DILATATIONS OF HOMEOMORPHISMS SATISFYING SOME MODULAR INEQUALITIES

SOME MODULAR INEQUALITIES

MIHAI CRISTEA

We study the dilatations of the balls generated by a homeomorphismf between two domains fromRⁿsatisfying some generalized modular inequalities. We obtain as a particular case some conditions of quasiconformality.

AMS 2010 Subject Classification: 30C65.

Key words: mappings satisfying generalized modular inequalities.

1. INTRODUCTION

If D⊂Rⁿ is a domain, we set A(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

γρds ≥ 1 for every γ ∈ Γ locally rectifiable}. We define for p ≥ 1, Γ ∈ A(D) and ω ∈L¹_loc(D) thepmodulus of weightω,Mω^p(Γ) = inf

ρ∈F(Γ)

R

Dω(x)·ρ^p(x)dxand for ω = 1 we obtain the classical p modulus Mp(Γ) = inf

ρ∈F(Γ)

R

Rⁿρ^p(x)dx. If D, D⁰are domains fromRⁿandf :D→D⁰is a homeomorphism so that there exists K ≥1 so that Mn(f(Γ)) ≤K·Mn(Γ) and Mn(Γ)≤K·Mn(f(Γ)) for every Γ∈A(D), we obtain the known class ofK-quasiconformal mappings. An important class of continuous, open discrete mappingsf :D⊂Rⁿ→Rⁿwhich generalizes quasiconformal mappings is the class of quasiregular mappings (see [22], [17], [18] the mains books concerning the theory of quasiconformal and quasiregular mappings). The important modular inequality of Poleckii also holds for quasiregular mappings and says that if f is K-quasiregular, then Mn(f(Γ)) ≤K·Mn(Γ) for every Γ∈ A(D), and is the key for proving most of the geometric properties of this class of mappings.

More general classes of continuous, open, discrete mappings (so called mappings of finite distortion) for which hold a generalized modular inequality of type “M_n(f(Γ))≤M_Kⁿ

I,n(f)(Γ) for every Γ∈A(D)” were studied in [2–6], [11–16], [19–21]. In some recent papers [7, 8] we studied continuous, open, dis- crete mappings f :D⊂Rⁿ→Rⁿ satisfying a generalized modular inequality of type M_q(f(Γ)) ≤ γ(Mω^p(Γ)) for every Γ ∈ A(D), with q > n−1, p > 1,

REV. ROUMAINE MATH. PURES APPL.,56(2011),4, 275–282

γ : [0,∞) → [0,∞) strictly increasing with lim

t→0γ(t) = 0 and some weight ω ∈L¹_loc(D). We gave Liouville, Montel, Picard type theorems, equicontinuity results and we gave estimates of the modulus of continuity. We extended in this way partially basic theorems from the theory of quasiregular mappings and from the class of continuous, open, discrete mappings mentioned before.

The basic tools we used before for proving this things was the modular in- equality M_q(f(Γ)) ≤ γ(Mω^p(Γ)) for every Γ ∈ A(D) together with the fact that lim

r→0M_ω^p(Γ_x,r,R) = 0 for every x ∈ D and every fixed R > 0 (see [7]).

We gave in [7, 8] examples of such mappings which are not quasiconformal showing in this way that our extensions are effective.

IfD, D⁰are domains fromRⁿ,f :D→D⁰is a homeomorphism,g=f⁻¹, x∈D,r >0 is so that B(x, r)⊂D, we set

L(x, f, r) = sup

|y−x|=r

|f(y)−f(x)|, l(x, f, r) = inf

|y−x|=r|f(y)−f(x)|, and we set

H(x, f) = lim sup

r→0

L(x, f, r)

l(x, f, r), H^∗(x, f) =H(f(x), g).

It is known that if there exists H ≥1 so thatH(x, f) ≤H for every x∈ D, or if H^∗(x, f)≤H for everyx∈D, then f is quasiconformal.

We shall show that if f : D → D⁰ is a homeomorphism between two domains from Rⁿ, p > 1, q > n−1, γ : [0,∞) → [0,∞) is strictly increas- ing with lim

t→0γ(t) = 0 so that M_q(f(Γ)) ≤ γ(Mω^p(Γ)) for every Γ ∈ A(D), x ∈ D and g = f⁻¹ and there exists K_x, r_x > 0, 0 ≤ α_x < p−1 so that R

B(x,δ)ω(z)dz≤Kx·δ^p ln ^e·r_δ^xαx

for every 0< δ≤rx, then we can calcu- late a bound for the local dilatation of the map g around the pointy =f(x), i.e. we can findρx>0, a function C: (0, δx)→(0,∞),

C(r) =C(r, p, q, γ, n, Kx, αx)

= exp exp

K_x·e^p·

∞

X

k=1

1

k^p−αx/γ⁻¹ C_n,q·r^{n−q 1}_p!!

e

for every 0< r≤δ_x so that ^L(y,g,r)_l(y,g,r) ≤C(r) for every 0< r≤ρ_x.

We say that a mapf :D⊂Rⁿ→ Rⁿ is ACL iff is continuous and for every cube Q⊂⊂ D with the sides parallel to coordinate axes and for every face S of Q it results that f|P_S⁻¹(y) ∩Q : P_S⁻¹(y)∩Q → Rⁿ is absolutely continuous for a.e.y∈S, whereP_S:Rⁿ→S is the projection onS. An ACL map has a.e. partial derivatives and ifp >1 we say thatf is ACL^p iff is ACL and the partial derivatives are locally in L^p. We say that f :D⊂Rⁿ → Rⁿ satisfies condition (N) ifµn(f(A)) = 0 for every set A⊂D with µn(A) = 0.

Here µn is the Lebesgue measure from Rⁿ. If A ∈ L(Rⁿ,Rⁿ), detA 6= 0, p > 1, we set K_0,p(A) = _|det^|A|p_A|, K_I,p(A) = ^|det_l(A)^A|p , where |A| = sup

|x|=1

|A(x)|,

l(A) = inf

|x|=1|A(x)| and |x| = _n

P

i=1

x²_i ¹₂

if x = (x1, . . . , xn) ∈ Rⁿ. If f : D → D⁰ is a homeomorphism between two domains from Rⁿ and p > 1, we define the map NI,p(f) : D → R+ by NI,p(f)(x) = KI,p(f⁰(x)) if f is differentiable in x and J_f(x)6= 0, N_I,p(f)(x) = 0 otherwise and we define the map N0,p(f) :D→ R+ by N0,p(f)(x) =K0,p(f⁰(x)) if f is differentiable in x and J_f(x) 6= 0, N_0,p(f)(x) = 0 otherwise. A quasiconformal map f :D→D⁰ is a.e. differentiable, is ACLⁿ and J_f(x) 6= 0 a.e. If D ⊂ Rⁿ is a domain and E, F ⊂ D, we set ∆(E, F, D) = {γ : [0,1]→ Rⁿ path|γ(0) ∈ E, γ(1)∈ F and γ((0,1)) ⊂ D} and if f : D → Rⁿ is continuous, we put L(x, f) = lim sup

h→0

|f(x+h)−f(x)|

|h| . If γ : [a, b] → Rⁿ is a rectifiable path, we set s_γ(t) = l(γ|[a, t]) for t ∈ [a, b] and we define a reparametrisation γ⁰ : [0, l(γ)] → Rⁿ of the path γ given by γ(t) = γ⁰(s_γ(t)) for t ∈ [a, b]. A domain A ⊂ Rⁿ is a ring if CA has exactly two components C₀, C₁ and we set A = R(C₀, C₁) and Γ_A = ∆(C0, C1,Rⁿ). Given p > 0 and r > 0, we put Φp(r) the set of all rings A = R(C₀, C₁) so that 0 ∈ C₀ and there exists a ∈ C₀ so that

|a| = 1 and so that ∞ ∈ C1 and there exists b ∈ C1 so that |b| = r and we denote H_n,p(r) = inf

A∈Φ_p(r)M(Γ_A). We see from Theorem 8 from [2] that if n−1 < p, then Hn,p(1) = Cn,p > 0. If x ∈ Rⁿ and 0 < r < R, we set Γ_x,r,R= ∆(B(x, r), S(x, R), B(x, R)\B(x, r)).

Using the idea from [15], page 94, Lemma 5.2, we extend Theorem 2 from [5].

Proposition 1. Let n ≥2, p >1, M > 0, 0 ≤ α < p−1, D ⊂Rⁿ a domain,x∈Dand0< r < Rso thatB(x, r)⊂D,ω :D→[0,∞]measurable and finite a.e. so that R

B(x,δ)ω(z)dz≤M·δ^p· ln ^Re_δ α

for every 0< δ < R and let C =M·e^p·

∞

P

k=1 1

k^p−α. Then Mω^p(Γ_x,r,R)≤C/ ln ln ^Re_r p

.

Proof. We can suppose that x= 0 and letρ:Rⁿ→[0,∞] be defined by ρ(z) = 1/

|z| ·ln

Re

|z|

·ln ln ^Re_r

ifz∈B(x, r)\B(x, r),ρ(z) = 0 otherwise.

Let a_k =Re^−k for k∈ N,B_k =B(0, a_k), k∈ N and letA_k =B_k\B_k+1 for k ∈N. We see thatρ ∈F(Γx,r,R) and we see that if z∈Ak, then _|z|^R ≤e^k+1

and 1/ln Re

|z|

≤ _k+1¹ for every k∈N. We find that M_ω^p(Γ_x,r,R)≤

Z

Rⁿ

ω(z)ρ^p(z)dz≤

1/

ln ln

Re r

p

· Z

B(x,R)\B(x,r)

ω(z)/

/

|z|^p·

ln Re

|z|

p dz≤

1/

ln ln

Re r

p

·

∞

X

k=0

Z

Ak

ω(z)/

/

|z|^p

ln Re

|z|

p dz≤

1/

ln ln

Re r

p

·

∞

X

k=0

Z

Bk

ω(z)· e^(k+1)p R^p(k+ 1)^pdz

≤

M·e^p/

ln ln Re

r

p

·

∞

X

k=0

e^kp

R^p(k+ 1)^p ·(Re^−k)^p·

ln Re

Re^−k α

=

=C/

ln ln

Re r

p

.

Theorem 1. Let n≥2, p >1, n−1 < q, γ : [0,∞) → [0,∞) strictly increasing with lim

t→0γ(t) = 0, D, D⁰ domains inRⁿ, ω:D→[0,∞]measurable and finite a.e., f :D→ D⁰ a homeomorphism so that M_q(f(Γ))≤γ(Mω^p(Γ)) for every Γ ∈ A(D) and let g = f⁻¹, x ∈ D and y = f(x). Suppose that there exists K_x > 0, r_x > 0, 0 ≤ α_x < p − 1 so that R

B(x,δ)ω(z)dz ≤ Kx ·δ^p · ln ^er_δ^xα

for every 0 < δ ≤ rx and let C(r) = exp(exp((Kx · e^p ·

∞

P

k=1 1

k^p−αx/γ⁻¹(C_n,q ·r^n−q))^1p))/e for r > 0. Then there exists ρ_x > 0 so that ^L(y,g,r)_l(y,g,r) ≤ C(r) for every 0 < r ≤ ρx and if q = n, then H(y, g) ≤ H = H(p, n, γ, Kx, αx). If D = D⁰ = Rⁿ and there exists Kx > 0 so that R

B(x,δ)ω(z)dz≤K_x·δ^p for every δ >0, then ^L(y,g,r)_l(y,g,r) ≤C(r) for every r >0.

Proof. Let ρx > 0 be so that L(y, g, r) ≤ rx for 0 < r ≤ ρx. Let 0< r≤ρxandLr=L(y, g, r),lr =l(y, g, r),Ar=R(B(x, lr), CB(x, Lr)) and let f(A_r) =R(B_r, C_r). We see that there existsa_r∈S(x, l_r),b_r ∈S(x, L_r) so that f(ar)∈S(y, r), f(br)∈S(y, r) andy, f(ar)∈Br,y, f(br)∈Cr. Letgr : Rⁿ→Rⁿ,gr(z) = ^z−y_r forz∈Rⁿ and hr=gr◦f. Then hr(Ar) =R(Dr, Er) and we can find points c_r ∈ D_r, d_r ∈ E_r so that c_r, d_r ∈ S(0,1), 0 ∈ D_r,

∞ ∈ E_r and let C_x = K_x·e^p·

∞

P

k=1 1

k^p−αx. Using Caraman’s result from [1], Theorem 8 and Proposition 1, we find that

0< C_n,q=H_n,q(1)≤M_q(Γ_hr(Ar)=M_q(Γ_gr(f(Ar))) = 1

r n−q

·M_q(Γ_f(Ar)) =

= 1

r n−q

·M_q(f(Γ_Ar))≤ 1

r n−q

·γ(M_ω^p(Γ_Ar)) = 1

r n−q

·γ(M_ω^p(Γ_x,lr,Lr))≤

≤ 1

r n−q

·γ

C_x/

ln ln

e·L_r lr

p . We therefore obtain that ^L_l^r

r ≤C(r) for every 0< r ≤ρ_x. IfD=D⁰ = RⁿandR

B(x,δ)ω(z)dz≤Kx·δ^p for everyδ >0, then, takingαx= 0, we obtain that ^L(y,g,r)_l(y,g,r) ≤C(r) = exp exp

Kx·e^p·

∞

P

k=1 1

k^p/γ⁻¹(Cn,q·r^n−q) ¹

p

!!

/e for every r >0.

Corollary 1. Let n ≥ 2, p > 1, D, D⁰ domains from Rⁿ, ω : D → [0,∞] measurable and finite a.e., γ : [0,∞) → [0,∞) strictly increasing with

t→0limγ(t) = 0, and f : D → D⁰ a homeomorphism so that M_n(f(Γ)) ≤ γ(Mω^p(Γ)) for every Γ ∈ A(D). Suppose that there exists K > 0, a > 0 so that for every x ∈ D there exists r_x > 0 and 0 ≤ α_x < p−1−a so that R

B(x,δ)ω(z)dz≤K·δ^p· ln ^e·r_δ^xα

for every 0< δ≤rx. Thenf is quasicon- formal and

H^∗(x, f)≤H=H(p, n, γ, K, a) =

= exp exp

K·e^p·

∞

X

k=1

1

k^1+a/γ⁻¹ C(n) ¹

p

!!

e

for every x∈D.

Proof. We take q = n and K = Kx in Theorem 1 and we see that

∞

P

k=1 1 k^p−αx ≤

∞

P

k=1 1

k^1+a and the theorem is proved.

Corollary 2. Let n ≥ 2, p > 1, n−1 < q, ω :Rⁿ → [0,∞] measu- rable and finite a.e., γ : [0,∞) →[0,∞) strictly increasing with lim

t→0γ(t) = 0, f : Rⁿ → Rⁿ a homeomorphism, let g = f⁻¹ and suppose that M_q(f(Γ))

≤ γ(M_ω^p(Γ)) for every Γ ∈ A(Rⁿ) and that there exists K > 0 so that R

B(x,δ)ω(z)dz≤K·δ^p for every x∈Rⁿ and every δ >0. Then L(y, g, r)

l(y, g, r) ≤C(r) = exp exp

K·e^p·

∞

X

k=1

1

k^p/γ⁻¹(C_n,q·r^n−q) _p¹!!

e

for every y∈Rⁿ and every r >0.

Remark1. We see that the homeomorphismg:Rⁿ→Rⁿfrom Corollary 2 satisfies a kind of quasisimmetry relation, i.e., if r > 0 there exists C(r) >0

so that ^L(y,g,r)_l(y,g,r) ≤ C(r) for every y ∈ Rⁿ and every δ ≥ r, hence for enough great balls B(y, r) their dilatation ^L(y,g,δ)_l(y,g,δ) is uniformly bounded on Rⁿ.

Theorem 2. Let n ≥ 2, n−1 < p ≤ n, D, D⁰ domains from Rⁿ, f :D→D⁰ a homeomorphism so thatg=f⁻¹is ACL^p, a.e. differentiable and Jg(y)6= 0a.e., letx∈D,y=f(x)and suppose that there exists Kx>0, rx>

0,0≤α_x < p−1so thatR

B(x,δ)N_I,p(f)(z)dz≤K_x·δ^p· ln ^e·r_δ^xαx

for every 0< δ≤r_xand letC(r) = exp

exp

K_x·e^p·

∞

P

k=1 1

k^p−αx/(C_n,pr^n−p)¹_p /e for r >0. Then there existsρx>0so that ^L(y,g,r)_l(y,g,r) ≤C(r)for every0< r≤ρx

and if p = n, then H(y, g) ≤ H(p, n, γ, Kx, αx). If D = D⁰ = Rⁿ and there exists K_x > 0 so that R

B(x,δ)N_I,p(f)(z)dz ≤ K_x·δ^p for every δ > 0, then

L(y,g,r)

l(y,g,r) ≤C(r) for everyr >0.

Proof. Let Γ ∈A(D), let ρ ∈ F(Γ) and letρ⁰ :Rⁿ → [0,∞] be defined by ρ⁰(z) =ρ(g(z))·L(z, g) if z∈ D⁰, ρ⁰(z) = 0 otherwise. Let A ={z ∈D | f is differentiable inz and J_f(z) 6= 0} and B = {z ∈ D⁰ | g is differentiable in z and J_g(z) 6= 0}. Then CB = 0 and A = g(B), hence A 6= φ and let Γ⁰ = {β ∈ f(Γ) |β is locally rectifiable and g◦β⁰ is absolutely continuous}.

We see from Theorem 5.3, page 12 from [22] thatρ⁰∈F(Γ⁰). Using Fuglede’s theorem (see Theorem 28.2, page 95, from [22]) and the change of variable formulae (3) from [10] we find that

Mp(f(Γ)) =Mp(Γ⁰)≤ Z

Rⁿ

ρ^0p(z)dz= Z

D⁰

ρ^p(g(z))·L(z, g)^pdz=

= Z

D⁰

ρ^p(g(z))· |g⁰(z)|^pdz= Z

B

ρ^p(g(z))·N_0,p(g)(z)· |J_g(z)|dz=

= Z

B

ρ^p(g(z))·NI,p(f)(g(z))· |J_g(z)|dz≤ Z

A

ρ^p(u)·NI,p(f)(u)du≤

≤ Z

Rⁿ

ρ^p(u)·N_I,p(f)(u)du.

It results thatMp(f(Γ))≤M_N^p

I,p(f)(Γ) for every Γ∈A(D) and we apply now Theorem 1 with ω=N_I,p^p (f) andγ(t) =tfor every y≥0.

Corollary 3. Let n ≥ 2, D, D⁰ domains from Rⁿ, f : D → D⁰ a homeomorphism so that g =f⁻¹ is ACLⁿ, a.e. differentiable and Jg(y) 6= 0 a.e. and suppose that there exists K > 0, a > 0 so that for every x ∈ D there exists rx > 0 and 0 ≤ αx < n−1−a, so that R

B(x,δ)NI,n(f)(z)dz ≤

Kx ·δⁿ· ln ^er_δ^xαx

for every 0 < δ ≤ rx. Then f is quasiconformal and H^∗(x, f)≤exp

exp

K·eⁿ·

∞

P

k=1 1

k^1+a/C(n) ¹_n

/e for everyx∈D.

Corollary 4. Letn≥2,n−1< p≤n,f :Rⁿ→Rⁿa homeomorphism so that g = f⁻¹ is ACL^p, a.e. differentiable and Jg(y) 6= 0 a.e. and so that there exists K >0 so thatR

B(x,δ)N_I,p(f)(z)dz≤K·δ^p for everyx∈Rⁿ and every δ > 0 and let C(r) = exp

exp

K·e^p·

∞

P

k=1 1

k^p/(C_n,pr^n−p)¹_p /e.

Then ^L(y,g,r)_l(y,g,r) ≤C(r) for everyy∈Rⁿ and every r >0.

Corollary 5. Let n ≥ 2, D, D⁰ domains from Rⁿ, f : D → D⁰ a homeomorphism satisfying condition(N)and having an ACLⁿ inverse so that there exists Q > 0 so that lim sup

r→0

R−

B(x,r)

N_I,n(f)(z)dz ≤ Q for every x ∈ D.

Then f isK(Q, n)-quasiconformal.

Proof. Let g=f⁻¹. We see from Lemma 6.7, page 190 from [17] that g is a.e. differentiable and satisfies condition (N). Let D_g ={y ∈D⁰ |g is not differentiable in y} andZ_g ={y∈D⁰ |g is differentiable iny and J_g(y) = 0}.

Using Sard’s lemma (see [2]) and the fact that g satisfies condition (N), we see that µ_n(g(D_g∪Z_g)) = 0 and since f also satisfies condition (N), we see that µn(Dg∪Zg) = 0. We proved that g is a.e. differentiable and Jg(y) 6= 0 a.e. We apply now Corollary 3.

A related result with Corollary 1 may be found in [9].

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Received 25 February 2012 University of Bucharest

Faculty of Mathematics and Computer Sciences Str. Academiei 14

010014 Bucharest, Romania [email protected]