=1kA(x)k.
We shall prove the following basic mean value theorem, extending some results from [7] and [8].
Theorem 1. Let D ⊂ Rn be open, v ∈ Sn, E ⊂ D with mn(E) = 0, f :D→Rm continuous on D,H ⊂Rm convex, U ⊂D open,[a, b]⊂U such that v= kb−akb−a , Df,v((D\E)∩U)⊂H andDf,v(x) compact in Rm for every x∈(D\E)∩U. Suppose that one of the following conditions holds:
1)f isv-ACL.
2) ∂f∂v is locally integrable onU andm1(f(E)) = 0.
3)E is of (n−1)-dimensional measure.
Then for every > 0 there exist v ∈ H and θ ∈ Rm with kθk <
such that f(b)−f(a) = vkb−ak+θ, hence there exists λ ∈ H¯ such that f(b)−f(a) =λkb−ak.
The following consequences of Theorem 1 are obvious.
Theorem 2. Let D ⊂ Rn be open, v ∈ Sn, E ⊂ D with mn(E) = 0, f :D → Rm continuous on D such that ∂f∂v exists on D\E, U ⊂ D open, [a, b]⊂U such thatv= kb−akb−a ,H ⊂Rm convex such that ∂f∂v((D\E)∩U)⊂H.
Suppose that one of the following conditions holds:
1)f isv-ACL,
2) ∂f∂v is locally integrable on U andm1(f(E)) = 0.
3)E is of (n−1)-dimensional measure.
Then for every > 0 there exists v ∈ H and θ ∈ Rm with kθk ≤ such that f(b)−f(a) = vkb−ak+θ, hence there exists λ ∈ H¯ such that f(b)−f(a) =λkb−ak.
Theorem 3. Let D ⊂ Rn be open, v ∈ Sn, E ⊂ D with mn(E) = 0, f :D→ Rm continuous such that there exists Lv >0 with Df,v+ (x) ≤Lv on D\E and suppose that one of the following conditions holds:
1)f isv-ACL.
2)m1(f(E)) = 0.
3)E is of (n−1)-dimensional measure.
Then if [a, b] ⊂ D is such that v = kb−akb−a we have kf(b) −f(a)k ≤ Lvkb−ak.
Theorem 4. Let D ⊂ Rn be open, v ∈ Sn, E ⊂ D with mn(E) = 0, f :D →Rm continuous such that ∂f∂v exists on D\E and is locally bounded
on D, [a, b]⊂ D such that v = kb−akb−a and suppose that one of the following conditions holds:
1)f isv-ACL.
2) ∂f∂v is locally integrable on D andm1(f(E)) = 0.
3)E is of (n−1)-dimensional measure.
Then there exists λ∈∂E(∂f∂v)([a, b]) such thatf(b)−f(a) =λkb−ak.
Theorem 5. Let D ⊂ Rn be open, v ∈ Sn, E ⊂ D with mn(E) = 0, f :D→Rm continuous such that ∂f∂v exists onD\E and there existsH ⊂Rm convex for which ∂E(∂f∂v)([a, b]) ⊂ H for every [a, b] ⊂ D with v = kb−akb−a . Suppose that one of the following conditions holds:
1)f isv-ACL.
2)∂f∂v is locally integrable andm1(f(E)) = 0.
3)E is of (n−1)-dimensional measure.
Then if [a, b] ⊂ D is such that v = kb−akb−a , there exists λ ∈ H¯ with f(b)−f(a) =λkb−ak.
A known mean value theorem of Lebourg [14] and Pourciau [18] says that if D ⊂ Rn is open, f :D → Rm is locally lipschitzian, E ⊂ D is such that mn(E) = 0 and f is differentiable on D\E, then, for [a, b]⊂D, there exists A∈∂Ef([a, b]) such thatf(b)−f(a) =A(b−a). The preceding theorems are the corresponding versions forv-ACL mappings. We notice that in Theorem 5 we do not ask the derivative ∂f∂v to be locally bounded. We also have
Theorem 6. Let D ⊂ Rn be open, v ∈ Sn, E ⊂ D with mn(E) = 0, f : D → Rm continuous on D and differentiable on D\E, U ⊂ D open, Q= co(f‘((D\E)∩U),[a, b])⊂U such that v= kb−akb−a and suppose that one of the following conditions holds:
1)f isv-ACL.
2) ∂f∂v is locally integrable on D andm1(f(E)) = 0.
3)E is of (n−1)-dimensional measure.
Then for every > 0 there exists A ∈ Q and θ ∈ Rm with kθk ≤ such that f(b)−f(a) =A(b−a) +θ. If f0 is locally bounded onD, we can find A∈∂Ef([a, b]) such thatf(b)−f(a) =A(b−a).
We know thatf :D⊂Rn→Rm is locally lipschitzian on Dif and only iff is GACL andf0 exists a.e. and is locally bounded on D. Our Theorem 6 brings some new information if f0 is locally bounded and f is not a GACL map, and this may happen in case 1), when we ask f to be v-ACL only on the direction v, and in case 3), when we only ask the “singular” set E to be
“thin” enough, i.e., to be of (n−1)-dimensional measure.
The following generalization of Denjoi-Bourbaki’s theorem can be proved using the classical proof:
Theorem 7. Let E, F be normed spaces,a, b∈E, v= kb−akb−a ,K ⊂[a, b]
at most countable, f : [a, b]→F continuous such that there exists M >0 with D+f,v(x)≤M for everyx∈[a, b]\K. Then kf(b)−f(a)k ≤Mkb−ak.
Using Theorem 7 can prove the following infinite dimensional version of Theorem 3.
Theorem 8. Let E be an infinite dimensional Banach space, v ∈ E with kvk= 1, F a normed space, D⊂E open,K =
∞
S
n=1
Kn with Kn compact sets for n ∈ N, f : D → F continuous such that there exists Lv > 0 with D+f,v(x) ≤ Lv on D\K. Then if [a, b] ⊂D is such that v = kb−akb−a , we have kf(b)−f(a)k ≤Lvkb−ak.
The second aim of this paper is to use the preceding mean value theo- rems to prove some univalence and local univalence results. A known theorem concerning the theory of the generalized derivative in the sense of Clarke is the lipschitzian local inversion theorem of Clarke. This theorem says that if D⊂Rn is open, x0 ∈D, f :D→Rn is locally lipschitizian on D,E ⊂Dis such that mn(E) = 0 andf is differentiable onD\E such that detA6= 0 for every A∈∂Ef(x0) (this last condition implies that 0 ∈/ ∂E(∂f∂v)(x0) for every v ∈Sn), thenf is a local homeomorphism atx0. We denote foru, v∈Rn\{0}
by a(u, v) the angle betweenuand vwhich is less or equal toπ, and ifv∈Sn and 0≤ϕ < π we set Cv,ϕ ={w∈Rn|a(v, w)< ϕ}, the cone of direction v and angle ϕ, centered at 0.
We can easily see that there exist continuous and not locally lipschitzian mappings f : D ⊂ Rn → Rm such that there exists v ∈ Sn and Lv > 0 with D+f,v(x) ≤ Lv on D. If for such a mapping the condition “detA 6= 0 for every A ∈ ∂Ef(x0)” is satisfied at a point x0 ∈ D, we use the fact that
∂E(∂f∂v)(x) is a compact subset ofRm forx∈D, that 0∈/ ∂E(∂f∂v)(x0) and the upper continuity of the multivalued mapx→∂E(∂f∂v)(x) to see that there exist rx0 > 0, w∈ Sn and δ > 0 such that ∂E(∂f∂v)(B(x0, rx0)) ⊂δw+Cw,π. This remark shows that the next theorem is an extension of Clarke’s lipschitzian local inversion theorem for v-ACL mappings (and also an extension of a result from [9]).
Theorem 9. Let D ⊂ Rn be a domain, E ⊂ D with mn(E) = 0, x0 ∈ D, f : D → Rm a GACL map such that ∂f∂v exists on D\E for every v ∈ Sn, and suppose that there exists rx0 > 0 such that B(x0, rx0) ⊂ D and
that for every v ∈Sn there exist w∈Sn and δ >0 depending on v such that
∂E(∂f∂v)(B(x0, rx0))⊂δw+Cw,π.
Then f is injective on B(x0, rx0) and if δ = δx0 does not depend on v ∈Sn, then kf(b)−f(a)k ≥δx0kb−ak for everya, b∈B(x0, rx0).
A known global inversion theorem of Hadamard, Levy and John [4], [12]
says that ifE, F are Banach spaces andf :E →F is a local homeomorphism such that there existsω: [0,∞)→[0,∞) continuous withD−f(x)≥ ω(kxk)1 for every x∈E, then f :E →F is a homeomorphism. Cristea [9] gave a version for a.e. differentiable GACL mappings, extending a result of Pourciau [18].
Another known global inversion theorem of Banach, Mazur and Stoilow [3]
says that ifE, F are pathwise connected Hausdorff spaces,F simply connected and f :E →F is a local homeomorphism which is a proper or a closed map, then f : E → F is a homeomorphism. A version of this theorem for a.e.
differentiable GACL mappings can be found in [9]. We prove here a version for GACL mappings not necessarily a.e. differentiable.
Theorem 10. Let E ⊂ Rn be such that mn(E) = 0, f : Rn → Rn a GACL map such that ∂f∂v exists onD\E for everyv∈Sn and letω: [0,∞)→ [0,∞) be continuous such that R∞
1 ds
ω(s) =∞. Suppose that for every x0 ∈Rn there exists rx0 >0 such that for every v∈Sn there exists w∈Sn depending on v such that ∂E(∂f∂v)(B(x0, rx0))⊂ ω(kx1
0k)w+Cw,π. Then f :Rn → Rn is a homeomorphism.
Theorem 11. Let D, F be domains in Rn,F simply connected,E ⊂D with mn(E) = 0, f : D → F a GACL map which is closed or proper such that ∂f∂v exists on D\E for every v ∈ Sn. Suppose that for every x0 ∈ D there exists rx0 >0 such thatB(x0, rx0)⊂D and for every v∈Sn there exist w∈Sn and δ >0 depending on v such that∂E(∂f∂v)(B(x0, rx0))⊂δw+Cω,π. Then f :D→F is a homeomorphism.
A basic complex univalence theorem of Warshawski says that if D⊂C is a convex domain and f ∈H(D) is such that Ref‘(z) >0 on D, then f is univalent on D. The result was generalized by Reade [19], who showed that if D ⊂ C is a ϕ-angular convex domain with 0 ≤ ϕ < π and f ∈ H(D) is such that |argf‘(z)|< π−ϕ2 on D, then f is univalent onD. Here, a domain D ⊂ Rn is ϕ-angular convex, with 0 ≤ ϕ < π, if for every z1, z2 ∈ D there exists z3 ∈D such that [z1, z3]∪[z2, z3]⊂Dand a(z1−z3, z2−z3)≥π−ϕ, and we see that a 0-angular convex domain is a convex domain. Mocanu [17, 16] extended these results to C1 mappings and Cristea [6], [7] and Gabriela Kohr [13] gave some extensions to continuous mappings. However, in [7] the setsDf,v(x) are supposed to be compact inRmfor everyx∈D. The theorem
of Rademacher and Stepanow shows that there existsE ⊂Dwithmn(E) = 0 such that ∂f∂v exists onD\E. The compactness of the sets Df,v(x) for every x ∈D implies that ∂f∂v is locally bounded onD, hence the sets∂E(∂f∂v)(x) are compact in Rm for every x∈D. We shall prove a version of these results in which we do not suppose the locally boundedness of the derivative ∂f∂v on D and for which the sets ∂E(∂f∂v)(x) may be unbounded for some pointsx∈D.
Theorem 12. Let 0 < ϕ < π, ψ = π−ϕ2 , D ⊂ Rn a ϕ-angular convex domain, E ⊂ D with mn(E) = 0, f :D → Rm a GACL map such that ∂f∂v exists on D\E for every v ∈ Sn, and suppose that for every v ∈ Sn there exists δ >0 only depending onvsuch that∂E(∂f∂v)([a, b])⊂δv+Cv,ψ for every [a, b]⊂D with v= kb−akb−a . Then f is injective on D.
The usefulness of the preceding theorems is that they are valid in the class of GACL mappings while such maps are not always locally lipschitzian, neither a.e. differentiable, although the directional derivatives ∂f∂v exist a.e.
on D for every v ∈Sn (but may be not locally bounded on D). One of the main subclass of the class of GACL mappings is the important class of con- tinuous Sobolev maps fromWloc1,1(D,Rm) (see [20, page 6]) and its well known subclass of quasiregular mappings (see [20] for a basic monograph regarding quasiregular mappings), hence our results hold in this classes of mappings.
Also, Theorem 9, which extends the lipschitzian local inversion theorem of Clarke, holds for mappings f :D⊂Rn →Rm withm 6=n. Also, we can re- place in Theorems 9, 10, 11, 12 the condition “f is a GACL map” by one of the conditions “∂f∂v is locally integrable onDfor everyv∈Snandm1(f(E)) = 0”
or “Eis of (n−1)-dimensional measure”, since in their proofs we use the mean value Theorem 1.
Finally, we shall use the mean value result from Theorem 6 to prove some implicit function theorems.
Theorem 13. Let U ⊂Rn and V ⊂Rm be open,E ⊂U×V such that mn+m(E) = 0, f :U ×V → Rm continuous on U ×V and differentiable on (U ×V)\E such that for every z = (x, y) ∈ U ×V there exist α > 0 with B¯(z, α) ⊂U ×V and m, M >0 such that k∂f∂x(u)k ≤M on ((U ×V)\E)∩ B(z, α), andl(C)≥m for every C∈co(∂f∂y((U ×V)\E)∩B(z, α)). Suppose that either f is GACL, or thatE is of(m+n−1)-dimensional measure. Then for every z= (a, b)∈U×V there existr, δ >0and a unique lipschitzian map ϕ :B(a, r) → B(b, δ) such that ϕ(a) = b and f(x, ϕ(x)) = f(a, b) for every x∈B(a, r).
Theorem 14. Let U ⊂Rn and V ⊂Rm be open,E ⊂U×V such that mn+m(E) = 0, f :U ×V → Rm continuous on U ×V and differentiable on
(U×V)\Esuch that ∂f∂x and ∂f∂y are locally bounded onU×V anddetC 6= 0for every C∈∂E(∂f∂y)(z) and everyz∈U×V. Suppose that either f is GACL, or thatE is of(m+n−1)-dimensional measure. Then for everyz= (a, b)∈U×V there exist r, δ > 0 and a unique lipschitzian map ϕ:B(a, r)→ B(b, δ) such that ϕ(a) =b andf(x, ϕ(x)) =f(a, b) for everyx∈B(a, r).
Theorems 13 and 14 can be connected to some earlier results of Cristea [5]. For some other recent results in this area see [10], [15], [23]. We end with an implicit function theorem for Sobolev mappings.
Theorem 15. Let U ⊂ Rn and V ⊂ Rm be open, f ∈ Wloc1,m+n(U × V,Rm) continuous such that det(∂f∂y(z)) > 0 a.e. on U ×V. Then for a.e.
(a, b)∈U×V there existr, δ >0 and a unique continuous mapϕ:B(a, r)→ B(b, δ) such that ϕ(a) = b, ϕ ∈ Wloc1,1(B(a, r),Rm) and f(x, ϕ(x)) = f(a, b) for every x∈B(a, r).
2. PROOFS
Proof of Theorem1. Let 0< < 12 be such thatkf(a)k< 161,kf(b)k<
1
16. We can find a ∈ B(a, ), b ∈ B(b, ) such that kf(a)−f(a)k < 32 , kf(b)−f(b)k < 32 , [a, b] ⊂ U, kb −ak = kb−ak, v = kbb−a
−ak and m1([a, b]∩E) = 0, and let g : [0,1] → [a, b], g(t) = a+t(b −a) for t∈[0,1] andh=f◦g.
Suppose first that f is v-ACL. Then we can choose a, b such that f|[a, b] : [a, b] → Rm is absolutely continuous. Hence we can find 0 <
δ < 2 such that if 0 ≤ a0 < b0 < a1 < b1 <, . . . , < am < bm = 1 are such that
m
P
q=0
(bq−aq)< δ, then
m
P
q=0
kh(bq)−h(aq)k< 16 .
Sincem1([a, b]∩E) = 0 andDf,v(x) is compact for everyx∈[a, b]\E, we can use Lemma 1 from [7] to find intervals Iq = (cq, dq), q = 0,1, . . . , m such that 1−Pm
q=0
(dq−cq)< δ2 and every intervalIqcan be covered by intervals Iqk= (tqk−qk, tqk+qk) such that (f(g(tqk)+tv)−f(g(tqk))/t∈B(H,8kb−ak ) for 0 < |t| < qk, k = 0,1, . . . , k(q), q = 0,1, . . . , m, and 0 ≤ c0 < d0 <
c1 < d1 <, . . . , < cm < dm ≤ 1. We can also suppose that we can find at least a point αqk ∈ Iqk∩Iq(k−1) for k = 1, . . . , k(q), q = 0,1, . . . , m. Let sq2k = tqk, k = 0,1, . . . , k(q), q = 0,1, . . . , m, sq2k−1 = αqk, k = 1, . . . , k(q), q = 0,1, . . . , m. We can suppose thatsqk+1−sqk≥0 fork= 0,1, . . . ,2k(q)−1, q = 0,1, . . . , m, and that cq = tq0 =sq0, dq = tqk(q) = sq2k(q), q = 0,1, . . . , m.
Let Zqk = h(s
q
k+1)−h(sqk)
sqk+1−sqk for q = 0,1, . . . ,2k(q)−1, q = 0,1, . . . , m, and let Zq=
2k(q)−1
P
k=0
(sqk+1−sqk)
dq−cq Zqk forq = 0,1, . . . , m. Then Zqk∈ kb−akB(H,8kb−ak ) fork= 0,1, . . . ,2k(q)−1,q= 0,1, . . . , m, henceZq∈ kb−akB(H,8kb−ak ) for q = 0,1, . . . , m.
Let ρ =
m−1
P
q=0
(h(cq+1) −h(dq)) and λ = m
P
q=0
(dq −cq)−1
. Since
m
P
q=0 dq−cq
λ Zq ∈ kb−akB(H,8kb−ak ), we can find v ∈ H and µ ∈ Rm with kµk< 8kb−ak such that
m
P
q=0 dq−cq
λ Zq=kb−ak(v+µ). We see thatkρk< 16 and 1< λ < 1−δ1
< 1−12 <1 + 22. Take θ =f(b)−f(a)− f(b)−f(aλ )−ρ
+
kb−akµ. We havekf(b)−f(a)−f(b)−fλ (a)
k ≤ (λλ−1)
kf(b)k+λλ−1
kf(a)k+
kf(b)−f(b)k
λ +kf(a)−fλ (a)k
≤ 8+8+16 +16 < 2, and this implies thatkθk< . We havef(b)−f(a) =
m
P
q=0
(h(dq)−h(cq)) +
m−1
P
q=0
(h(cq+1)−h(dq)) = λ m
P
q=0 dq−cq
λ Zq
+ρ =λvkb−ak+λµkb−ak+ρ, hence f(b)−f(a) = vkb−ak+θ.
Suppose now that condition 2) holds. Let Hv ={x ∈ Rn | hx, vi = 0}
andP :Rn→Hvthe projection onHv. By the theorem of Fubini, there exists B ⊂Hv withmn−1(B) = 0, so thatm1(P−1(y)∩E) = 0 andD+f,v(x)<∞on P−1(y)\Efor everyy∈Hv\B, and the theorem of Rademacher and Stepanov implies that ∂f∂v exists a.e. in P−1(y)∩D for every y∈Hv\B. This implies that if F = {x ∈ D | ∂f∂v does not exist atx}, then mn(F) = 0. Since ∂f∂v is locally integrable on D, we can choose a, b as before such that in addition m1([a, b]∩F) = 0 and ∂f∂v is integrable on [a, b], i.e., h‘ is integrable on [0,1]. Since lim sup
t→0
kf(x+tv)−f(x)k
t <∞ on [a, b]\E, we have m1(f(A)) = 0 for every A⊂ [a, b]\E withm1(A) = 0, and since m1(f(E)) = 0, we have m1(h(A)) = 0 for every A⊂[0,1] with m1(A) = 0 (i.e. h satisfies condition (N) on [0,1]). We use now a theorem of Bary [21, page 285] to deduce thath
is absolutely continuous on [a, b]. We use from now on the same argument as before.
Suppose now that condition 3) holds. ForA⊂Dwe denote byω(f, A) = sup
x∈A
f(x)−inf
x∈Af(x) the oscillation off on the setA. Using a theorem of Gross [22, page 106], we can takea, b such that the set E= [a, b]∩E is at most
countable for >0. Let E = (ai)i∈N,E=g(F), with ai =g(ti),i∈N.
Let Ji be open intervals centered at ai, so that ω(f, Ji) < 2i+7 for every i∈Nand
∞
P
i=1
l(Ji)≤2kb−ak. LetIi ⊂[0,1] be such that Ji =g(Ii) for every i∈N.
Ift /∈F, sinceDf,v(x) is compact on [a, b]\E, we can use Lemma 1 from [7] to find αt>0 such that f(g(t)+sv)−f(g(t))
s ∈B(H,8kb−ak ) for 0 <|s|< αt
and let B = S
t∈[0,1]\F
(t−αt, t+αt). Then [0,1]\B is a compact subset of F, hence [0,1]\B⊂
∞
S
i=1
Ii. Extracting if necessary a finite subcovering, we can find 0< c0< d0 < c1 < d1 <, . . . , < cm< dm≤1 such that every interval [dq, cq+1] is the union of a finite number of intervalsIi forq = 0,1, . . . , m−1, and [0,1]\B ⊂
m−1
S
q=0
[dq, cq+1]. Since
∞
P
i=1
ω(f, Ji) ≤
∞
P
i=1
2i+7 < 16, we have
m−1
P
q=0
kh(cq+1)−h(dq)k ≤ 16 . Let Iq = (cq, dq) for q = 0,1, . . . , m. Then 1−
m
P
q=0
(dq−cq)≤
∞
P
i=1
l(Ii)< 2and every intervalIqcan be covered by intervals Iqk = (tqk−qk, tqk+qk) such that f(q(tqk)+tv)−f(gt (tqk)) ∈B(H,8kb−ak ) for 0 <|t|< qk, k= 1, . . . , k(q), q = 0,1, . . . , m. We use from now on the same argument as before.
Proof of Theorem 4. Let H = ∂E(∂f∂v([a, b])). Then H is a compact convex subset of Rm and for > 0, we can find δ > 0 such that ∂f∂v((D\ E)∩B([a, b], δ))⊂B(H, ). LetH= co(∂f∂v((D\E)∩B([a, b], δ))) for >0.
ThenH⊂B(H, ) for >0. By Theorem 1, we can findv ∈Handθ ∈Rm with kθk < such that f(b)−f(a) =vkb−ak+θ and, letting → 0, we can find λ∈H such thatf(b)−f(a) =λkb−ak.
Example1. Letf : [0,1]→[0,1] be continuous with f(0) = 0, f(1) = 1 and f‘(t) = 0 a.e. in [0,1] and let F : [0,1]2 → R2 be defined by F(x, y) = (f(x),0) for x, y ∈ [0,1]. Then F is note1-ACL and there exists E ⊂[0,1]2 with m2(E) = 0, so that F is differentiable on [0,1]2 \E and F‘(z) = 0 on [0,1]2\E. We see that∂EF([0,1]2) ={0} and kF(1,0)−F(0,0)k= 1, hence conditions 2) and 3) imposed in Theorems 1, 2, 3, 4, 5, 6 are necessary.
Proof of Theorem6. LetH =Q(v) ={w∈Rm|there existsA∈Qsuch that w =A(v)}. We use Theorem 1 to see that for >0 there exist A ∈Q andθ∈Rm withkθk< such thatf(b)−f(a) =A(b−a) +θfor [a, b]⊂D with v= kb−akb−a . If f‘ is locally bounded on D, we use Theorem 4.
Proof of Theorem7. LetK={rn}n∈Nandαs=a+s(b−a) fors∈[0,1].
Let >0 andA=n
t∈[0,1]| kf(αs)−f(a)k ≤(M+)kαs−ak+ P
rn∈[a,αs) 1 2n
for every s ∈ [0, t) o
. Then A 6= 0, A is an interval and let c = supA. Then kf(αc)−f(a)k ≤ (M +)kαc −ak+ P
rn∈[a,αc) 1
2n, hence A = [0, c].
Suppose that 0 < c < 1. If αc ∈/ K, we can find δ > 0 such that c <
c+δ < 1 and kf(αt)−f(αc)k ≤(M +)kαt−αck forc ≤ t < c+δ, hence kf(αt)−f(a)k ≤ kf(αt)−f(αc)k+kf(αc)−f(a)k ≤(M+)(kαt−αck+kαc− ak) + P
rn∈[a,αc) 1
2n ≤ (M +)kαt−ak+ P
rn∈[a,αt) 1
2n for every t ∈ [c, c+δ).
If αc ∈ K, αc = rm, we use the continuity of f at αc to find δ > 0 such that c < c +δ < 1 and kf(αt) −f(αc)k ≤ 2m for t ∈ [c, c +δ). Then kf(αt)−f(a)k ≤ kf(αt)−f(αc)k+kf(αc)−f(a)k ≤ 2m+ (M+)kαc−ak+
P
rn∈[a,αc) 1
2n ≤(M+)kαt−ak+ P
rn∈[a,αt) 1 2n.
We obtained in both cases that t ∈ A for every t ∈ [c, c+δ) and this contradicts the definition ofc= supA. It follows thatc= 1 and letting→0 we get kf(b)−f(a)k ≤Mkb−ak.
Proof of Theorem8. Let M ={w ∈ E |there exists x ∈K and t∈ R such that w=x+tv}. ThenM also is a countable union of compact sets and since dimE=∞, we have intM =∅. Let p∈Nand ap∈B(a,1p)\M,bp ∈ B(b,1p)\M be such that [ap, bp]⊂Dand v= kbbp−ap
p−apk. Then [ap, bp]∩K=∅ and using Theorem 7 on [ap, bp] we find that kf(bp)−f(ap)k ≤Lvkbp−apk.
Letting p→ ∞, we obtainkf(b)−f(a)k ≤Lvkb−ak.
Remark 1. We can easily obtain some Lipschitz conditions using Theo- rem 3 or Theorem 8. Suppose that D is a c-convex domain (i.e. for every a, b ∈ D there exists γ : [0,1] → D rectifiable such that γ(0) = a, γ(1) = b and l(γ)≤ckb−ak), and thatD+f,v(x)≤LonD\E for everyvwithkvk= 1.
If the map f is as in Theorems 3 or 8, than f iscL-Lipschitz on D. Indeed, let a, b∈Dand γ : [0,1]→Da rectifiable path such that γ(0) = a, γ(1) =b and l(γ)≤ckb−ak, and let ∆ = (0 =t0< t1<, . . . , < tm= 1)∈ D([0,1]) be such that [γ(tk), γ(tk+1)]⊂D fork= 0,1, . . . , m−1. Then kf(b)−f(a)k ≤
m−1
P
k=0
kf(γtk+1)−f(γtk)k ≤L
m−1
P
k=0
kγ(tk+1)−γ(tk)k ≤Ll(γ)≤Lckb−ak.
Also, if in Theorem 3 we takeD={x∈Rn|αi< xi < βi,i= 1, . . . , n}
and there existsL >0 such thatD+f,e
i(x)≤LonD\Efori= 1, . . . , n, thenf is√
nL-Lipschitz onD. Indeed, leta, b∈D,a= (a1, . . . , an),b= (b1, . . . , bn), and let zi = (b1, . . . , bi−1, bi, ai+1, . . . , an) for i = 0,1, . . . , n, with z0 = a,
zn=b. Thenzi∈Dfori= 0,1, . . . , n−1,ei+1 = kzzi+1−zi
i+1−zik fori= 0,1, . . . , n1 and we have kf(b) −f(a)k ≤
n−1
P
i=0
kf(zi+1 −f(zi)k ≤ L
n−1
P
i=0
kzi+1 −zik = L
n
P
i=1
|bi−ai| ≤√
nLkb−ak.
Proof of Theorem9. Leta, b∈B(x0, rx0) be such thatv= kb−akb−a and let w∈Sn and δ >0 be such that ∂E(∂f∂v)(B(x0, rx0))⊂δw+Cw,π and let H= δw+Cw,π. By Theorem 5 we can findλ∈H¯ such thatf(b)−f(a) =λkb−ak, hence kf(b)−f(a)k ≥δkb−ak. This implies thatf is injective onB(x0, rx0).
Ifδdoes not depend on the directionv, it is obvious thatD−f(x0)≥δx0. Proof of Theorem10. It follows from Theorem 9 thatf is a local homeo- morphism and D−f(x) ≥ ω(kxk)1 for every x ∈ Rn. By Theorem 6 from [4], f :Rn→Rn is a homeomorphism.
Proof of Theorem11. It follows from Theorem 9 thatf is a local homeo- morphism which is a proper or a closed map, and we apply Banach-Stoilow’s theorem (see [3] for a proof).
Proof of Theorem 12. Let z1, z2 ∈ D, z1 6= z2 be such that f(z1) = f(z2). Since D is ϕ-angular-convex, there exists z3 ∈ D such that [z1, z3]∪ [z2, z3] ⊂ D and a(z2−z3, z1−z3) ≥ π−ϕ. Let u = kzz1−z3
1−z3k, v = kzz2−z3
2−z3k. The hypothesis and Theorem 5 imply that there exist δu, δv > 0 such that
f(z1)−f(z3)
kz1−z3k ∈ δuu+Cu,ψ ⊂ Cu,ψ and f(zkz2)−f(z3)
2−z3k ∈ δvv+Cv,ψ ⊂ Cv,ψ. Then f(z1)−f(z3) =f(z2)−f(z3)∈Cu,ψ∩Cv,ψ. This implies that a(z1−z3, z2− z3) =a(u, v)< 2ψ =π−ϕ, and we reached a contradiction. It follows that f(z1)6=f(z2) for everyz1, z2 ∈D, hence f is injective onD.
Proof of Theorem13. Letz= (a, b),α, m, M >0 be such that ¯B(z, α)⊂ D, k∂f∂x(u)k ≤ M for every u ∈ (((U ×V)\E)∩B(z, α)) and l(C) ≥ m for everyC∈co(∂f∂y((U×V)\E)∩B(z, α)). LetF :U×V →Rn×Rmbe defined by F(x, y) = (x, f(x, y) +b−f(a, b)) for (x, y)∈U×V. ThenF is continuous onU×V, differentiable on (U×V)\E, and a GACL map iff is a GACL map.
We show that there exists l >0 such that l(A)≥l for every A∈co(F‘((U× V)\E)∩B(z, α)). Indeed, letA∈co(F‘((U×V)\E)∩B(z, α)). Then there existB ∈co(∂f∂x((U×V)\E)∩B(z, α)) andC∈co(∂f∂y((U×V)\E)∩B(z, α)) such that
A=
IdRn 0
B C
.
Let (u, v) ∈ Rn×Rm be such that kuk2 +kvk2 = 1. Then kA(u, v)k2 = kuk2+kB(u) +C(v)k2. Let 0< < m
2√
2 and l= min{,M ,√1
2}. Ifkuk ≥ M ,
we see that kA(u, v)k2 ≥ kuk2 ≥ M22 ≥l2, hence kA(u, v)k ≥l. Suppose now that kuk ≤ M . We have |kB(u) +C(v)k − kC(v)k| ≤ kB(u)k ≤ Mkuk ≤ , hence kB(u) +C(v)k ≥ kC(v)k − ≥ mkvk −. In the case kvk ≤ 2m, we have kA(u, v)k2 ≥ kuk2 = 1− kvk2 ≥1− 4m22 ≥ 12 ≥l2, hencekA(u, v)k ≥l.
In the case kvk> 2m, we have kA(u, v)k2 ≥ kB(u) +C(v)k2 ≥(mkvk −)2≥ (m2m −)2 =2 ≥l2, and we also have kA(u, v)k ≥l. It follows thatl(A)≥l for every A∈co(F‘((U×V)\E)∩B(z, α)).
We show now thatF is a local homeomorphism atz. Letx, y∈B(z, α), x 6=y, such that F(x) =F(y) and let 0 < < lky−xk. Using Theorem 6, we can find A ∈ co(F‘((U ×V)\E)∩B(z, α)) and θ ∈ Rn+m such that kθk< and F(y)−F(x) =A(y−x) +θ. It follows thatkF(y)−F(x)k= kA(y−x) +θk ≥ kA(y−x)k − kθk ≥lky−xk − >0, henceF(y)6=F(x).
We proved thatF is injective onB(z, α) and also thatD−F(u)≥lonB(z, α).
Let now W ∈ V(z) and δ > 0 such that B(a, δ) ⊂ U, B(b, δ) ⊂ V and F :B(a, δ)×B(b, δ) → W is a homeomorphism, and let g = (g1, g2) : W → B(a, δ)×B(b, δ) be its inverse. Letl >0 be such thatB(a, l)×B(b, l)⊂W and r = min{l, δ}. We have (x, y) =F(g(x, y)) = (g1(x, y), f(g1(x, y), g2(x, y)) + b−f(a, b)) for every (x, y)∈B(a, r)×B(b, δ) and we deduce thatx=g1(x, y) and f(x, g2(x, y)) = y −b+f(a, b) for every x ∈ B(a, r) and y ∈ B(b, δ).
Define ϕ : B(a, r) → B(b, δ) by ϕ(x) = g2(x, b) for x ∈ B(a, r). We see that f(x, ϕ(x)) = f(a, b) for every x ∈ B(a, r). We also see that F(a, b) = (a, b) = (a, f(a, g2(a, b)) +b−f(a, b)) =F(a, g2(a, b)) =F(a, ϕ(a)), and using the injectivity of the map F on B(a, r)×B(b, δ), we see that ϕ(a) = b. If ψ:B(a, r)→B(b, δ) is a map such that ψ(a) =band f(x, ψ(x)) =f(a, b) for everyx∈B(a, r), we haveF(x, ϕ(x)) = (x, f(x, ϕ(x)) +b−f(a, b)) = (x, b) = (x, f(x, ψ(x)) +b−f(a, b)) = F(x, ψ(x)) for every x ∈B(a, r). Using again the injectivity of the map F onB(a, r)×B(b, δ), we deduce thatϕ(x) =ψ(x) for every x∈B(a, r). Since D−F(u)≥lon B(a, r)×B(b, δ), the mapping g is 1l lipschitzian, henceϕis 1l-lipschitzian.
Proof of Theorem14. Let z(=a, b) and α >0 be such that there exists M >0 withk∂f∂x(u)k ≤M,k∂f∂y(u)k ≤M for everyu∈((U×V)\E)∩B(z, α).
LetQ={A∈ L(Rm,Rm) |detA6= 0}. ThenQ is open in L(Rm,Rm) and since ∂f∂x and ∂f∂y are bounded near z, the set ∂E(∂f∂y)(z) is a compact, convex subset of Q. We can choose α > 0 as before such that there exists δ >0 withB(∂E(∂f∂y)(z), δ)⊂Qand∂E(∂f∂y)(B(z, α))⊂B(∂E(∂f∂y)(z), δ)⊂Q.
Let F : U ×V → Rm ×Rm, F(x, y) = (x, f(x, y) +b−f(a, b)) for (x, y)∈U ×V. We show that F is injective onB(z, α). Letz1, z2 ∈B(z, α).
SinceF is continuous onU×V, differentiable on (U×V)\E, and a GACL map iff is and a GACL map andF0 is bounded onB(z, α), we can use Theorem 6
to find A∈∂EF([z1, z2]) such that F(z2)−F(z1) =A(z2−z1). Then A=
IdRn 0
B C
,
whereB ∈∂E(∂f∂x([z1, z2]),C∈∂E(∂f∂y)([z1, z2]), henceC∈∂E(∂f∂y)(B(z, α))⊂ Q. It follows that detA= detC6= 0 and this implies thatF(z2)6=F(z1). We proved that F is injective on B(z, α) and we argue now as in the proof of Theorem 13.
Proof of Theorem15. Let F :U ×V →Rn×Rm,F(x, y) = (x, f(x, y)) for (x, y)∈U×V. ThenJF(z)>0 a.e. inU×V,F ∈Wloc1,m+n(U×V,Rn+m).
By Theorem 6.1 in [11, page 150], F is locally invertible with a local inverse in the Sobolev class Wloc1,1 around a.e. pointsz∈U×V. We argue now as in the proofs of Theorem 13 and 14.
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Received June 2008 University of Bucharest
Faculty of Mathematics and Computer Sciences Str. Academiei 14
010014 Bucharest, Romania, mcristea@fmi.unibuc.ro
