The radius of injectivity of local ring $Q$-homeomorphisms

0.15 MB·English

by  Evgeny Sevost'yanov

THE RADIUS OF INJECTIVITY OF LOCAL
RING –HOMEOMORPHISMS
Q
E. SEVOST’YANOV
2
1 September 18, 2012
0
2

p
e Abstract
S
The paper is devoted to the study of mappings with non–bounded characteristics
4
1 of quasiconformality. The analog of the theorem about radius injectivity of locally

quasiconformal mappings was proved for some class of mappings. There are found
]
V
sharp conditions under which the so called local Q–homeomorphisms are injective in
C
some neighborhood of a fixed point.
.
h
t
a
m 2010 Mathematics Subject Classification: Primary 30C65; Secondary 30C62
[

1 1 Introduction
v
1
4
3 Here are some definitions. Everywhere below, D is a domain in Rn, n ≥ 2, m be a measure
3 of Lebesgue in Rn, and dist(A,B) is the Euclidean distance between the sets A and B in Rn.
.
9 The notation f : D → Rn assumes that f is continuous on its domain. In what follows (x,y)
0
2 denotes the standard scalar multiplication of the vectors x,y ∈ Rn, diamA is Euclidean
1
diameter of the set A ⊂ Rn,
:
v
i
X B(x ,r) = {x ∈ Rn : |x−x | < r} , Bn := B(0,1),
0 0
r
a
S(x ,r) = {x ∈ Rn : |x−x | = r}, Sn 1 := S(0,1),
0 0 −
ω denotes the square of the unit sphere Sn 1 in Rn, Ω is a volume of the unit ball Bn in
n 1 − n
Rn−. A mapping f : D → Rn is said to a local homeomorphism if for every x ∈ D there is a
0
number δ > 0 such that a mapping f| to be a homeomorphism.
B(x0,δ)
Recall that a mapping f : D → Rn is said to be a mapping with bounded distortion, if the
following conditions hold:
1) f ∈ W1,n,
loc
2) a Jacobian J(x,f) := detf (x) of the mapping f at the point x ∈ D preserves the sign
′
almost everywhere in D,
3) kf (x)kn ≤ K ·|J(x,f)| at a.e. x ∈ D and some constant K < ∞, where
′
kf (x)k := sup |f (x)h|,
′ ′
h Rn:h=1
∈ | |
1
THE RADIUS OF INJECTIVITY OF LOCAL RING Q–HOMEOMORPHISMS 2
see., e.g., §3 Ch. I in [5], or definition 2.1 of the section 2 Ch. I in [6]. In this case we also
say that f is K–quasiregular, where K is from condition 3) meaning above.
The following result was proved in the work [3] by O. Martio, S. Rickman and J. Va¨isa¨la¨,
see Theorem 2.3 in [3] or Theorem 3.4 of Ch. III in [6], see also the paper [1].
Statement 1. If n ≥ 3 and f : Bn → Rn is a K–quasiregular local homeomorphism,
then f is injective in a ball B(0,ψ(n,K)), where ψ is a positive number depending only on
n and K.
A goal of the present paper is a proof of the analog of the Statement 1 for more general
classes of mappings of ring Q–homeomorphisms. To introduce this class of the mappings,
we give some definitions.
A curve γ in Rn is a continuous mapping γ : ∆ → Rn where ∆ is an open, closed or half–
open interval inR.Given a familyΓ ofpaths γ inRn, n ≥ 2, a Borelfunction ρ : Rn → [0,∞]
is called admissible for Γ, abbr. ρ ∈ admΓ, if
ρ(x)|dx| ≥ 1
Z
γ
for each γ ∈ Γ. The modulus of Γ is the quantity
M(Γ) = inf ρn(x)dm(x).
ρ admΓ
∈ RZn
Given a domain D and two sets E and F in Rn, n ≥ 2, Γ(E,F,D) denotes the family of
all paths γ : [a,b] → Rn which join E and F in D, i.e., γ(a) ∈ E, γ(b) ∈ F and γ(t) ∈ D for
a < t < b.
Let D be a domain in Rn, Q : D → [0,∞] be a (Lebesgue) measurable function. Set
A(x ,r ,r ) = {x ∈ Rn : r < |x−x | < r }.
0 1 2 1 0 2
We say that a mapping Rn is a ring Q–mapping at a point x ∈ D if
0
M (f(Γ(S(x ,r ),S(x ,r ),A(x ,r ,r )))) ≤ Q(x)·ηn(|x−x |) dm(x) (1.1)
0 1 0 2 0 1 2 0
Z
A(x0,r1,r2)
for every ring A(x ,r ,r ), 0 < r < r < r = dist(x ,∂D), and for every Lebesgue
0 1 2 1 2 0 0
measurable function η : (r ,r ) → [0,∞] such that
1 2
r2
η(r)dr ≥ 1.
Z
r1
Ifthecondition(1.1)holdsateverypointx ∈ D,thenwealsosaythatf isaringQ–mapping
0
in the domain D, see section 7 in [4].
In what follows q (r) denotes the integral average of Q(x) under the sphere |x−x | = r,
x0 0
1
q (r) := Q(x)dS, (1.2)
x0 ω rn 1
n−1 − x xZ0=r
| − |
THE RADIUS OF INJECTIVITY OF LOCAL RING Q–HOMEOMORPHISMS 3
where dS is element of the square of the surface S.
One of the main results of the paper is following.
Theorem 1.1. Let n ≥ 3 and f : Bn → Rn is a local ring Q–homeomorphism at the
point x = 0, such that Q ∈ L1 (Bn) and
0 loc
1
dt
= ∞. (1.3)
1/(n 1)
Z tq0 − (t)
0
Then f is injective in a ball B(0,δ(n,Q)), where δ is a positive number depending only on
n and function Q. From other hand, the condition (1.3) is precise, in fact, for every δ > 0
and every Q ∈ L1 (Bn) with Q(x) ≥ 1 a.e. and
loc
1
dt
< ∞ (1.4)
1/(n 1)
Z tq0 − (t)
0
there exists a mapping f = f : Bn → Rn which is local ring Q–homeomorphism at the
Q
point x = 0 and which is not injective in B(0,δ).
0
2 The main Lemma
A set Q ⊂ Rn is said to be relatively locally connected if every point in Q has arbitrary small
neighborhoods U such that U ∩Q is connected.
We also need following statements, see Lemma III.3.1–Lemma III.3.3 in [6].
Proposition 2.1. Let f : G → Rn be a local homeomorphism, let Q be a simply
connected and locally pathwise connected set in Rn, and let P be a component of f 1(Q)
−
such that P ⊂ G. Then f maps P homeomorphically onto Q. If, in addition, Q is relatively
locally connected, f maps P homeomorphically onto Q.
Proposition 2.2. Let f : G → Rn be a local homeomorphism and let F be a compact
set in G such that f is injective. Then f is injective in a neighborhood of F.
F
Proposition 2.3. Let f : G → Rn be a local homeomorphism, let A,B ⊂ G, and let
f be homeomorphic in A and B. If A ∩ B 6= ∅ and f(A) ∩ f(B) is connected, then f is
homeomorphic in A∪B.
Finally, we need the following statement of P. Koskela, J, Onninen and K. Rajala, see
Lemma 3.1 in [1].
Proposition 2.4. Let n ≥ 1 and r > 0. Let a 6= b, a,b ∈ S(0,r). Then there exists a
point p = p(a,b) ∈ B(0,r) such that for every t ∈ r, √3r either
2 2
(cid:16) (cid:17)
0,b ∈ B(p,t) and a 6∈ B(p,t)
or
a,b ∈ B(p,t) and 0 6∈ B(p,t).
THE RADIUS OF INJECTIVITY OF LOCAL RING Q–HOMEOMORPHISMS 4
The following Lemma plays the main role in the following.
Lemma 2.1. Let n ≥ 3, Q : Bn → [0,∞] and f : Bn → Rn is a local ring Q–homeo-
morphism at the point x = 0. Suppose that there exist a function ψ : (0,1) → [0,∞] and a
0
constant C = C(n,Q,ψ) such that
r2
0 < I(r ,r ) := ψ(t)dt < ∞ ∀r ,r ∈ (0,1) (2.1)
1 2 1 2
Z
r1
and for some α > 0
Q(x)ψ(|x|)dm(x) ≤ C ·In α(r ,r ). (2.2)
− 1 2
Z
r1<x<r2
| |
Let
1
I(0,1) := ψ(t)dt = ∞, (2.3)
Z
0
then f is injective in a ball B(0,δ(n,Q,ψ)), where δ is a positive number depending only
on n, functions Q and ψ.
Proof. We may assume f(0) = 0. Let r = sup{r ∈ R : r > 0,U(0,r) ⊂ Bn}, where
0
U(0,r) is the 0–component of f 1(B(0,r)). Clearly r > 0. Fix r < r and set U = U(0,r),
− 0 0
l = l (0,f,r) = inf{|z| : z ∈ ∂U},
∗ ∗
L = L (0,f,r) = sup{|z| : z ∈ ∂U}.
∗ ∗
By Proposition 2.1, f maps U homeomorphically onto B(0,r). Thus f is injective in B(0,l )
∗
and it suffices to find a lower bound for l .
∗
Note that L → 1 as r → r . In fact, let L 6→ 1 as r → r . Note that L is increasing
∗ 0 ∗ 0 ∗
function by r and, consequently, the limit of L as r → r exists. Now L → ε as r → r ,
∗ 0 ∗ 0 0
ε ∈ (0,1). Then U(0,r) ⊂ B(0,ε ) for all r ∈ (0,r ). Note that B(0,r ) ⊂ f(B(0,ε )).
0 0 0 0 0
Now B(0,r ) ⊂ f(B(0,ε )) and hence there exists r > r such that U(0,r ) ⊂ Bn that
0 0 ′ 0 ′
contradicts to the definition of r . The above contradiction shows that L → 1 as r → r
0 ∗ 0
that we need to prove.
Pick x and y ∈ ∂U such that |x| = L and |y| = l . Note that, by the definition of U,
∗ ∗
f(x), f(y) ∈ S(0,r). By Proposition 2.4 there exists a point p ∈ B(0,r) such that, for every
t ∈ r, √3r , f(x) ∈ B(p,t) and either 0 ∈ B(p,t) and f(y) 6∈ B(p,t), or 0 6∈ B(p,t) and
2 2
f(y)(cid:16)∈ B(p(cid:17),t). Fix such a t. Note that 0,f(y) and f(x) ∈ f(B(0,l )) and, consequently,
∗
f(B(0,l ))∩B(p,t) 6= ∅ 6= f(B(0,l ))\B(p,t). Since f(B(0,l )) is connected, this implies
∗ ∗ ∗
that there exists a point z ∈ S(p,t)∩f(B(0,l )), see Theorem 1.I.46.5 of [2].
t ∗
Let z be the unique point in f 1(z )∩B(0,l ). Let C (ϕ) ⊂ S(p,t) be the spherical cap
t∗ − t ∗ t
with center z and opening angle ϕ,
t
C (ϕ) = {y ∈ Rn : |y −p| = t,(z −p,y −p) > t2ϕ}.
t t
THE RADIUS OF INJECTIVITY OF LOCAL RING Q–HOMEOMORPHISMS 5
Let ϕ be the supremum of all ϕ for which the z –component of f 1(C (ϕ)) gets mapped
t t∗ − t
homeomorphically onto C (ϕ). Let C = C (ϕ ) and let C be the z –component of f 1(C ).
t t t t t∗ t∗ − t
We claim that C meets S(0,L ).
t∗ ∗
Suppose this is not true. Since C is connected and C ∩B(0,L ) 6= ∅, this implies that
t∗ t∗ ∗
C ⊂ B(0,L ), see Theorem 1.I.46.5 in [2]. Remark that, in this case, C is a compact
t∗ ∗ t∗
subset of U and by Proposition 2.1 f maps C homeomorphically onto C . By Proposition
t∗ t
2.2 f is injective in a neighborhood of C . Thus ϕ = π, C = S(p,t) and C is a topological
t∗ t t t∗
(n − 1)–sphere. Note that bounded component D of Rn \ C contained in B(0,L ). Now
t∗ ∗
f(D) is a compact subset of f(Bn) and, since the mapping f is open, ∂f(D) ⊂ f(∂D).
Remark that f(D) ⊂ B(p,t). In fact, let f(D) 6⊂ B(p,t), then there exists y ∈ f(D) \
B(p,t). Now we have f(Bn)\B(p,t) ∩f(D) 6= ∅ and, since f(D) is compact subdomain
of f(Bn), f(Bn)\B((cid:16)p,t) \f(D) 6= ∅(cid:17). Since f(Bn)\B(p,t) is connected, this implies that
there exist(cid:16)s z ∈ ∂f(D)∩ (cid:17)f(Bn)\B(p,t) , see Theorem 1.I.46.5 of [2], that contradicts to
the inclusion ∂f(D) ⊂ B(cid:16)(p,t). Now f(D)(cid:17)⊂ B(p,t). Remark that B(p,t) ⊂ f(D). Indeed,
let there exists a ∈ B(p,t) \ f(D). Since B(p,t) is connected and B(p,t) ∩ f(D) 6= ∅ this
implies that ∂f(D)∩B(p,t) 6= ∅, see Theorem 1.I.46.5 of [2]. The last relation contradicts
to the inclusion ∂f(D) ⊂ B(p,t). Thus f(D) = B(p,t). By the definition, D is a component
of f 1(B(p,t)). By Proposition 2.1 f maps D onto B(p,t) homeomorphically.
−
Since z ∈ C ∩ U, D ∩ U 6= ∅. Since f maps U homeomorphically onto B(0,r), f is
t∗ t∗
injective in U ∪D. This is impossible, because in view of the equality f(D) = B(p,t) and
that f(x) ∈ B(p,t) there exists a point x 6= x, x ∈ D, such that f(x ) = f(x). Thus C
1 1 1 t∗
meets S(0,L ).
∗
Let k ∈ C ∩S(0,L ) and k = f(k ). Let Γ be the family of all curves connecting k
t∗ t∗ ∗ t t∗ t′ t
and z in C . Moreover, let Γ be the union of the curve families Γ , t ∈ r, √3r . Denote
t t ′ t′ 2 2
by ft the restriction of f to Ct∗. Then ft maps Ct∗ homeomorphically onto(cid:16)Ct. Fur(cid:17)thermore,
denote
Γ = f 1 ◦γ : γ ∈ Γ .
t− t′
t∈(cid:16)[2r,√23r(cid:17)(cid:8) (cid:9)
Since for every t ∈ r, √3r , z ∈ B(0,l ) and k ∈ S(0,L ), be the definition of ring
2 2 t∗ ∗ t ∗
Q–mapping we have (cid:16) (cid:17)
M(f(Γ(S(0,l ),S(0,L ),A(0,l ,L )))) ≤ Q(x)·ηn(|x|)dm(x) (2.4)
∗ ∗ ∗ ∗
Z
A(0,l ,L )
∗ ∗
for every function η : (l ,L ) → [0,∞] with
∗ ∗
L∗
η(r)dr ≥ 1.
Z
l
∗
Setting η(t) = ψ(t)/I(l ,L ), where ψ is the function from the condition of Lemma, we
∗ ∗
observe that η satisfies the above condition. Now from (2.2) and (2.4) we obtain that
M(Γ ) = M(f(Γ(S(0,l ),S(0,L ),A(0,l ,L )))) ≤
′ ∗ ∗ ∗ ∗
THE RADIUS OF INJECTIVITY OF LOCAL RING Q–HOMEOMORPHISMS 6
≤ Q(x)·ηn(|x|)dm(x) ≤ C/Iα(l ,L ). (2.5)
∗ ∗
Z
A(0,l ,L )
∗ ∗
On other hand, by [8], Theorem 10.2,
C
ρn(x)dS ≥ n (2.6)
t
Z
S(p,t)
for every ρ for which ≥ 1 for every γ ∈ Γ . The integration of (2.6) over t yields
t′
γ
R
M(Γ ) ≥ C . (2.7)
′ n
We obtain from (2.5) and (2.7) that
C ≤ C/Iα(l ,L ) ≤ C/Iα(l (0,f,r ),L (0,f,r)) (2.8)
n ∗ ∗ ∗ 0 ∗
because I(ε ,ε ) > I(ε ,ε ) as ε > ε . Letting into the limit as r → r in (2.8), we have
1 2 3 2 3 1 0
C ≤ C/Iα(l (0,f,r ),1). (2.9)
n ∗ 0
First of all, from the (2.9) follows that I(ε,1) < ∞ for every ε ∈ (0,1). Follow, let
l (0,f,r ) → 0, then it follows from (2.3) that the right hand of (2.9) tends to zero, that
∗ 0
contradicts to (2.9). Thus, l (0,f,r ) ≥ δ for all such f. The proof is complete. ✷
∗ 0
3 Proof of the main result
The following statement would be very useful, see Theorem 3.15 in [7].
Proposition 3.1. Let D be a domain in Rn, n ≥ 2, and Q : D → [0,∞] a locally
integrable measurable function. Ahomeomorphism f : D → Rn is a ring Q–homeomorphism
at a point x if and only if for every 0 < r < r < r = dist(x ,∂D),
0 1 2 0 0
ω
n 1
M (Γ(f(S1),f(S2),f(D))) ≤ In−1
−
where ω is the area of the unit sphere in Rn, q (r) is the average of Q(x) over the sphere
n 1 x0
|x−x | =−r, S = {x ∈ Rn : |x−x | = r }, j = 1,2, and
0 j 0 j
r2
dr
I = I(r ,r ) = .
1 2 1
rZ1 rqxn0−1(r)
Proof of Theorem 1.1. Given 0 < r < r < r = 1 consider the function
1 2 0
1
1/[tqn 1(t)] , t ∈ (r ,r ) ,
ψ(t) = 0− 1 2 (3.1)
( 0 , t ∈/ (r ,r ) ,
1 2
THE RADIUS OF INJECTIVITY OF LOCAL RING Q–HOMEOMORPHISMS 7
r2
Note that ψ satisfies all the conditions of Lemma 2.1, in particular, dt < ∞ by
r1 tq01/(n−1)(t)
Theorem 1 in [9], and by Fubini theorem, Q(x) ·ψn(|x|) dm(x)R = ω ·I(r ,r ).
n 1 1 2
−
r1<x<r2
Now the first part of the Theorem follows from|R| Lemma 2.1.
To prove second part of the Theorem, we take δ > 0 and some function Q ∈ L1 (Bn)
loc
satisfying (1.4). Set
x
f(x) = ρ(|x|),
|x|
where
1
dt 1
ρ(r) = exp − , q (r) := Q(x)dS,
 Zr tq01/(n−1)(t) 0 ωn−1rn−1 xZ=r
e | | e
 e  Q(x), |x| > δ ,
Q(x) =
1 , |x| ≤ δ.
(cid:26)
Note that a mapping f is a rineg Q–homeomorphism at x = 0. In fact, we have f(S(0,r)) =
0
1
S(0,R), where R := exp − dt . Now
(cid:26) r tq01e/(n−1)(t)(cid:27)
R e
f(Γ(S(0,r ),S(0,r ),A(0,r ,r ))) = Γ(S(0,R ),S(0,R ),A(0,R ,R )),
1 2 1 2 1 2 1 2
1
where R := exp − dt , i = 1,2. Now by 7.5 in [8],
i ( ri tq01/(n−1)(t))
R e
ω
n 1
M(f(Γ(S(0,r1),S(0,r2),A(0,r1,r2)))) = − n 1 .
r2 −
dt
r1 tq01/(n−1)(t)!
R e
Now f is a ring Q–homeomorphism at the point x = 0 by Proposition 3.1. Note that under
0
δ → 0 the image f(B(0,δ)) includes the ball B(0,σ), where σ does not depend on δ. Now we
map the ball B(0e,σ) by some map g, which is K–quasiregular and local homeomorphism for
some K ≥ 1, but not injective in B(0,σ); for instance, let g is a winding map, whose axes of
rotation does not contain a ball f(B(0,1)), see 5.1 in [5]. Remark that K does not depend
on δ. Now we construct a local ring K ·Q(x)–homeomorphism f at zero, f = g ◦f, which
2 2
is not injective in B(0,δ). Since Q is arbitrary function with Q ≥ 1 satisfying (1.4), we can
replace Q on the Q/K in the start of the second part of the proof. So, we obtain a local ring
Q(x)–homeomorphism with the properties meaning above. The proof is complete. ✷
References
[1] P.Koskela, J.OnninenandK.Rajala, Mappings of finite distortion: Injectivity radius of
a local homeomorphism, pp. 169–174 in ”Future Trends in Geometric Function Theory
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University of Jyva¨skyla¨, 2003.
THE RADIUS OF INJECTIVITY OF LOCAL RING Q–HOMEOMORPHISMS 8
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Evgenii A. Sevost’yanov
Institute of Applied Mathematics and Mechanics,
National Academy of Sciences of Ukraine,
74 Roze Luxemburg str., 83114 Donetsk, UKRAINE
Phone: +38 – (062) – 3110145,
Email: [email protected]