2 STANISLAV HENCL AND PEKKA KOSKELA The most recent result in this direction is due to Hencl and Malý [7]. They showed that, for a quasi-light mapping

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1 MAPPINGS OF FINITE DISTORTION: DISCRETENESS AND OPENNESS FOR QUASI-LIGHT MAPPINGS STANISLAV HENCL AND PEKKA KOSKELA Abstract. Let f 2 W ;n (; R n ) be a continuous mapping so that the components of the preimage of each y 2 R n are compact. We show that f is open and discrete if jdf(x)j n» K(x)J f (x) a.e where K(x) and K n =Φ(log(e + K)) 2 L () for a function Φ that satisfies R =Φ(t)dt = and some technical conditions. This divergence condition on Φ is shown to be sharp.. Introduction Let be a connected, open subset of R n with n 2. In this paper we consider continuous mappings f 2 W ;p loc (; Rn );p : We suppose that there is a measurable function K :! [; ) so that the distortion inequality jdf(x)j n» K(x)J f (x) holds almost everywhere in ; where Df(x) is the differential matrix of f at x; jdf(x)j is the operator norm of this matrix, and J f (x) is the determinant of Df(x): We say that f is of finite distortion K if furthermore J f is locally integrable. If above K is bounded, then necessarily f 2 W ;n loc (; Rn ); and we recover the class of mappings of bounded distortion, also called quasiregular mappings (cf. [22], [23], [9], [26]). One of the fundamental properties of mappings of bounded distortion is the remarkable result by Reshetnyak [2] that such a mapping is either constant or both discrete and open. This means that the preimage of each point is a discrete set of points and that f maps open sets to open sets. A principal goal in the theory of mappings of finite distortion has been to try and obtain analogs of Reshetnyak's result. In [0] Iwaniec and»sverák proved in dimension two, using the Beltrami equation, that each non-constant mapping f 2 W ;2 (; R 2 ) of finite distortion K 2 L () is both open and discrete. Subsequently, Heinonen and Koskela [6] proved in higher dimensions that a quasi-light mapping f 2 W ;n loc (; Rn ) of finite distortion K 2 L p (); p>n ; is open and discrete. Here the quasi-lightness means that that the components of the preimage of each y 2 R n are compact. Manfredi and Villamor [20] then showed that the quasi-lightness assumption can be disposed of. 99 Mathematics Subject Classification. 30C65, 26B0, 73C50. Key words and phrases. Mapping of finite distortion, dicreteness, openness.

2 2 STANISLAV HENCL AND PEKKA KOSKELA The most recent result in this direction is due to Hencl and Malý [7]. They showed that, for a quasi-light mapping f 2 W ;n loc (; Rn ) of finite distortion, the integrability assumption K 2 L n () is sufficient for discreteness and openness. The general case remains open. It is then natural to inquire if the integrability ofk could be further relaxed. To this end, let us first recall a construction by Ball []. He gives an example of a non-constant, Lipschitz continuous quasi-light mapping of finite distortion K; defined in a domain ; so that f maps a line segmenttoapoint and with K 2 L p () for all p<n: Moreover, the preimage of every other point consists of at most a single point. Regarding the distortion function K; one can in fact check that (.) K n Φ(log(e + K)) 2 L () dt whenever <. Iwaniec and Martin [9] have conjectured that Φ(t) each non-constant mapping f 2 W ;n loc (; Rn ) of finitedistortion that dt satisfies (.) for some (sufficiently regular Φ) with Φ(t) = is in fact both open and discrete; in fact their conjecture is slightly stronger because it involves a different distortion function. However, this far there have been no results under assumptions weaker than K 2 L n (); even for quasi-light mappings. We give the first step towards to this conjecture by establishing the following sharp result. Suppose that we are given a function Φ : [; )! (0; ) such that (.2) (i) Φ is continuous and non-decreasing; (ii) dt Φ(t) = ; t n (iii) the function t! is increasing ; Φ(log(e + t)) (iv) for every c > 0 there is c 2 > 0such that Φ(c t)» c 2 Φ(t): Notice that these conditions are satisfied for example for Φ (t) =, Φ 2 (t) =t, Φ 3 (t) =t log(e + t) and so on. Theorem.. Let ρ R n be a connected open set. Suppose that f 2 W ;n loc (; Rn ) is a quasi-light mapping of finite distortion K that satisfies (.) with a function Φ that satisfies (.2). Then f is open and discrete. Before discussing the proof of Theorem., let us briefly commenton the regularity assumptions on f: First of all, there exists a quasi-light mapping f of finite distortion K that fails to be discrete and open, satisfies f 2 W ;p (; R n ) for all p<n and satisfies K 2 L p () for all p < : Thus the regularity assumption f 2 W ;n loc (; Rn ) cannot be

3 MAPPINGS OF FINITE DISTORTION: DISCRETENESS AND OPENNESS 3 substantially relaxed. Such mappings have been constructed in [2], [4]. We prove a stronger version of Theorem. in Section 5 that comes with an optimal regularity assumption. Secondly, wehave taken continuity as a standing assumption for mappings of finite distortion. Under the regularity assumptions referred to above, continuity follows from the other assumptions, as was shown in [4], [8], [4]. All the proofs of discreteness and openness for n 3 that we are aware of rely on the following idea. One first proves that the mapping in question is sense-preserving. After that one verifies that the preimage of each y 2 R n is totally disconnected. The claim then follows by invoking the Titus-Young theorem [24]. We follow this procedure. The fact that f be sense-preserving in the setting of Theorem. is already due to Reshetnyak and the more general case can be essentially found in [2], [4]. Thus we are reduced to showing that the preimage of each y 2 R n is totally disconnected. This will be guaranteed by our following theorem. Theorem.2. Let ρ R n be a connected open set and suppose that f 2 W ;s loc (; Rn ); n < s» n; is a mapping of finite distortion K. Furthermore, assume that K satisfies (.) with a function Φ that satisfies (.2) and that the multiplicity of f is essentially bounded ona neigborhood of 0. Then either f 0 or H (f (0)) = 0: The essential boundedness of the multiplicity means that there is an integer k so that the cardinality of f (y) is at most k for almost all y in the given neigborhood of 0: The fact that we can bound the multiplicity under our assumptions is based on certain results in [2], [3] and [7]. We prove Theorem.2 by first establishing a sharp generalization of an oscillation estimate given in [7]. We believe that this oscillation estimate, given in Section 3, is of its own interest. Indeed, the original version has already found applications [8]. Our estimate is, in a sense, a substitute for the usual bounds on capacity in terms of Hausdorff measures. The usual bounds are not subtle enough for our purposes. Theorem.2 is then obtained by combining the oscillation estimate with a delicate integrability result on jfj : Theorem.2 is very sharp. The integrability assumption on K cannot be relaxed because of the example due to Ball, mentioned above. Moreover, we cannot take s = n at least when n = 2; as is seen x by considering the mapping defined by f(x) = log(e=(jxj)) for x 2 B(0; 2) n B(0; ) and by f(x) = 0 for x 2 B(0; ): Indeed, f is of K finite distortion K with Φ(log(e + K)) 2 L (B(0; 2)) for, say, Φ(t) =t; and the multiplicity of f is essentially bounded by one in any neighborhood of 0.

4 4 STANISLAV HENCL AND PEKKA KOSKELA The paper is organized as follows. In Section 2 we recall some definitions and preliminary results. Section 3 is devoted to the proofs of oscillation estimates. In Section 4, we prove Theorem.2. Finally, in Section 5, we discuss discreteness and openness and give the proof of Theorem.. 2. Preliminaries 2.. Quasicontinuous representatives. In anticipation of future applications of our oscillation estimates, we will formulate them without the continuity assumption. This will be done in terms of quasicontinuous representatives. Let us point out that each continuous Sobolev mapping is quasicontinuous. In this paper, the precise definition of such a representive of a Sobolev mapping is not needed, because we will only employ the following fact [7, Proposition ]. See [9] for more on quasicontinuity. Proposition 2.. Let» p < ~p» n. Let u 2 W ;~p () be ; ~pquasicontinuous. Then for H np -a.e. point z 2 we have where fi = p ~p. lim sup r!0 r fi B(z;r) ju u(z)j dx < ; 2.2. Topological properties. A mapping f :! R n is said to be discrete if the preimage of each point of R n is locally finite in, and light if the preimage of each point of R n is totally disconnected. We say that f :! R n is quasi-light if for each y 2 R n the components of the set f (y) are compact. We call a continuous mapping f :! R n sense-preserving if deg(f; 0 ;y 0 ) > 0 for all domains 0 ρρ and all y 0 2 f( 0 ) n f(@ 0 ), where deg(f; 0 ;y 0 ) is the topological degree of f at y 0 with respect to 0. For the definition of the topological degree see e.g. [3] Area formula. We denote by jej the Lebesgue measure of a set E ρ R n. We will use the well-known area formula. Let f 2 W ; (; loc Rn ). The multiplicity function N(f;;y)off is defined as the number of preimages of y under f in. Let be a nonnegative Borel measurable function on R n. Without any additional assumption we have (2.) (f(x)) jj f (x)j dx» (y) N(f;;y) dy: R n This follows from the area formula for Lipschitz mappings, from the a.e. approximate differentiability of f [2, Theorem 3..4], and a general property of a.e. approximately differentiable functions [2, Theorem 3..8], namely that that can be exhausted up to a set of measure zero by sets the restriction to which of f is Lipschitz continuous.

5 MAPPINGS OF FINITE DISTORTION: DISCRETENESS AND OPENNESS Fine properties of Hausdorff measure. In the proof of the oscillation estimate we need the following set functions: d ffi : E 7! inf By [2, ], nx ff χ E» X ff a ff (diam E ff ) d : a ff 0; diam E ff» ffi; a ff χ Eff o; ffi>0: (2.2) lim d ffi(e) =H d (E) ffi!0 for any set E ρ R n, where H d is the usual Hausdorff measure. 3. Divergence criterium If we assume, for p < n; that a non-negative function u 2 W ;p () vanishes sufficiently fast on a set of positive (n p)-dimensional Hausdorff measure (it suffices to assume a certain power-like decay of integral means of u over B(x; r) as r! 0), then we can find pairwise disjoint sets E i such that je i j > 0 and (3.) sup u p» C E i E i dx (see [7, Theorem 3 and Theorem 4] for the exact statement, and also [7]). By a small modification of the proofs of those results one can obtain inf Ei u>csup Ei u; and, therefore u p dx = : We generalize this fact in the next lemma by obtaining a sharp divergence statement (3.4). Lemma 3.. Let» p<n, μ>0, fi 2 (0; ) and fl > 0. Let ρ R n be an connected open set and u 2 W ;p (). Suppose that u > 0 a.e. and let (3.2) = n z 2 : lim sup r!0 o r fi udx<fl : B(z;r) Suppose that H np () > μ. Suppose further that a function Φ : [; )! (0; ) satisfies (3.3) (i) for every c > 0 there is c 2 > 0 such that for every l 2 N Φ(c (l + )) c 2 Φ(c l); dt (ii) Φ(t) = :

6 6 STANISLAV HENCL AND PEKKA KOSKELA Then, for each 0 <ffi<e e ; (3.4) 0<u<ffi dx = : u p log =u Φ(log log =u) Proof. Our argument is an improvement on [7, Theorem 3]. We will omit the parts of the reasoning there that need not be altered. Set (3.5) fi := 2 fi=4 : For j; m 2 we denote m = W j m = n z 2 : B(z; 2 m ) ρ ; 2 m0fi n z 2 : B(z;2 m0 ) udx» 2fl for all m 0 = m; m+;::: fi fi fu>fi j g B(z; 2 m ) fi fi > j m = m W j m : 2 jb(z; 2m )j As in [7] we can find k 2 N and a compact set Λ ρ k such that (3.6) log(4fl) log fi <k and H np ( Λ ) >μ: In view of (2.2) we can also suppose that (3.7) np 2 k+ ( Λ ) >μ and log(=ffi) log fi <k (this can only increase the value of k and therefore (3.6) remains valid). From now on this k 2 N S is fixed. Since Λ is compact and Λ ρ j W j k ; where W ρ k W 2 k ρ ::: are open sets, we infer that there is i 2k such that (3.8) Λ ρ W i k : We denote P j k = [ mk j m n j m+ : With each z 2 P j we associate a ball k B z = B(z; 2 m ) where m is such that z 2 A j m, m k. By the Besicovitch covering theorem we find a countable system B j k ρfb z : z 2 m j n j m+ ; m kg such that (3.9) χ P j k» X B2B j k χ B» N: Using the definition of the sets j m we can deduce with some work that np 2 k+ ( Λ )» a X X a<j»3a B2B j k (diam B) np o o ; ;

7 MAPPINGS OF FINITE DISTORTION: DISCRETENESS AND OPENNESS 7 for any a 2 N, a i. In fact this was proved in [7] (two linesdown from formula (8) there) only for a = i but it is easy to see that everything works well also for a i. From (3.7) we obtain (3.0) μ» a X X a<j»3a B2B j k Fix j 2fa +;:::;3ag and set (diam B) np : E j := [ ` := fi j+ ; B fu» `g fu>fi `g: B2B j k Note that the sets E j are clearly pairwise disjoint. Let B = B(z; r) 2 B j and k v = maxφ Ψ fi `; minfu; `g. Since z 2 P j k, there exists m k such that z 2 m j nm+. j Therefore we can use the Poincaré inequality in a standard way (see [7] for details) to deduce that `p» Cr pn jrvj p dx = Cr B E pn dx j B where C = C(n; p; fi). Since, for every x 2 E j ; we have u(x) ο ` ο fi j and a<j» 3a this implies (3.) C a» rpn B E j u p log =u dx: We multiply both sides of (3.) by r np =μ ο (diam B) np =μ and sum over B 2B j k and then sum over j 2fa +:::3ag. Then, with the aid of (3.0), we arrive at (3.2) 3aX C» Cμ X 3aX (diam B) np» C 0 jruj μ a Ej p u p log =u dx j=a+ B2B j k j=a+ with C 0 = C 0 (n; p; fi; μ) (the constant C 0 involves also the constant N from the Besicovitch covering theorem). It follows from the estimate u» fi j+ on E j, i k and (3.7) that f0 <u<ffigff [ m=i+ E m :

8 8 STANISLAV HENCL AND PEKKA KOSKELA Therefore, the estimate fi j»juj»fi j+ on E j, (3.3) (i), (3.2) and (3.3) (ii) imply 0<u<ffi u p log =u Φ(log log =u) dx X m=i+ X l=0 j=3 l i+ X X l=0 l=0 Em 3X l+ i u p log =u Φ(log log =u) dx Ej u p log =u Φ(log log =u) dx X C Φ log(3 l+ i log(fi)) 3 l+ i C Φ(C(l + )) = : j=3 l i+ Ej u p log =u dx Theorem 3.2. Assume that» p < ~p» n. Let ρ R n be a connected open set and let u 2 W ;~p () be ~p-quasicontinuous. Suppose that u > 0 a.e. and H np (fu = 0g) > 0. Suppose further that a function ~Φ satisfies (.2) (i) and (ii). Then, for each 0 <ffi<e e ; (3.3) 0<u<ffi u p Φ(log ~ dx = : =u) Proof. By Proposition 2. there exists fl 2 (0; )suchthath np ( fl ) > 0, where n o fl = z 2 : lim sup r fi r!0 jfj dx < fl : B(z;r) Recall that fi = p. ~p Define Φ(exp Φ(t) = ~ t) : exp t From (.2) (i) for ~Φ we obtain (3.3) (i); and clearly dt Φ(t) = exp tdt ~Φ(exp t) = ds ~Φ(s) = : Now Lemma 3. yields = 0<u<ffi u p log =u Φ(log log =u) 0<u<ffi dx = u p Φ(log ~ =u) dx: Λ e Λ The following elementary example shows that our assumption (.2) (ii) is essential in Theorem 3.2.

9 MAPPINGS OF FINITE DISTORTION: DISCRETENESS AND OPENNESS 9 Example 3.3. Let n; p 2 N, = p<n. Set u(x) = ep ; ep q x 2 + :::+ x 2 p. Clearly H np (fu =0g) =H np f0g p Suppose that Then u p Φ(log ~ =u) =2np» C =e 0 ds ~Φ(s) < : (=ep;=ep) p dt t ~Φ(log =t) = C n and suppose that» ep ; ep np > 0: dx jxj p ~ Φ(log =jxj)» ds ~Φ(s) < : 4. Hausdorff measure of f (0) We need the following elementary inequalities that can be viewed as variants of Young's inequality. Lemma 4.. Let Ψ:[; )! [; ) be a differentiable concave function and set ψ(t) :=Ψ 0 (t). Suppose that the function t! t n ψ log(e + t) is increasing. Then (4.) c n ψ log(e +=a) a n Ψ log(e +=a)» cn ψ log(e +=a) ba n Ψ n n for every a>0, b>0 and c>0. log(e +=a) + b n ψ log(e + b=c) : Proof. If the first term on the right-hand side of (4.) is greater or equal to the left-hand side then the inequality is obvious. Otherwise c b log(e +=a) : aψ n Since the function t n ψ log(e + t) is increasing, ψ is non-increasing and Ψ ; this implies b n ψ log(e + b=c) c n a n Ψ log(e +=a) ψ log e + c n a n Ψ log(e +=a) ψ log(e +=a) : aψ n (log(e +=a)) Λ

10 0 STANISLAV HENCL AND PEKKA KOSKELA Lemma 4.2. Suppose that Φ : [; )! (0; ) satisfies (.2) for n =2. Then (4.2) a» a2 Φ log(e + a=c) + C b b Φ log(e + b=c) : for every a 0, b>0 and c>0. Proof. If the first term on the right-hand side of (4.2) is greater or equal to the left-hand side then the inequality is obvious. Otherwise (4.3) b a Φ log(e + a=c) : From (.2) (i) and (iv) it is easy to see that Φ increases at most like a power function. Therefore Φ log(e + t)» C + Ct for every t > 0. With the help of (.2) (iv) this implies (4.4) Φ log e + tφ log(e + t)» CΦ log(e + t) : From (.2) (iii), (4.3) and (4.4) we have b Φ log(e + b=c) a Φ log(e + a=c) Φ log e + a=cφ log(e + a=c) Ca: Proof of Theorem.2. Suppose that f is not identically 0 and that H (f (0)) > 0. We know that there is 0 < ffi < e e such that the multiplicity off is bounded almost everywhere on B(0;ffi)by constant M > 0. Set (4.5) Ψ(t) =+ t ds Φ(s) and ψ(t) =Ψ 0 (t) = Φ(t) for t : From (.2) (ii) we know that lim t! Ψ(t) = and therefore also t ψ(s) ds = lim log Ψ(t) = lim = t! t! Ψ(s) ds Φ(s)Ψ(s) : Hence the function ~Φ(t) = Φ(t)Ψ(t) satisfies assumptions (.2) (i) and (ii): We wish to apply Theorem 3.2 to jfj for p = n : In order to do this we still need to check that jf (0)j = 0: When n = 2 this follows from Lemma 4.3 below and for n 3 from formula (2.3) in [5]; notice that this result can be applied because (.) and (.2) imply that K =(n) 2 L loc (): We thus obtain from Theorem 3.2, for p = n and u = jfj, that jdfj n ψ log(e +=jfj) (4.6) jfj n Ψ log(e +=jfj) = : 0<jfj<ffi Denote 0 = fjdf(x)j 6= 0g f0 < jfj < ffig. It is not difficult to verify from (4.5) and (.2) that Ψ satisfies all the assumptions of Λ

11 MAPPINGS OF FINITE DISTORTION: DISCRETENESS AND OPENNESS Lemma 4.. We use inequality (4.) for a(x) = jf(x)j, b(x) = K(x) and c(x) = jdf(x)j to obtain the estimate jdfj 0 n ψ log(e +=jfj) (4.7) jfj n Ψ log(e +=jfj)» jdfj» 0 n ψ log(e +=jfj) + Kjfj n Ψ n K n log(e +=jfj) n ψ log(e + K=jDfj) : 0 Since ψ is non-increasing, part (iv) of (.2) and our integrability assumptions give us (4.8) 0 K n ψ log(e + K=jDfj)»»» ψ()» C K»jDfj s=(n) + jdfj s + jdfj s + C K>jDfj s=(n) K n ψ log(e + K sn+ s ) K n ψ log(e + K)» C: Using the distortion inequality, (2.) and lim s! Ψ(s) = we conclude that jdfj 0 n ψ log(e +=jfj)» Kjfj n Ψ n n log(e +=jfj) J f (x)ψ log(e +=jfj)» 0 jfj n Ψ n n log(e +=jfj) ψ log(e +=jyj) (4.9)» CM B(0;ffi) jyj n Ψ n n log(e +=jyj) dy ffi ψ log(e +=t) = C log(e +=t) dt 0» C Ψ» C: tψ n n n log(e +=ffi) lim Ψ n t!0+ Combining (4.7), (4.8) and (4.9) we arrive at jdfj n ψ log(e +=jfj) 0<jfj<ffi jfj n Ψ log(e +=jfj) = jdfj 0 n ψ log(e +=jfj) = jfj n Ψ log(e +=jfj)» C: This clearly contradicts (4.6). log(e +=t) We close this section by verifying the following result that was employed in the proof above. Λ

12 2 STANISLAV HENCL AND PEKKA KOSKELA Lemma 4.3. Suppose that n = 2 and that f is as in Theorem.2. Then jf (0)j =0: Proof. By Lemma 5. in [] we may find a (radial) function u 2 W ;n 0 (B(0; )) so that lim y!0 u(y) = and B(0;) jruj 2 Φ log(e + jruj) < : Therefore we can find a decreasing sequence of numbers R k & 0 and sequence of functions u k 2 W ;n 0 (B(0;R k )) so that u k onb(0;r k+ ) and (4.0) lim k! B(0;R k ) jru k j 2 Φ log(e + jru k j) =0: Since f is quasi-light, we may assume that is bounded, f (0) is compact and there exists ffi > 0 such that f (B(0;ffi)) ρρ (cf. [25, Theorem 3.]). For k 2 N we write B k = B(0;R k );A k = f B k n B k+ and ~ Ak = A k fjdfj 6=0g: Fix k 2 N large and denote v = u k ffi f. If R k <ffi; then the function v has zero boundary values and thus we may use the Sobolev inequality for v. Since u k (0) on B k+ we obtain from the Sobolev inequality and (4.2) for a = jrvj, b = K and c = jdfj that (4.) jf (0)j»» ju k ffi fj»c jrvj 2 Φ log(e + jrvj=jdfj) + C K Analogously to (4.8) we obtain (4.2) Φ log(e + K=jDfj)» K» C jdfj s + C jrvj = jrvj» K Φ log(e + K=jDfj) : K Φ log(e + K) : The right-hand side of (4.2) tends to zero when k!because the sets A k are clearly pairwise disjoint. For k large enough, the multiplicity of f is essentially bounded by M on B k. Therefore we can use the distortion inequality, (2.) and (4.0) to obtain (4.3) Φ log(e + jrvj=jdfj)» K jrvj 2»» M j(ru k ) ffi fj 2 J f Φ log(e + j(ru k ) ffi fj) B k jru k j 2 Φ log(e + jru k j)! k! 0:

13 MAPPINGS OF FINITE DISTORTION: DISCRETENESS AND OPENNESS 3 From (4.), (4.2) and (4.3) we obtain jf (0)j =0. 5. Openness and discreteness In this section we prove discreteness and openness of a mapping under a weaker integrability condition on Df than Df 2 L n. We use an Orlicz-type condition that was introduced to this setting in [6], [8], and [4]. Theorem 5.. Let ρ R n be a connected open set and suppose that n <p» n. Let f 2 W ;p (; loc Rn ) be a continuous, sense-preserving mapping that has essentially bounded multiplicity. Suppose that f is a mapping of finite distortion K so that K satisfies (.) with a function Φ that satisfies (.2). Then f is either constant or both discrete and open. Proof. Any sense-preserving, light and continuous mapping is both discrete and open, see [24] or [23, Lemma 5.6]. Hence it remains to show that f is light. However, by Theorem.2, H (f (y)) = 0 for each y 2 R n, which easily implies lightness. Λ Now let us state the main result of our paper. Theorem. follows from Theorem 5.2 by choosing Ψ(t) =t n. Theorem 5.2. Let ρ R n be a connected open set. Let Ψ be a nonnegative, strictly increasing and continuously differentiable function on [0; ) satisfying the conditions Ψ(t) dt = ; t+n tψ 0 (t) lim inf t! Ψ(t) >s with s > n. Suppose that f is a quasi-light mapping of finite distortion K that satisfies (.) with a function Φ that satisfies (.2), and suppose further that Ψ(Df) 2 L loc (): Then f is discrete and open. Proof. Using results from [9] (or [5],[6]) and [2] we may obtain analogously to [7, Theorem 5 and 7] that f is sense-preserving, and that each point x 0 2 is contained in a subdomain 00 ρ suchthatn(f; 00 ; ) is essentially bounded. Thus we may use Theorem 5. to show that f is open and discrete on 00. Since these properties are local, the proof is complete. Λ Let us close the paper by commenting on the sharpness of our assumptions. As discussed in the introduction, our assumption on K is optimal. Moreover, an example constructed in [4] gives us, given Ψ with Ψ(t) dt < ; t+n a non-discrete, non-open, quasi-light mapping of finite distortion K so that K 2 L p () for all p; in particular for p = n ; and so that Λ

14 4 STANISLAV HENCL AND PEKKA KOSKELA Ψ(Df) isinl loc (): Thus the integrability assumption on jdfj is also optimal. References [] J. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh Sect. A 88 no. 34 (98), [2] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 53 Springer-Verlag, New York, 969 (Second edition 996). [3] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Clarendon Press, Oxford, 995. [4] V. Gol'dstein and S. Vodop'yanov, Quasiconformal mappings and spaces of functions with generalized first derivatives, Sibirsk. Mat.. 7 (976), [5] L. Greco, Sharp integrability of nonnegative Jacobians, Rend. Mat. 8 (998), [6] J. Heinonen and P. Koskela, Sobolev mappings with integrable dilatations, Arch. Ration. Mech. Anal. 25 no. (993), 897. [7] S. Hencl and J. Malý, Mappings of finite distortion: Hausdorff measure of zero sets, Math. Ann. 324 (2002), [8] T. Iwaniec, P. Koskela and J. Onninen, Mappings of finite distortion: Monotonicity and continuity, Invent. Math. 44 (200), [9] T. Iwaniec and G. Martin, Geometric function theory and nonlinear analysis, Oxford Mathematical Monographs, Clarendon Press, Oxford 200. [0] T. Iwaniec and V.»Sverák, On mappings with integrable dilatation, Proc. Amer. Math. Soc. 8 (993), 888. [] J. Kauhanen, P. Koskela and J. Malý, On functions with derivatives in a Lorentz space, Manuscripta Math. 00 (999), 870. [2] J. Kauhanen, P. Koskela and J. Malý, Mappings of finite distortion: Discreteness and openness, Arch. Ration. Mech. Anal. 60 (200), 355. [3] J. Kauhanen, P. Koskela and J. Malý, Mappings of finite distortion: Condition N, Michigan Math. J. 49 (200), 698. [4] J. Kauhanen, P. Koskela, J. Malý, J. Onninen and X. hong, Mappings of finite distortion: Sharp Orlicz-conditions, Rev. Mat. Iberoamericana 9 (2003), [5] P. Koskela and J Malý, Mappings of finite distortion: the zero set of the Jacobian, J. Eur. Math. Soc. (JEMS) 5 (2003), [6] P. Koskela and X. hong, Minimal assumptions for the integrability of the Jacobian, Ricerche Mat. LI (2002), [7] J. Malý and O. Martio, Lusin's condition (N) and mappings of the class W ;n, J. Reine Angew. Math. 458 (995), 936. [8] J. Malý, D. Swanson and W.P. iemer, Coarea formula for Sobolev mappings, Trans. Amer. Math. Soc. 355 no. 2 (2003), [9] J. Malý andw.p. iemer, Fine regularity of solutions of elliptic partial differential equations, AMS, Providence, R.I., 997. [20] J. Manfredi and E. Villamor, An extension of Reshetnyak's theorem, Indiana Univ. Math. J. 47 no. 3 (998), 345. [2] Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Sibirsk. Mat.. 8 (967), [22] Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Trans. of Mathematical Monographs, Amer. Math. Soc, vol. 73, 989.

15 MAPPINGS OF FINITE DISTORTION: DISCRETENESS AND OPENNESS 5 [23] S. Rickman, Quasiregular Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 26. Springer- Verlag, Berlin, 993. [24] C. Titus and G. Young, The extension of interiority, with some applications, Trans. Amer. Math. Soc. 03 (962), [25] J. Väisälä, Minimal mappings in Euclidean spaces, Ann. Acad. Sci. Fenn. Ser. AI 366 (965), 22. [26] M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math. 39, Springer-Verlag, Berlin, 988. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FIN-4004, Jyväskylä, Finland address: hencl@maths.jyu.fi address: pkoskela@maths.jyu.fi