on the occasion of her 85th birthday
SOME METRIC RELATIONS OF THE HOMEOMORPHISMS SATISFYING GENERALIZED MODULAR INEQUALITIES (I)
MIHAI CRISTEA
We study some metric relations of some homeomorphismsf :D→D‘ between two domains fromRnsatisfying a generalized modular inequality. These relations are in connection with the property of local quasisymmetry of quasiconformal mappings.
AMS 2010 Subject Classification: 30C65.
Key words: metric properties of the homeomorphisms satisfying generalized modular inequalities.
IfD⊂Rnis a domain, we denoteA(D) the set of all path families Γ from D and if Γ ∈ A(D) we put F(Γ) = {ρ :Rn → [0,∞] Borel maps|R
γ
ρds ≥ 1 for every γ ∈ Γ locally rectifiable}. We define for p > 1, Γ ∈ A(D) and ω∈D→[0,∞] measurable and finite a.e. thep modulus of weightω,
Mωp(Γ) = inf
ρ∈F(Γ)
Z
Rn
ω(x)ρp(x)dx For ω= 1 we obtain the classicalp modulus
Mp(Γ) = inf
ρ∈F(Γ)
Z
Rn
ρp(x)dx.
The systematic utilization of the arbitrary weight pmodulus in the map- ping theory was initiated by Cabiria Andreian Cazacu in [1].
Let D, D‘ be domains in Rn and f : D → D‘ a homeomorphism. We say that f satisfies condition (N) if µn(f(A)) = 0 for every A ⊂ D with µn(A) = 0 (here µn denotes the Lebesgue measure in Rn). If x ∈ D, we set L(x, f) = lim sup
h→0
|f(x+h)−f(x)|
|h| . We denote for x ∈ D and r > 0 such that B(x, r)⊂Dby L(x, f, r) = sup
|y−x|=r
|f(y)−f(x)|,l(x, f, r) = inf|y−x|=r|f(y)− f(x)|,Hf(x, r) = L(x,f,r)l(x,f,r) and Hf(x) = lim sup
r→0
Hf(x, r).
MATH. REPORTS15(65),4(2013), 387–396
We say that f is quasiconformal if there existsK ≥1 such that MnK(Γ) ≤ Mn(f(Γ)) ≤KMn(Γ) for every Γ ∈ A(D) (the geometric definition of quasi- conformality). This definition is equivalent with the metric definition of qua- siconformality, which says that f is quasiconformal if there exists H≥1 such that Hf(x)≤H for every x∈D. We recommend the reader the book [27] for some basic facts concerning quasiconformal mappings.
We say that f is η-quasisymmetric if there exists a homeomorphism η : [0,∞) → [0,∞) such that |f(y)−f(x)||f(z)−f(x)| ≤ η(|y−x||z−x|) if x, y, z ∈ D, x 6= y 6=z. If f is η-quasisymmetric, then Hf(x, r) ≤η(1) for every x ∈Dand every r >0 such thatB(x, r)⊂D, hencef is quasiconformal. Also, iff is quasiconformal, and B(x,3r) ⊂D, then f|B(x, r) : B(x, r) → f(B(x, r)) is η-quasisymmetric and if f :Rn→Rn is a homeomorphism, thenf is quasiconformal if and only iff is η-quasisymmetric.
An important class of continuous, open, discrete mappingsf:D⊂Rn→Rn which generalizes quasiconformal mappings is the class of quasiregular map- pings (see [20, 21] or [28] for some basic facts concerning this theory), for which the important modular inequality of Poleckii holds. This says that iff :D→ Rn isK-quasiregular, then Mn(f(Γ))≤KMn(Γ) for every Γ∈A(D) and this is the key for proving most of the geometric properties of this class of mappings.
More general classes of continuous, open, discrete mappings (the so called mappings of finite distortion) were intensively studied in the last years in [3–
6], [12–19], [22–26] and several conditions were imposed to the dilatation K of the function f or to the function f, like K ∈BM O(D), or such that there exists an Orlicz functionA such thatexp(A◦K)∈L1loc(D) or such thatf has locally ACLn inverses. For all of them, the modular inequality ”Mn(f(Γ))≤ MKnn−1(f)(Γ)” holds for every Γ∈A(D) and this is the main instrument used in studying this functions.
In some recent papers [7–10] we studied classes of open, discrete mappings f :D→Rnfor which a modular inequality of type “Mq(f(Γ))≤γ(Mωp(Γ)) for every Γ∈A(D)” holds, where 1< p,n−1< q,γ : [0,∞)→ [0,∞) is strictly increasing with lim
t→0γ(t) = 0 and ω ∈ L1loc(D). We extended partially basic theorems from the theory of quasiregular mappings and from the classes of open, discrete mappings mentioned before. We gave Liouville, Picard, Montel type theorems, equicontinuity results and we gave estimates of the modulus of continuity. The basic tool for proving these things was the modular inequality
”Mq(f(Γ)) ≤ γ(Mωp(Γ))” together with the fact that Mωp(x) = 0 for some points x ∈ D. Using the modulus method, we developed an unified theory which contains all the classes of continuous, open, discrete mappings studied before. We continue this researches in the present paper.
LetD⊂Rna domain andf :D→Rna map. We say thatf isACLiff is continuous and for every cubeQ⊂⊂Dwith the sides parallel to coordinate axes and for every faceS of Qit results thatf|PS−1(y)∩Q:PS−1(y)∩Q→Rn is absolutely continuous for a.e. y∈S, wherePS:Rn→S is the projection on S. AnACLmap has a.e. first partial derivatives and ifq >1, we say thatf is ACLq iff isACL and the first partial derivatives are locally in Lq. If q >1, we denote byWloc1,q(D,Rn) the Sobolev space of all functionsf :D⊂Rn→Rn which are locally inLqtogether with their first order distributional derivatives.
We see from Prop. 1.2, page 6 in [21] that iff ∈C(D,Rn), thenf isACLq if and only iff ∈Wloc1,q(D,Rn). IfE, F ⊂D, we set ∆(E, F, D) ={γ : [0,1]→Rn path |γ(0) ∈ E, γ(1) ∈ F and γ((0,1)) ⊂ D}. We denote by Vn the volume of the unit ball in Rn. If A ∈ L(Rn,Rn), we set ||A|| = sup
|x|=1
|A(x)|, l(A) =
|x|=1inf |A(x)|.
If γ : [a, b] → Rn is rectifiable, we set γ0 its normal representation (see Def. 2.5, page 5 in [27]).
Let D⊂Rn be a domain andϕ:B(D)→[0,∞]. We say that ϕ is a set function if ϕ(A) < ∞ for every A ⊂ D compact and ϕ(
∞
S
i=1
Ai) =
∞
P
i=1
ϕ(Ai) if A1, ..., Ai, ...are disjoint Borel sets inD. We say thatϕis absolutely continuous if for every >0 there existsδ>0 such thatϕ(A)< for everyA∈ B(D) with µn(A)< δ. Ifϕassumes only finite values, then ϕis absolutely continuous if and only ifϕ(A) = 0 wheneverA∈ B(D) is such thatϕ(A) = 0. We say thatϕ has a derivative ϕ‘(x) in the point x∈D if there existsϕ‘(x) = lim
r→0
ϕ(B(x,r)) µn(B(x,r)). A set function ϕ has a.e. a finite derivative and ifϕis absolutely continuous, thenϕ(A) =R
A
ϕ‘(x)dx for everyA∈ B(D) (see [27], page 81–83).
Let now D, D‘be domains in Rnandf :D→D‘a homeomorphism. We set µf :B(D)→[0,∞) byµf(A) =µn(f(A)) for everyA∈ B(D). Then µf is a set function onD, hence µ‘f(x) exists and is finite a.e. inDand µ‘f is Borel measurable. If in addition f satisfies condition (N) and g : Rn → [0,∞] is a Borel map, then R
A
g(f(x))µ‘f(x)dx= R
f(A)
g(y)dy for everyA∈ B(D).
Ifq >1,f :D→D‘is a homeomorphism andg=f−1, we define a.e. the q inner dilatation off,KI,q(f) :D→[0,∞] by KI,q(f)(x) =L(f(x), g)µ‘f(x).
ThenKI,q(f) is Borel measurable and if f is differentiable in xandJf(x)6= 0, thenKI,q(f)(x) = l(f|Jf‘(x))(x)|q.
We shall need the following result (see Theorem 4, page 516 in [2]).
Lemma A. Let n−1< q, let D=B(x, b)\B(x, a) ⊂Rn, let E, F ⊂D be disjoint such that S(x, t)∩E 6= φ, S(x, t)∩F 6= φ for every a < t < b.
Then Mq(∆(E, F, D))≥C(n, q)(bn−q−an−q) if q 6=nand Mn(∆(E, F, D))≥ C(n) lnab, where C(n, q) is a constant depending only on n and q and C(n) is a constant depending only on n.
We give first some sufficient conditions in order that a homeomorphism f :D→ D‘ to satisfy some generalized modular inequalities. The result is a slight extension of Theorem 1 in [10].
Theorem 1. Let n ≥ 2, 1 < q < p, D, D‘ be domains in Rn, let f : D → D‘ be a homeomorphism satisfying condition (N), let g be its in- verse and suppose that g is ACLq. Then Mq(f(Γ)) ≤ MKq
I,q(f)(Γ) for every Γ ∈ A(D) and Mq(f(Γ)) ≤ CMp(Γ)q/p for every Γ ∈ A(D), where C = (R
D
KI,q(f)(x)p/(p−q)dx)
p−q p .
Proof. Since g is an ACLq homeomorphism, there exists a constant C(n, q) depending only on n and q such that L(y, g) ≤ C(n, q)|g‘(y)| a.e.
Let G ⊂⊂ D be open. Then R
G
KI,q(f)(x)dx = R
G
L(f(x), g)qµ‘f(x)dx
= R
f(G)
L(y, g)qdy ≤ C(n, q)q R
f(G)
|g‘(y)|qdy < ∞, hence KI,q(f) is finite a.e.
Let Γ ∈ A(D) and let ∆ = {γ ∈ Γ|β = f ◦γ is rectifiable and g◦β0 is ab- solutely continuous on every compact, nonempty interval I ⊂[0, l(β)]}. Using Fuglede’s theorem (see Theorem 28.2, page 95 in [27]), we see thatMq(f(∆)) = Mq(f(Γ)). Let ρ ∈ F(∆) and let η : Rn → [0,∞), η(y) = ρ(g(y))L(y, g) if y ∈D‘, η(y) = 0 if y 6∈D‘. Using Theorem 5.3, page 12 in [27], we see that η∈F(f(∆)) and using relation (1), we see that
Mq(f(Γ)) =Mq(f(∆))≤ Z
Rn
η(y)qdy= Z
D‘=f(D)
ρ(g(y))qL(y, g)qdy=
= Z
D
ρ(g(f(x)))qL(f(x), g)qµ‘f(x)dx= Z
D
ρ(x)qKI,q(f)(x)dx≤
≤MKq
I,q(f)(∆)≤MKq
I,q(f)(Γ).
Using H¨older’s inequality, we have Mq(f(Γ)) ≤ R
D
ρ(x)qKI,q(f)(x)dx ≤ (R
D
ρp(x)dx)q/pCand sinceρ∈F(∆) was arbitrarily chosen, we see thatMq(f(Γ))≤ CMp(∆)q/p ≤CMp(Γ)q/p.
The following example from [1] is aC∞diffeomorphism which is not qua- siconformal and satisfies a generalized modular inequality of type “Mq(f(Γ))≤ CMp(Γ)qp for every Γ∈A(D) and some 1< q < p,q, p6=n”.
Example 1. Let D = (0,1)n, 1 < q < p, 0 < c < (p −q)/(pq−p), f :D→Rn,f(x1, ..., xn) = (x1, ..., xn−1,x1+c1+cn ) forx = (x1, ..., xn) ∈D. Then f : D → f(D) is a C∞ diffeomorphism, Jf(x) = xcn = l(f‘(x)), ||f‘(x)|| = 1 and Hf(x) = ||fl(f‘‘(x)||(x)) =x−cn → ∞ifx →0 and hence f is not quasiconformal.
We see that KI,q(f)(x) =xc(1−q)n and if C= (R
D
KI,q(f)(x)p/(p−q)dx)
p−q q , then C < ∞ and we see from Theorem 1 that Mq(f(Γ)) ≤ CMp(Γ)q/p for every Γ∈A(D).
Theorem 2. Let n≥2, n−1< q ≤n, p >1, D, D‘ be domains in Rn, ω∈L1loc(D),γ : [0,∞)→[0,∞) be increasing withlim
t→0γ(t) = 0, letx∈Dand R >0 be such that B(x, R)⊂D, let Qx,R= R
B(x,R)
ω(u)du and let f :D→D‘ be a homeomorphism such that Mq(f(Γ)) ≤ γ(Mωp(Γ)) for every Γ ∈ A(D).
Then, if a∈B(x,R4) and b, c∈B(a,R4) are such that |b−a| ≤ |c−a|, then (2) |f(c)−f(a)|
|f(b)−f(a)| ≤exp( 1
C(n)γ( Qx,R
|b−a|p))if q =n (3) |f(c)−f(a)|n−q− |f(b)−f(a)|n−q≤ 1
C(n, q)γ( Qx,R
|b−a|p)if n−1< q < n If Cx,R,p = sup
0<r≤R
R
B(x,r)
ω(u)du/rp <∞, then, if 0<|y−x| ≤ |z−x| ≤ R3, we have
(4) |f(z)−f(x)|
|f(y)−f(x)| ≤exp( 1
C(n)γ(Cx,R,p(2 + |z−x|
|y−x|)p))if q =n
|f(z)−f(x)|n−q− |f(y)−f(x)|n−q≤
(5) ≤ 1
C(n, q)γ(Cx,R,p(2 + |z−x|
|y−x|)p)if n−1< q < n
Here the constants C(n, q) and C(n) are the constants from Lemma A.
Proof. Suppose first that Cx,R,p < ∞ and let y, z ∈ B(x,R3) be such that 0 < |y−x| ≤ |z−x| ≤ R3. Let w ∈ S(x,|z−x|+|y−x|) and S ∈ S(x,2|y−x|+|z−x|) be such that L(x, f,|z−x|+|y−x|) =|f(w)−f(x)|, L(x, f,2|y−x|+|z−x|) =|f(S)−f(x)|. ThenB(x,2|y−x|+|z−x|)⊂B(x, R) and let P be the plane determined by the points x, y, w. Let γ : [0,1] → B(x,|x−z|+|y−x|)\B(y,|y−x|) be a path such that γ(0) = x,γ(1) =w and let d be the line perpendicular on the plane P in the point y and let M ∈d∩S(x,2|y−x|+|z−x|). LetE=Imγ,F = [y, M]∪S(x,2|y−x|+|z−x|)
and let Γ0 = ∆(E, F, B(x,2|y−x|+|z−x|)). Letρ= |y−x|1 XB(x,2|y−x|+|z−x|). Then ρ∈F(Γ0) and
(6) Mωp(Γ0)≤ Z
Rn
ω(u)ρp(u)du= 1
|y−x|p
Z
B(x,2|y−x|+|z−x|)
ω(u)du
hence Mωp(Γ0)≤Cx,R,p(2 +|y−x||z−x|)p.
We see that f(E) is a path joining f(x) with f(w) and f(F) is com- pact, connected and joins f(y) with f(S). We see that |f(y) − f(x)| ≤ L(x, f,|y−x|) ≤ L(x, f,|z−x|+|y−x|) = |f(w)−f(x)| ≤ L(x, f,2|y− x| +|z − x|) = |f(S) −f(x)|. Let Γ‘ = ∆(f(E), f(F)), B(f(x),|f(w) − f(x)| \B(f(x),|f(y) −f(x)|)) and let Γ be the family of all maximal lift- ings of some paths from Γ‘ starting from some points of E. Then Γ‘ > f(Γ) and since f(B(x,|z−x|+|y−x|))⊂B(f(x),|f(w)−f(x)|) and f is a home- omorphism, we see that Γ ⊂ Γ0. Using Caraman’s result from Lemma A, we obtain that Mn(Γ‘) ≥ C(n) ln(|f(w)−f|f(y)−f(x)|(x)|) ≥ C(n) ln(|f(z)−f|f(y)−f(x)|(x)|) if q = n and Mq(Γ‘)≥C(n, q)(|f(w)−f(x)|n−q− |f(y)−f(x)|n−q) ≥C(n, q)(|f(z)− f(x)|n−q− |f(y)−f(x)|n−q) if n−1< q < n.
It results that
C(n) ln(|f(z)−f(x)|
|f(y)−f(x)|)≤Mn(Γ‘)≤Mn(f(Γ))≤γ(Mωp(Γ))≤
≤γ(Mωp(Γ0))≤γ(Cx,R,p(2 + |z−x|
|y−x|)p) ifq =nand
C(n, q)(|f(z)−f(x)|n−q− |f(y)−f(x)|n−q)≤Mq(Γ‘)≤Mq(f(Γ))≤
≤γ(Mωp(Γ))≤γ(Mωp(Γ0))≤γ(Cx,R,p(2 + |z−x|
|y−x|)p) ifn−1< q < n.
We proved in this way relations (4) and (5).
We can remark that in the case 0<|y−x| ≤ |z−x|2 ≤ R4, the estimates in (4) and (5) can be improved, i.e.
(7) |f(z)−f(x)|
|f(y)−f(x)| ≤exp( 1
C(n)γ(Cx,R,p(1 + |z−x|
|y−x|)p)) ifq=n
|f(z)−f(x)|n−q− |f(y)−f(x)|n−q≤
(8) ≤ 1
C(n, q)γ(Cx,R,p(1 + |z−x|
|y−x|)p) ifn−1< q < n
Indeed, in this case we can takew∈S(x,|z−x|) such that|f(w)−f(x)|= L(x, f,|z−x|) and we use the same proof.
Suppose now that a∈B(x,R4) and b, c∈B(a,R4) are such that|b−a| ≤
|c−a|. Then B(a,2|b−a|+|c−a|)⊂B(x, R) and hence
1
|b−a|p
R
B(a,2|b−a|+|c−a|)
ω(u)du≤ |b−a|Qx,Rp. Replacing x, y, z witha, b, c in the pre- ceding arguments, we see from (6) that Mωp(Γ0) ≤ |b−a|Qx,Rp and we apply then the same arguments.
Remark 1.If we take |y−x|= r = |z−x| in relations (4) and (5), we obtain
Hf(x, r) = L(x,f,r)l(x,f,r) ≤exp(C(n)1 γ(3pCx,R,p)) and (9) Hf(x)≤exp(γ(3pCx,R,p)
C(n) )if q=n,0< r≤ R 3 0≤L(x, f, r)n−q−l(x, f, r)n−q≤ (10) ≤ γ(3pCx,R,p)
C(n, q) ifn−1< q < n and0< r≤ R 3
Also, if we take|c−a|=|b−a|=r in relations (2) and (3) we obtain Hf(a, r) = L(a, f, r)
l(a, f, r) ≤exp( 1
C(n)γ(Qx,R
rp ))if a∈B(x,R 4),
(11) 0< r≤ R
4 andq =n L(a, f, r)n−q−l(a, f, r)n−q ≤ 1
C(n, q)γ(Qx,R
rp )if a∈B(x,R 4),
(12) 0< r≤ R
4 andn−1< q < n
If we fix δ >0 and we take in (11)δ ≤r≤ R4, we find that Hf(a, r)≤exp( 1
C(n)γ(Qx,R
δp )) =C(n, p, δ, R) for every
(13) a∈B(x,R
4) and everyδ≤r≤ R 4
The last relation shows that for large enough balls centered in points a ∈ B(x,R4) the dilatation Hf(·,·) is uniformly bounded on B(x,R4)×(δ,R4) by a constant C=C(n, p, δ, R).
We show now that the homeomorphismsf :D→D‘satisfying condition (N) and having ACLq inverses are so that the metric relations from Theorem 2 are verified by them.
Theorem 3. Let n ≥ 2, D, D‘ be domains in Rn, n−1 < q ≤ n, f : D → D‘ be a homeomorphism satisfying condition (N) such that its inverse is ACLq, let x ∈ D and R > 0 be such that B(x, R) ⊂ D and suppose that KI,q(f)∈L1loc(D) and that Cx,R,q = sup
0<r≤R
R
B(x,r)
KI,q(f)(x)dx/rq<∞. Then
|f(z)−f(x)|
|f(y)−f(x)|≤exp(Cx,R,q
C(n) (2 + |z−x|
|y−x|)n)if q =n (14) and0<|y−x| ≤ |z−x| ≤ R
3
|f(z)−f(x)|n−q− |f(y)−f(x)|n−q≤ Cx,R,q
C(n, q)(2 +|z−x|
|y−x|)q (15) if n−1< q < n and0<|y−x| ≤ |z−x| ≤ R
3 Proof. We see from Theorem 1 that Mq(f(Γ)) ≤ MKq
I,q(f)(Γ) for every Γ∈A(D). We apply now Theorem 2, relation (4) and (5).
Theorem 4. Let n ≥ 2, D, D‘ be domains in Rn, n−1 < q ≤ n, f : D→D‘ be a homeomorphism satisfying condition(N) such that its inverse is ACLq. Then, if q =n and p > n and C = (R
D
KI,n(f)(x)p/(p−n)dx)
p−n p <∞, we have
|f(z)−f(x)|
|f(y)−f(x)| ≤exp( C
C(n)(Vn(2|y−x|+|z−x|)n
|y−x|p )np)f or every x∈D and (16) R >0such that B(x, R)⊂D and0<|y−x| ≤ |z−x| ≤ R
3 and
Hf(x, r)≤exp( C
C(n)(3nVnrn−p)np)f or (17) every x∈D and every0< r < d(x, ∂D)
3 .
If n−1< q < nandC= (R
D
KI,q(f)(x)n/(n−q)dx)n−qn <∞, it results that
|f(z)−f(x)|n−q− |f(y)−f(x)|n−q ≤ C C(n, q)V
q
nn(2 +|z−x|
|y−x|)qf or every x∈D and every R >0such that
(18) B(x, R)⊂D and0<|y−x| ≤ |z−x| ≤ R 3
and
L(x, f, r)n−q−l(x, f, r)n−q ≤ C C(n, q)3qV
q
nn if x∈D
(19) and every0< r < d(x, ∂D) 3
Proof. Suppose that q = n. We see from Theorem 1 that Mn(f(Γ)) ≤ CMp(Γ)np for every Γ∈A(D) and keeping the notations from Theorem 2 we see that Mp(Γ0) ≤ R
Rn
ρp(u)du ≤ |y−x|1 p
R
B(x,2|y−x|+|z−x|)
du = Vn(2|y−x|+|z−x|)n
|y−x|p . Taking γ : [0,∞) → [0,∞), γ(t) = Ctn/p for t ≥ 0 and using the same arguments as in Theorem 2 we find that C(n) ln(|f(z)−f|f(y)−f(x)|(x)|) ≤ Mn(Γ‘) ≤ Mn(f(Γ))≤γ(Mp(Γ))≤γ(Mp(Γ0))≤C(Vn(2|y−x|+|z−x|)n
|y−x|p )np and relation (16) is proved.
Suppose now thatn−1< q < n. We see from Theorem 1 thatMq(f(Γ))≤ CMn(Γ)nq for every Γ ∈ A(D). Taking γ : [0,∞) → [0,∞), γ(t) = Ctq/n for t ≥ 0 and ω = 1 in Theorem 2, we apply this theorem and we obtain relation (18).
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Received 11 February 2013 University of Bucharest,
Faculty of Mathematics and Computer Sciences, Str. Academiei 14, R-010014,
Bucharest, Romania mcristea@fmi.unibuc.ro
