On distortion of the moduli of rings under locally quasiconformal mappings in r^n Текст научной статьи по специальности «Математика»

32

Probl. Anal. Issues Anal. Vol. 4(22), No. 2, 2015, pp. 32-44

DOI: 10.15393/j3.art.2015.2952

UDC 517.54

S. Yu. Graf

ON DISTORTION OF THE MODULI OF RINGS UNDER LOCALLY QUASICONFORMAL MAPPINGS IN

Abstract. Some of the earlier results of the author concerning distortion of the moduli of ring domains under planar locally qua-siconformal mappings are generalized on the case of locally quasi-conformal mappings in , n > 2. The main result of the article represents the sharp double-sided estimation of modulus M(D) of the image D of the concentric spherical ring K(r, R) = {x G £ : r < |x| < R} under locally quasiconformal homeomor-phism f:

JR Pf1/(1-n)(t)^ < Mod(D) < JR Py(n-1)(t)

Here the function Pf is the majorant of dilatation of mapping f and Pf is well defined as

Pf (t) = lim essup{pf (x) : t — s < |x| < t + s}.

As the consequence of the main inequalities the sharp estimations of the derivative f '(0) of the normalized locally quasiconformal automorphisms f of the unit ball in terms of majorant of the dilatation of function f are proved. The sharpness of the results is demonstrated by examples of non-trivial locally quasiconformal mappings with unbounded dilatation that provide equalities in the estimations. The main theorems were obtained by means of method of moduli of families of curves and hypersurfaces in .

Key words: locally quasiconformal mapping, modulus of ring domain

2010 Mathematical Subject Classification: 3062, 3065, 3075, 3080

©Petrozavodsk State University, 2015 IMlIiHI

1. Let r be a family of at least locally rectifiable curves in Rn, n > 2. We remind [1, 2, 3], that modulus of the family of curves r is defined as

M (r) = inf / pn (x)dvn,

P JRn

where dvn is an n-dimensional volume element and infimum is taken over all admissible measurable non-negative a. e. functions p such that fY p(x)ds > 1 for any curve y G r, ds is a differential of arclength on 7.

The domain D C Rn is called ring domain, if its complement consists of exactly two disjointed components of connectivity. Within this article we need not more detailed specification of types of this components because we'll deal with locally quasiconformal images of spherical rings.

Let us consider the family r of all possible curves 7 C D, joining boundary components of ring domain D. The modulus of ring domain D is a number

Mod(D) = (MM V/(1-"), VWn-1 )

where un-1 is an area of the hypersphere of radius 1 in the space Rn. Particularly, the modulus of the concentric spherical ring K(r, R) = {x G G Rn : r < |x| < R} is equal to

R

Mod(K(r,R))=ln r. (1)

r

This relation and a couple of following properties are available, for example, in [2], chapter 8, or in [3], chapter 2. As in the well-known particular case n = 2 (i.e. in the case D C C) moduli of the ring domains possess the property of monotonicity:

Mod(Di) < Mod(D2), if Di C D2, (2)

and also satisfy the inequality

m

Mod (D) > ^ Mod(Dk), (3)

k=1

if ring domains Dk are pairwise disjoint and domain D = int{Um=1Dk} is also a ring domain. Here int A means the interior of a set A, and A is the closure of A.

The term of modulus of family of surfaces (see for example [3], chapter 2) is closely associated with notions of modulus of family of curves and ring domains. Let £k be the family of k-dimensional locally Lipschitz hypersurfaces in Rn. The modulus of £k is the value

M(£k) = inf / pn(x)dvn, P JRn

where the infimum is taken over all admissible measurable non-negative functions p such that Ja pk (x)dsk > 1 for any surface a E £k, dsk is an element of k-dimensional area on the surface a.

Let us consider a ring domain D C Rn and the family of all possible curves r C D, joining boundary components of D as in the definition of modulus of ring domain D. Then the modulus M(r) can be expressed via the modulus of the family £ of all possible (n — 1)-dimensional hypersurfaces, separating the boundary components of domain D:

M (r) = ( mW ■ (4)

This result follows from investigation of F. Gehring and W. Ziemer and can be found in [3]. The families r and £ in this case are called conjugate.

2. A homeomorphism f is called quasiconformal in a domain D C Rn, if there exists a constant Q > 1 such that for any family of curves r C D

QM(r) < M(f (r)) < QM(r). (5)

There is much literature devoted to the properties of spatial quasiconformal mappings (see for example monographs [2, 3, 1]). The quantity Q, as in the planar case, is called a coefficient of quasiconformality of f. Hence the quasiinvariance of the moduli of the families of curves is placed in definition (5) of quasiconformal mappings. The mapping f is conformal if and only if Q = 1. It should be noted that unlike the planar case the class of spatial conformal mappings is very narrow whereas the set of quasiconformal mappings is still wide and useful for many applications also in the case n > 2.

Every quasiconformal homeomorphism f is differentiable almost everywhere in the domain D and consequently the dilatation of the mapping f (see for example [2]) is defined a.e. as

( x ( Ln (x) | Jf (x)|

pf (x) = max < , -' . ,, . .

f ( ) I |Jf (x)|, in (x)

where Jf (x) is the Jacobian of f,

lf(x) = min |f/(x) ■ h1, Lf(x) = max |f/(x) ■ h|. |h|=1 |h|=1

Here f /(x) denotes the Jacobi matrix, i. e., the derivative of function f at point x. The coefficient of quasiconformality is connected with dilatation by means of relation

Q = essup pf (x).

xeD

A homeomorphism f is called locally quasiconformal in the domain D, if f is quasiconformal on every compact set K C D. It is obvious that such mappings can distort the moduli of families of curves indefinitely. Nevertheless as in the planar case [4, 5, 6], it is possible to obtain some useful estimations of distortion of moduli in terms of dilatation majorant of locally quasiconformal mappings.

Consider a locally quasiconformal mapping f of the ring K(r, R) and define for t e (r, R) the majorant of dilatation of the mapping f in the following way:

Pf (t) = li^essup{pf (x) : t — e < |x| < t + e}. (6)

It is obvious that function Pf (t) is well defined for all t e (r, R) and Pf (t) > 1. Moreover, the following theorem is true:

Theorem 1. For any locally quasiconformal mapping f of the ring K (r, R), 0 < r < R < to, the function Pf (t) is measurable on (r, R).

Proof. Let f satisfy the condition of the theorem. Consider the partition of the segment [r, R] by the points tm,i = r + (R — r)//2m, m e N, l = = 0,..., 2m, and define the sequence of functions Pf,m(t) on the interval (r, R) such that

P (t) = J essup {pf (x) : tm,l-1 < |x| < tm,l} for t e (tm,l-1,tm,l ],

Pf,m(t) = ^ I = 1 2m

, , (7)

It is clear that Pf,m (t) > 1 and Pf,m(t) is non-increasing in m for any fixed t e (r, R). Hence there exist the limits

Qf (t) = lim Pf,m (t), t e (r,R),

m

at that function Qf (t) is measurable as the limit of the step-functions (see, for example [7], chapter V, §4).

Let us show that

Qf (t) = Pf (t) a.e. on (r,R).

Let t = tm,i for any pair m, l. Then there exists the sequence of points tmk,ik, such that t E (tmk,ik-1, tmk,ik) for all k E N and tmk,ik ^ t as k ^ ro. Consider £k = min{t — tmk,ik — 1, tmk,ik — t} > 0. Then

essup{pf (x) : t — £k < |x| < t + £k} < Pf,mk(t).

Now passing to the limit as k ^ ro in the last inequality, we conclude that

Pf (t) < Qf (t).

On the other hand, if we choose £k = max{t — tmk ,ik—1 ,tmk ,ik — —t} + 1/k, then we obtain, in the similar way, the opposite inequality Pf (t) > Qf (t). Therefore,

Pf (t) = Qf (t)

for all t = tm,i, i.e. almost everywhere on (r, R). So the function Pf is measurable as coincident a.e. with the measurable function Qf. □

The next theorem allows us to estimate the distortion of the moduli of rings under locally quasiconformal mappings in terms of the dilatation majorant Pf of the function f. This theorem of course goes back to the famous Belinsky lemma for planar quasiconformal mappings [8]. In the case of planar locally quasiconformal mappings corresponding generalization of Belinsky lemma was proved by the author in 2009 [4]. Also it should be remarked that the next theorem may be deduced from results recently obtained by M. Brakalova, I. Markina and A. Vasilev [9]. But independent proof proposed in this article differs from the cited above and is shorter and direct. Related results were also considered in [10, 11] for Heisenberg groups and for some special cases for n = 2.

The sharpness of the estimation in forthcoming theorem 2 is regarded in the sense that there are locally quasiconformal mappings of rings with unbounded and constant on a.e. hyperspheres dilatation for which equalities hold in left or right hand sides of (8).

Theorem 2. Let f be a locally quasiconformal mapping of the ring K(r, R), 0 < r < R < to, onto the ring domain D C Rn and the function P/ (t) is defined by equality (6). Then

Pf1/(1-n)(t)- < Mod(D) < iR P//(n-1)(t)- (8)

J r t J r t

Both inequalities are sharp.

Proof. Let the conditions of the theorem be fulfilled. In the view of the measurability of the function P/ (t) and boundedness of the function P//(1-n) (t)/t on the interval (r, R) for r > 0, R < to, both integrals in (8) exist and integral in the left hand part of this inequality is finite and positive.

a) At first, let us prove the lower estimation. Consider the partition of the segment [r, R] by the points tm,i = r + (R — r)l/2m, m G N, l = = 0,..., 2m, and represent the ring K(r, R) as the finite union of the rings K(tm,1-1, tm,i), l = 1,..., 2m, with the attached appropriate hyperspheres. Then in the view of homeomorphism of f, inequality (3) and property (5) of the quasiinvariance of moduli under quasiconformal mapping f of the each of rings K(tm,1-1,tm,1), l < 2m, we obtain that

Mod(D) = Mod(f (K(r, R))) = Mod U f (K(tw_1,tw))

>

a=1

2

> ? Modi/ IK ( l_1

1=1 1=1

^ Mod(f (K(tm,1_1, tm,1))) > ^ P^d1 n)(tm,l)Mod(K(tm,i_1,tm,i)),

where P/,m are defined by equalities (7). Taking into account formulae (1) for moduli of rings K(tm,1-1,tm,1) we obtain

__ R dt

/ ^ /,m (tm,i) ln t I P /,m (t) t

1 = 1 tm,i_1 Jr ^

Mod(D) > VP!,m1-n)(tm,i= P!/i1-n)(t)dt

Passing to the limit in the integrals of step functions and taking into account that P/,m ^ P/ for a.e. t G (r, R), we arrive to a required inequality in the left side of (8).

b) Now we are going to prove the upper estimation. In the view of definition of the modulus of a ring domain and the property (4), connecting the moduli of the conjugated families of curves and hypersurfaces, we

have:

Mod(f (K(r, R))) = (f (E)) < (9)

where E is the family of all (n — 1)-dimensional hypersurfaces, separating the boundary components of the concentric ring K(r, R), and p(y) is an arbitrary measurable non-negative function admissible in the definition of modulus of family f (E).

As p(y) we will consider the function

( ) = ( (po/.n-(n-1)) ◦ f-1 (y), for y e f (K(r,R)), \o, if y ef (K(r,R)),

where p0 is an arbitrary admissible function for a modulus of family E of (n — 1)-dimensional hypersurfaces, pn-1 (x) is the minimal distortion of the (n — 1)-dimensional volume at the point x under the mapping f. Defined in this way, function p is admissible in the problem of finding the modulus of the family of surfaces f (E). Indeed, for any surface a e E and its image f (a) let dsn—1 and ds^-i. be elements of (n — 1)-dimensional area correspondingly. Then we obtain

\ n — 1

pn-1(y)d<_ 1 >/ [ 1 p(x) _ | Pn—1(x)dsn —1 =

Iff \ ' "" ^ / v ,,1/(n —

'/H \Pn— 1 (x)

(x)dsn—1 > 1

n— n1

Note that for almost all y the preimage of the infinitesimal ball {y e e f (K(r, R)) : |y — f (x)| < 5} of radius 5 under the mapping f is a domain, close to ellipsoid with semiaxes a1 > a2 > ... > an, where

5 5

a1 = i-TT, an = ^TT. (10)

l/(x) L/(x)

Then

5n—1

pn—1 (x) =-, whereas J/ (x) =

a1a2■■ ■ an-1 a1a2■■ ■ an-1an

n

5

n

Using (10) and the latter equality, we obtain

J/ (x) = (a1 a2 ••• fln_1)n/(n-1) = ^n_(n_1)(x) a1 a2 ••• an_1an

X 1/(n_1) / rn(x) x 1/(n_1) (11)

«1«2 ■ ■ ■ a- \ _ / L/ (x) N

ai / VJ/(x)

After the change of variables under the sign of integral in the right hand side of (9), in view of equality (11) we obtain the estimations

Mod(f (K(r, R))) < ^n/_(n_1^ P-(x) Jn_Xl ^ dvn =

•>K(r,R) P„-1 ) (x)

r /L-(x)\ 1/(n_1)

= "i«?-1'/ PS(*)(Try) dv„ <

./K(r,R) VJ/(x) /

2m

< ^r^ V / PS(x) p}/(n_1)(x) dvn <

1 = 1 J K (tm,l-1,tm,l ) 2m

(12)

< ^/_(n_1) £ P/1,mn_1) (tm,1) P-(x)dvn,

1 = 1 J K(tm,l- 1 ,tm,l)

where the rings K(tm,1_1 , tm,i) are the same that in the part a) of the proof of this theorem, and the functions P/,m are defined via equality (6).

It remains to note that function p0(x) = (^S^- 1) |x|)_1 is admissible in the problem of finding of the modulus of family £ in the ring K(r, R) and after the changeover to the spherical coordinates

[' 1 dt [' 1 dt

P- (x)dvn = ——- / — ds„_1 =

,,"/("_ 1) t t /, , =1 -_1 ^1/(n_1) t t '

- Wn_1 m,l 1 ' */|x|=1 Wn_1 'l 1 '

K (tm,l-1,tm,l)

Taking into account the last equalities we obtain from (12) that Mod(f (K(r, R))) < £ P/1,/in_1) (tm,1) itm'1 ^ = /R P/1,/in_1) (t)^

1 = 1 Jtm,l-1 t -'r t

Passing to the limit in the integrals of the step-functions as in the part a) we obtain the desired upper estimation in (8).

c) In order to prove the sharpness of inequalities (8) we consider the sequence of points rk = 1 — 1/2k, k = 0,1, 2,.... For each k we define the quasiconformal homeomorphism of the set Kk := {x E Rn : rk < |x| < < rk+i} by the formulae

|x|ak-1

fk (x) = x-,

Ck

where exponents ak := 2-k ^ 0 as k ^ ro, and coefficients ck are defined by recurrent ratios

co = 1, ck+i = ck r^1 -ak.

Then the function

f (x) = {fk(X), if X E Kk, (13)

7 \k = 0,1,2,... V ;

is a locally quasiconformal homeomorphism of the unit ball Bn = {x E E Rn : |x| < 1} onto the ball B(R) = {x E Rn : |x| < R} with radius

rak tt , X ak

Ck

k = 1

R = lim = r^ K^1 < ro. (14)

k^tt Ck f-*- V rk )

Here the convergence of the infinite product follows from the inequality rk+1 /rk < 3/2 for any k > 1 and from the fact that ln R is majorized by the converging series

tt

ln ri + ln3/2^ ak = ln ri + ln3/2^2-k. k=1 k=1

The mapping (13) is identical in the neighborhood of the origin and maps the ring K(r1,1) locally quasiconformally onto the ring K(r1 ,R). With a glance of formulae (1) for a moduli of rings

R

Mod(f (K(n, 1)) = Mod(K(n, R)) = ln —.

r1

The function f maps each ring Kk in one-to-one manner onto the rings {x E Rn : r^ /ck < |x| < r0+1/ck}. The mapping f, inside every ring Kk, is quasiconformal and is known as the radial mapping. It is possible to calculate (see, for example, [1], chapter 16, example 16.2, or [3], chapter 4, §2, 4) that in Kk the radial mapping f has a constant dilatation pj (x) = = ak-n = 2k(n-1). Therefore Pf (t) = ak-n for any t E (rk,rk+1), k > 1.

In such a way

f1 pV<1-"> (t) dt = £ Ok frk+I dt = tm — = m R - in n,

I 1 k=1 ^fc 4 k=1 rk

i.e. Mod(f (K(ri, 1)) = fr[ P71/(1-n>(t) dtt. The sharpness of the lower estimation in inequality (8) has been proved.

In order to prove the sharpness of the upper estimation in (8) it is sufficient to change the exponents ak in the definition of function (13) putting, for example, ak = (3/2)k. Then the radius R, determined by (13), is finite as before since, in the view of relation ln(rk±1/rfc) ~ ln(1 + 2—k) « « 2—k, the series

ln R = ln r1 + ak ln rk±1

rk

k=1 k

has the same asymptotic as the converging series it=1(3/4)k. In addition, the dilatation of the mapping f in every ring Kk is equal to p/ (x) =

= a

n —1 k

= (3/2)k(n—1> and

p1 j, ^

p1/(n—1> (t)= y ak ln =ln R - ln r1 = Mod(f (K(n, 1))), Jri f 4 k=1rk

that illustrates the sharpness of the upper estimation in (8). □

Remark. If the mapping f in theorem 2 is Q-quasiconformal, then P/ (t) < < Q and the classical estimations (5) of the distortion of moduli follow easily from inequalities (8).

3. In this section we will apply theorem 2 to the proof of the estimation of the derivative of locally quasiconformal automorphism of the ball Bn = = {x e Rn : |x| < 1}.

Theorem 3. Let f be a locally quasiconformal automorphism of the unit ball Bn such that f (0) = 0, function f be differentiable at the origin and l/(0) = L/(0) =: |f'(0)|, i. e. f be conformal at the point x = 0. Then

11 (1 - p;/(n—1>(t)) f < ln if'(0)i < l1 (1 - p;/(1—n>(t)) dtt. (15)

Estimations are sharp.

Proof. Let the conditions of the theorem be fulfilled. Consider the small value e > 0. Function f maps the ring K(e, 1) onto ring domain De, whose outer boundary component coincides with the unit hypersphere S«-1 = {x E Rn : |x| = 1}, whereas inner boundary component, in view of conformality of f at origin, coincides accurate to o(e) with the hypersphere of radius e|f (0)| with the center at point x = 0.

First we prove the upper estimation in (15). In view of property (2) of monotonicity of modulus of ring domains

Mod(D) < Mod(K(e|f (0)| — o(e), 1)) = — ln(e|f'(0)| — o(e))

for any small e > 0. Here o(e) is some positive infinitesimal function that decreases faster than e when e ^ 0+.

From the other side in view of theorem 2

Mod(De) > J1 Pf1/(1-n)(t)dtt.

Therefore,

— ln(e|f(0)| — o(e)) > i1 Pf1/(1-n)(t)dtt,

i.e.

^ ' >1/(1-«) (t) — ^ dt

ln |f(0)|< — / (Pf1/(1-n) (t) — 1) | + o(1),

Passing to the limit as e ^ 0, we obtain the required upper estimation in (15).

The proof of the lower estimation may be obtained in the similar way with the help of upper estimation in (8). Indeed, there exists a positive infinitesimal function o(e) such that K(e|f(0)| + o(e), 1) C De and in view of (1), (2), (8)

ln + ln--o(1) = ln

lf'(0)| e ^ e|f (0)| + o(e)

dt 1

i dt = Mod(K(e|f (0)| + o(e), 1)) < Mod(De) < I Pf1/(n-1)(t)

The required lower estimation of |f'(0)| follows from this relationship.

In order to prove the sharpness of the inequality (15) we consider the function F(x) = f (x)/R, where f, R are determined by equalities (13)

and (14) respectively. The mapping F is a locally quasiconformal automorphism of the ball B", having in the neighborhood of the origin representation F(x) = x/R. Hence f' (0) = I/R, where I is a Jacobi matrix of the identical transformation and |F'(0)| = (0) = LF (0) = 1/R.

From the other side, if ak = 2-k then, as was mentioned above, PF(t) = ak-" for any t G (rk, rk+), k > 1, and PF(t) = 1 for all t G [0,r1). Therefore

i1 (P^-"» (t) - 1) f = t(ak - 1)/dtt

k = 1 ^ t

TO

= ak ln rk+1 + ln r1 = ln R. ^^ rk

k=1 k

Consequently

ln|F'(0)|=ln R = - ^ (P^1-"» (t) - 1) f,

and the sharpness of the upper estimation of |f (0)| is proved.

The sharpness of the lower estimation in inequality (15) may be demonstrated in a similar way if we choose the values ak = (3/2)k. □

Remark. Note that the convergence of the integrals in inequalities (15) depends on the velocity of tending of function Pf (t) to 1 as t ^ 0+ and on the velocity of growth of Pf (t) as t ^ 1-. In particular case the values |f' (0)| are bounded and separated from zero, if there exist positive numbers a and ft such that

Pf (t) - 1 = o (ta) ift ^ 0+, Pf (t)(1 - t)"-1 = o ((1 - t)^) if t ^ 1 - .

For example, if in the assumption of theorem 3 the majorant Pf (t) < < ((1 +1)/(1 - t))"-1, then |f (0)| < 4, but in the case of Pf (t) > (1 --1/ lnt)"-1 we have only the trivial estimation |f'(0)| < to.

Acknowledgment. The research was supported by the Russian Foundation for Basic Research (project 14-01-92692).

References

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Received July 11, 2015.

In revised form, November 17, 2015.

Tver State University

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Petrozavodsk State University

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E-mail: [email protected]