CC BY
18Probl. Anal. Issues Anal. Vol. 5(23), No. 2, 2016, pp. 33-37
DOI: 10.15393/j3.art.2016.3330
33
UDC 517.54
A. P. Kopylov
QUASI-ISOMETRIC MAPPINGS AND THE p-MODULI OF
PATH FAMILIES
Abstract. In this article, we study a connection between quasi-isometric mappings of n-dimensional domains and the p-moduli of path families. In particular, we obtain explicit (and sharp) estimates for the distortion of the p-moduli of path families under K-quasi-isometric mappings.
Key words: p-modulus of path families, p-capacity of the condenser, quasi-isometric mappings
2010 Mathematical Subject Classification: 30C65
1. Introduction. The article is devoted to the study of problems connected with the search for a complete descritpion of quasi-isometric mappings of n-dimensional domains in terms of the p-moduli of families of paths (curves). Note that this problem (for quasi-isometric mappings and also for quasiconformal mappings, space mappings with bounded distortion, mappings with finite distortion, homeomorphisms with finite mean dilatations, mappings with (p, q)-distortion etc) was successfully solved by many mathematicians (see, for example, [1]-[3j; see also [4]-[9]). Our main goal is to obtain explicit (and sharp) estimates for the distortion of the p-moduli of families of paths and curves under K-quasi-isometric mappings. Here we use the following, metric definition of such mappings:
Definition 1. Let K G [1, A homeomorphism f: Ui ^ U2 of domains U\ and U2 in Rn is called K-quasi-isometric if
K-1 < liminf ifM-fM < Umsup iM-/M < k y^x |y - x| |y - x|
for any x G U\. A homeomorphism f: U\ ^ U2 is called quasi-isometric if it is K-quasi-isometric for some K G [1,
©Petrozavodsk State University, 2016
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Our main result is
Theorem 1. Suppose that f: U\ ^ U2 is a K-quasi-isometric homeo-morphism of bounded domains Ui and U2 in Rn, where n > 2 (1 < K < < to). Then
K2-p-nMp(r) < Mp(f (r)) < Kp+n-2Mp(r) (1)
for every p e]1, to[ and any family r of paths 7 such that Im 7 C cl Ui.
Remark 1. The quantity Mp (r), where 1 < p < to, is called the p-
modulus of the path family r and defined as
Mp(r) = / Wx)]p dx
Rn
where R(r) is the set of all nonnegative Borel measurable functions p: Rn ^ IR such that J pds > 1 for every rectifiable path 7 e r.
It should be noted that our main result (Theorem 1) is conceptually most close to the results on quasi-isometries in [1].
For example, using Theorem 1 in [10] and our result, Corollary 3 to Theorem 4.4/ in [1], Chapter 5, Section 4, can be supplemented by the following assertion:
Theorem 2. Under the conditions of Theorem 1, K2-p-n Cp1 (Fo ,Fi; Ui) < Cp (f (Fq ),f (Fi);U2) < Kp+n-2Cp(Fi, F2; Ui) for every p e]1, to[ and any condenser (F0, Fi; U).
Remark 2. Cp(F0, Fi, U) is the p-capacity of the condenser (F0, Fi; U)
(F0 and Fi are closed disjoint nonempty sets in cl U, where U C Rn = = Rn U {to} is an open set), i.e.,
Ci(Fo, Fi; U)=inf^ |Vu|pdx,
u i
where infimum is taken over all functions u E C^(U) H Lp (U) that are
equal to unity (zero) in some neighborhood of F0 (Fp) (see [11]).
In what follows, for x E Rn and E C Rn, dist(x,E) = inf |x — y|,
yeE
all paths 7: [a, ^] ^ Rn, where a, 0 E R, are assumed continuous and non-constant, and /(7) means the length of a path 7.
2. Proof of Theorem 1. The proof of Theorem 1 follows the lines of the proof of the second claim of Theorem 6.5 in [12].
Let r be a family of paths in the domain U1 (i.e., of paths 7: [a, b] ^ ^ Rn such that Im7 C cl Ui). Consider the subfamily r* of r consisting of all locally rectifiable paths 7 E r such that f is absolutely continuous on every closed subpath of 7. Since f is a quasi-isometry, f E ACLp for all p > 1 (see, for example, [13, 12], for the definition of the class ACLp); therefore, Mp(r0) = 0 for the family To of all locally rectifiable paths in Ui having subpaths on which the mapping f is not absolutely continuous ([13]). The fact that r \ r* C r0 and the properties of moduli imply the equality Mp(r \ r*) = 0. Consequently, Mp(r*) = Mp(r). Therefore, for proving, for example, the left-hand inequality in (1), which we will do below, it suffices to show that Mp(r*) < Kp+n-2Mp(f (r)), where f (r) = {f o y : Y E r}.
Let E be a Borel subset in U1 that contains all points x E U1 at which f is not differentiable and all those points x in U1 at which f is differentiable but the Jacobian J(x, f) = 0, moreover, mes E (= mesn E) = 0. Here we use the facts that a quasi-isometric mapping is quasiconformal and the set of points of nondegenerate differentiability of a quasiconformal mapping is a set of full measure with respect to its domain of definition.
Assume that 7 E R(f (r*)) (f (^ = {f ◦ y : Y E r*}), i.e., /- 7(x)ds > > 1 for every locally rectifiable path 7 E f (r*). Define a function p : Rn ^ ^ Rn by setting p(x) = 7(f (x))||f'(x)|| if x E U1 \ E, p(x) = to if x E E, and p(x) = 0 if x E Rn \ U1. Arguing as in the proof of the second part of Theorem 6.5 in [12] (or of Theorem 32.3 in [14], which is the n-dimensional variant of the first theorem), we further infer that p E R(r*), and hence
Mp(r) = Mp(r*) < Jppdx = y[7(f(x))]p||f(x)||pdx =
Rn Ui
= J [7(f (x))]p Jf |J (x,f )|dx < Kp+n-2 J [7(f (x))]p|J (x,f )|dx =
Ui ' Ui
= Kp+n-2 J [7(y)]pdy = Kp+n-2 J[7(y)]pdy. (2)
U2 Rn
In (2), we have used the fact that, since f is K-quasi-isometry, it is easy to verify the inequality |||f/(Xf)|)|p < Kp+n-2 for x E U1 \ E. Taking (2)
into account and recalling that the inverse mapping f-i is also K-quasi-isometric, we finally get (1).
3. Sharpness of estimates (1). Suppose that nn =]0,1[n, K E E [1, to[, and
f: x = (xi,... ,xn-i,xn) ^ (Kxi,.. .,Kxn,K-ixn), x E nn.
Then f: nn ^ f (nn) is a K-quasi-isometric homeomorphism, and if p E]1, to[ and r is the family of paths joining the sets ]0,1[n-ix{0} and ]0,1[n-ix{1} in nn, f (r) = {f o y: y e r} then Mp(r) = 1, and
K n-i
Mp (f (r))=(K--^ = K p+n-2.
Thus, the rightmost estimate in (1) is sharp. Similarly, the leftmost estimate is also sharp.
Remark 3. It is worth noting that estimates (1) were previously unknown.
Acknowledgment. The author was partially supported by the Russian Foundation for Basic Research (Grant 11-01-00819-a), the Interdisciplinary Project of the Siberian and Far-Eastern Divisions of the Russian Academy of Sciences (2012-2014 no. 56), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-921.2012.1) and the Exchange Program between the Russian and Polish Academies of Sciences (Project 2014-2016).
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Received October 05, 2016.
In revised form, December 05, 2016.
Accepted December 02, 2016.
Sobolev Institute of Mathematics
4, Acad. Koptyuga pr., Novosibirsk 630090, Russia;
Novosibirsk State University
2, Pirogova st., Novosibirsk 630090, Russia
E-mail: [email protected]
