COMPOSITION OF q-quasiconformal MAPPINGS AND FUNCTIONS IN ORLICZ-SOBOLEV SPACES

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1 COMPOSITION OF -QUASICONFORMAL MAPPINGS AND FUNCTIONS IN ORLICZ-SOBOLEV SPACES STANISLAV HENCL AND LUDĚK KLEPRLÍK Abstract. Let Ω R n, n and α 0 or < n and α 0. We prove that the composition of -uasiconfomal mapping f and function u W L log α L loc fω)) satisfies u f W L log α L loc Ω). Moreover each homeomorphism f which introduces continuous composition operator from W L log α L to W L log α L is necessarily a -uasiconformal mapping. As a new tool we prove a Lebesgue density type theorem for Orlicz spaces.. Introduction Let Ω, Ω 2 R n be domains and let f : Ω Ω 2 be a homeomorphism. Given a function space X we would like to characterize mappings f for which the composition operator T f : T f u) = u f maps XΩ 2 ) into XΩ ) continuously. This problem has been studied for many function spaces and one the most important is the following well-known result: The composition operator T f : T f u) = u f maps W,n loc Ω 2) into W,n loc Ω ) continuously if f : Ω Ω 2 is a uasiconformal mapping [2], [23], [5, Lemma 5.3]). Moreover each homeomorphism f which maps W,n loc Ω 2) into W,n loc Ω ) continuously is necessarily a uasiconfomal mapping. Similarly it is possible to characterize homeomorphism for which the composition operator is continuous from W,, loc to Wloc and we obtain a class of -uasiconformal mappings [5] see also [3]). Here homeomorphism f W, loc Ω, Rn ) is called a -uasiconformal mapping if there is a constant K such that.) Dfx) K J f x) for a.e. x Ω. For the properties and further applications of n-uasiconformal mappings see [], [2], [5] and [2]. Let us note that we do not assume that J f 0 a.e. as usual, i.e. on the right-hand side we have J f and not J f. This does not seem to be an essential restriction since all homeomorphisms f that are regular enough satisfy either J f 0 a.e. in Ω or J f 0 a.e. in Ω see [] for details). That is up to a simple reflection we have the usually considered definition. In general one could expect that different function spaces have a different class of morphisms unless the answer is somehow trivial. Surprisingly this is not the case as 2000 Mathematics Subject Classification. 46E35,46E30,30C65. Key words and phrases. Orlicz-Sobolev spaces, Quasiconformal mappings, Lebesgue density theorem. The first author was supported by the RSJ algorithmic trading grant and the work of the second author was supported by the grant SVV

2 2 STANISLAV HENCL AND LUDĚK KLEPRLÍK many examples indicate. For example n-uasiconformal mappings serve as the best class of morphisms not only for W,n loc functions but also for other function spaces that are close to W,n loc. Let us mention for example the stability under uasiconformal mappings for the BMO space [9], fractional Sobolev spaces Ṁn/s, s, s 0, ], [4, Theorem.3] see also [22] and [9]), absolutely continuous functions of several variables ACλ n [7] or exponential Orlicz space exp LΩ) in the plane [3]. We would like to explore this in detail and we would like to know if there is some general principle that somehow close spaces have the same class of morphisms. We show that the same phenomenon occurs for some Orlicz-Sobolev spaces see Preliminaries for the definition and basic properties) that are close to W, and that -uasiconfomal mappings are the best class of morphisms also for some of those function spaces. In particular see Section 3 and 4 for the general statement) we prove Theorem.. Let n and α 0 or < n and α 0 and suppose that f : Ω Ω 2 is a -uasiconformal mapping. Then T f maps W L log α L loc Ω 2 ) CΩ 2 ) into W L log α L loc Ω ) for > n and T f maps W L log α L loc Ω 2 ) into W L log α L loc Ω ) for n. Moreover the -uasiconformal mappings are the best class of homeomorphisms for these Orlicz-Sobolev spaces if n and α 0 or in the second case n and α 0. Theorem.2. Let f : Ω Ω 2 be a homeomorphism, and let α R. For n we moreover assume that f is differentiable a.e. Suppose that T f maps W L log α LΩ 2 ) into W L log α LΩ ) continuously, that is Du f L log α LΩ ) C Du L log α LΩ 2 ) for every u W L log α LΩ 2 ) CΩ 2 ). Then f is a -uasiconfomal mapping. It follows that also in the remaining cases < n and α > 0 or > n and α < 0 we get that the morphisms of W L log α L spaces are subclass of -uasiconformal mappings. On the other hand for each < n and α > 0 or for each > n and α < we give an explicit construction of -uasiconformal mapping f and a function u W L log α L C such that u f / W L log α L loc. Thus an analogy of Theorem. does not hold for these values of parameters and the exact description of the class of morphisms must be different. We would like to know if the similar examples can be constructed also for > n and < α < 0. For the proof of the Theorem.2 we use the usual approach. We construct a suitable test functions in the small neighborhood of the point x and after passing to the limit we use a Lebesgue density type theorem to conclude that the derivative satisfies.). We believe that this Lebesgue density theorem for Orlicz functions is of independent interest and may find application elsewhere. Theorem.3. Suppose that Φ is a Young function and let f L Φ Ω) be nonnegative. Then fχ Bx,r) L Φ.2) lim inf fx) for almost every x Ω. r 0+ χ Bx,r) L Φ

3 COMPOSITION AND ORLICZ-SOBOLEV SPACES 3 If we moreover assume that our Φ satisfies.3) Φab) CΦa)Φb) for every a, b 0, then.4) lim r 0+ fχ Bx,r) L Φ χ Bx,r) L Φ = fx) for almost every x Ω. Surprisingly we cannot have.4) for general Young functions, because the term χ Bx,r) does not necessarily scale well for small r. For α < 0 we construct a function f L log α L such that the limit in.4) is infinite everywhere. The additional condition.3) is the so called -condition and it is known to be important for other properties in the theory of Orlicz spaces see [8]). This paper is organized as follows. In Section 2 we recall some basic properties of uasiconformal mappings and Orlicz spaces. In Section 3 we prove Theorem.3 and we also give a simple counterexample to such a statement for L log α L spaces for α < 0. We prove a general version of Theorem. in Section 4 and general version of Theorem.2 in Section 5. Finally in Section 6 we construct examples showing that -uasiconformal mappings do not map W L log α L to W L log α L for all values of and α. 2. Preliminaries We use the usual convention that C denotes a generic positive constant whose exact value may change from line to line. For two functions g, h : Ω [0, ) we write g h on I, if there is C > 0 such that gx) Chx) for every x Ω. If g h and h g, we write g h. For a function h : Ω R we denote by supp h its support. By A Ω we denote the fact that the closure of A lies inside Ω, i.e. A Ω. The Lebesgue measure of a set A is denoted by L n A) or for short A. 2.. Orlicz spaces. A function Φ : R + R + is a Young function if Φ0) = 0, Φ is increasing and convex. Denote by L Φ A) the corresponding Orlicz space with Young function Φ on a set A with measure L n. This space is euipped with the Luxemburg norm 2.) f L Φ A) = inf{λ > 0 : Φ fx) /λ) dx }. For and α R we denote by L log α LA) the Orlicz space with a Young function such that Φt) lim t t log α t =. For an introduction to Orlicz spaces see e.g. [8]. We define the Orlicz-Sobolev space W L Φ A) as the set euipped with the norm W L Φ A) := {u: u, Du L Φ A)} u W L Φ A) := u L Φ A) + Du L Φ A), A

4 4 STANISLAV HENCL AND LUDĚK KLEPRLÍK where Du is the weak derivative of u. Let Φ be a Young function and let us define 2.2) Ht) = Φ t ) = The standard computation gives us 2.3) χ E L Φ = Φ t ) ). Φ E ) = H E ) for any measurable set E Ω. We say that a function Φ satisfies the 2 -condition, if there is C > such that Φ2t) C Φt) whenever t 0. Analogously we say that a function Φ satisfies the 2 -condition if there is C > 2 such that Φ2t) C Φt) whenever t 0. It is not difficult to show that if Φ satisfies 2 condition, then 2.4) h k k L Φ Ω) 0 Φ h k ) dx k On -uasiconformal mapping. We will need the following version of the derivative of composed function see [3, Theorem.3] for special choice p = ). Theorem 2.. Let and let f : Ω Ω 2 be a homeomorphism of finite - distortion. Then the operator T f is continuous from W, loc Ω 2) CΩ 2 ) to W, loc Ω ) for > n and continuous from W, loc Ω 2) to W, loc Ω ) for n. Moreover for every u W, loc Ω 2) we have 2.5) Du f)x) = Du fx) ) Dfx) for a.e. x Ω if we use the convention that Du fx) ) 0 = 0 even if Du does not exist or it is infinity at fx). It is easy to see from the definition of -uasiconformal mappings that 2.6) Dfx) K J f x) K Dfx) n and therefore each such a map lies in W, for > n. By [3] we know that - uasiconformal mapping f for < n satisfies Luzin N ) condition, i.e. f maps sets of zero measure onto sets of zero measure. Therefore for < n it cannot happen that J f = 0 on a set of positive measure and we can use 2.6) to obtain L. Df It is also well-known that each uasiconformal mappings has better integrability see e.g. [5]). Theorem 2.2. Let Ω be an open set. Suppose that f is n-uasiconformal mapping on Ω. Then there exist p > n and r > 0 such that Df p L loc Ω) and L Df r loc Ω). Ω

5 COMPOSITION AND ORLICZ-SOBOLEV SPACES Volume derivative. Let us denote the volume derivative by f vx) = lim r 0 fbx, r)) Bx, r) We shall need the following connection between f v and the Jacobian of f [7, Theorem 24.2 and Theorem 24.4]. Theorem 2.3. Let f : Ω R n be a homeomorphism. Then f v is a measurable function and f v < almost everywhere. Moreover f vx) = J f x) for every point x where f is differentiable Area formula. We will use the well known area formula for homeomorphisms in W,, loc Ω). It is known that each f Wloc Ω) is approximatively differentiable almost everywhere [2, Thm 3..4] and that the set of approximative differentiability can be exhausted up to a set of measure zero by sets the restriction to which of f is Lipschitz [2, Thm 3..8]. Hence we can decompose Ω into pairwise disjoint sets 2.7) Ω = Z such that Z = 0 and f Ωi is Lipschitz. Let f W, loc Ω; Rn ) be a homeomorphism and let B Ω be a Borel set. Let η be a nonnegative Borel measurable function on R n. Without any additional assumption we have 2.8) ηfx)) J f x) dx ηy) dy. B k= Ω k fb) This follows from the area formula for Lipschitz mappings and 2.7). 3. Lebesgue density theorem for Orlicz spaces Let us note that a Young function such that clearly satisfies.3) since Φt) t n log α e + t) for some α 0 loge + ab) log e + a)e + b) ) 2 loge + a) loge + b) for every a, b 0. Lemma 3.. Suppose that a Young function Φ satisfies.3). Then H h L Φ Ω)) C Φ hx) ) dx for every function h L Φ Ω). supp h Proof. Let us denote λ = h L Φ Ω). By the definition of the Luxemburg norm and.3) we obtain hx) ) 3.) = Φ dx C Φ hx) )Φ dx. λ λ) Ω supp h.

6 6 STANISLAV HENCL AND LUDĚK KLEPRLÍK Using 2.2) and 3.) we get Hλ) = Φ λ ) C supp h Φ hx) ) dx. Proof of Theorem.3. Let us first prove.2) for arbitrary Φ. Let us fix > 0, Q, and x Ω such that x is a density point of a set Using 2.3) it is easy to estimate fχ Bx,r) L Φ χ Bx,r) L Φ M := {y Ω : fy) > }. χ Bx,r) M L Φ χ Bx,r) L Φ Since Φ is a concave function we obtain It follows that Φ ) = Bx,r) M Φ lim inf r 0+ fχ Bx,r) L Φ χ Bx,r) L Φ Bx,r) Bx,r) M = χ Bx,r) M L Φ χ Bx,r) L Φ ) Bx,r) lim inf r 0+ Bx, r) M Bx, r) = Φ ) Bx 0,r) Φ ). Bx,r) M Bx,r) Bx,r) M Φ Bx,r) ). Almost every point x M is a density point and we have only countably many sets M. Therefore almost every x Ω is a density point of all sets M such that fx) > and therefore we obtain.2). Now assume that our Φ satisfies.3) and let us prove.4). Let us fix α > 0. We want to show that the measure of the set { S α = x Ω : lim sup fχ } Bx,r) L Φ fx) > 2α χ Bx,r) L Φ r 0+ is zero. It is easy to see that.4) is valid for every continuous function. The - condition implies that our Φ satisfies the 2 -condition see [8, Chapter 2.2]) and therefore continuous functions are dense in L Φ. Hence we can find g continuous such that fx) = gx) + hx) and h L Φ Ω) < ε. Clearly and fχ Bx,r) L Φ χ Bx,r) L Φ fχ Bx,r) L Φ χ Bx,r) L Φ fx) gχ Bx,r) L Φ χ Bx,r) L Φ fx) gχ Bx,r) L Φ χ Bx,r) L Φ Since.4) is valid for g it is easy to see that =. gx) + hχ Bx,r) L Φ χ Bx,r) L Φ gx) hχ Bx,r) L Φ χ Bx,r) L Φ S α N α M α where N α = {x Ω : hx) α} { hχ Bx,r) } L Φ and M α = x Ω : lim sup > α. r 0+ χ Bx,r) L Φ hx) hx).

7 COMPOSITION AND ORLICZ-SOBOLEV SPACES 7 It is easy to estimate the measure of N α by 3.2) N α = Φα) dx Φα) Φα) Nα) Ω Φhx)) dx. It remains to estimate M α. Using a Besicovitch covering Theorem we obtain balls B i r i ) such that M α i B i r i ), i χ Bi r i ) C and hχ B i r i ) L Φ χ Bi r i ) L Φ > α. From 2.3) and the last ineuality we obtain αh B i r i ) ) = α χ Bi r i ) L Φ < hχ Bi r i ) L Φ. Using 2.2) twice and.3) we thus get H hχ Bi r i ) L Φ) > HαH B i r i ) )) = Using Lemma 3. we now obtain that C Φ α )Φ H B i r i ) ) ) = Φ αh B i r i ) ) ) C Φ α ) B ir i ). M α B i r i ) CΦ ) H hχ α Bi r i ) L Φ) i i CΦ ) Φ hx) ) dx CΦ ) α α i B i r i ) Ω Φ hx) ) dx. Using this estimate and 3.2) we may use h L Φ < ε and 2.4) to obtain S α N α + M α ε Counterexample. Example 3.2. Let and α < 0. Then there is f L log α L0, ) such that fχ Bx,r) L lim log α L r 0+ χ Bx,r) L log α L Proof. Let us consider the Young function such that Let us fix 0 < ε < α and set = for every x 0, ). Φt) t log α t) for t 2. r k = 2 2k 2 kα+) k +ε and define the function f : [0, ] R by fx) = 2 2k for x [ j 2 k, j 2 k + r k ], k N and j {0,,..., 2 k }.

8 8 STANISLAV HENCL AND LUDĚK KLEPRLÍK If the two intervals intersect for different k and k 2 then we define f as the bigger number. It is easy to see that f L Φ [0, ]) since Φf) 2 k r k Φ2 2k ) C 2 k 2 2k 2 kα+) 22k log α 2 2k ) = k+ε k <. +ε 0 k= k= Now let us fix x 0, ) and pick a radius r = 2 k 0. It is not difficult to see that Φ t) t log α t for large t and hence we can use 2.3) to get 3.3) χ Bx,r) L Φ = Φ ) C2r) α log 2r Bx,r) C2 k0 α k 0. Let us denote λ = fχ Bx,r) L Φ and we may assume that r is so small that λ <. For k k 0 we have at most C2 k k 0 points of the type j in the interval x r, x + r). 2 k Using definition of Luxemburg norm and α < 0 we thus get = Φ f ) C 2 k k0 r λ k Φ 22 k ) C 2 k0 2 kα λ Bx,r) k +ε λ logα 22 k ) λ k=k 0 k=k 0 C 2 k0 2 kα k +ε λ logα 2 2k ) = C2 k 0 λ k. +ε k=k 0 k=k 0 This ineuality implies an estimate of λ which gives us 3.4) log 22k λ ) C log22k ) for each k k 0. Moreover for each k k 0 we have at least C2 k k 0 points of the type j in the interval 2 k x r, x + r). Further the value 2 2k is attained on each interval [ j, j + r 2 k 2 k k ] on a set of measure at least r k 2 j k r j r k 2. j=k+ Using all these estimates and the definition of Luxemburg norm we get = Φ f ) C 2 r k k0 k k 22 Φ λ Bx,r) 2 ) C 2 k0 2 kα λ k +ε λ logα 22 k k=k 0 k=k 0 C 2 k0 2 kα k +ε λ logα 2 2k ) = C2 k 0 λ k. +ε k=k 0 k=k 0 It follows that fχ Bx,r) L Φ = λ C2 k 0 k ε 0. Now we can use this estimate, 3.3) and ε < α to obtain k= λ ) fχ Bx,r) L Φ lim r 0+ χ Bx,r) L Φ 2 k 0 C lim k 0 k ε 0 2 k α 0 k 0 = C lim k 0 k ε+α 0 =.

9 COMPOSITION AND ORLICZ-SOBOLEV SPACES 9 4. Stability of W L Φ under -uasiconformal mappings Theorem 4.. Let Ω, Ω 2 R n be domains, n and let f W, loc Ω, Ω 2 ) be a -uasiconformal homeomorphism. Suppose that Φ is a Young function such that Φt) = t αt) where 4.) αt) i) α is non-decreasing, ii) lim = 0 for every δ > 0, t t δ iii) αt β ) Cβ)αt) for every β and t. Then the operator T f is continuous from W L Φ loc Ω 2) into W L Φ loc Ω ), i.e. for every open set A Ω we have 4.2) DT f u L Φ A) C Du L Φ fa)) for every u W L Φ Ω 2 ) CΩ 2 ) or for every u W L Φ Ω 2 ) for = n. Proof. Let u W L Φ loc Ω 2) CΩ 2 ) and A Ω. By 4.) i) we know u W, loc Ω 2) and therefore we may use Theorem 2. to conclude that u f W, loc Ω ) and Du f) = Du) f ) Df. To obtain 4.2) it is enough to show that the modular of Du f) is bounded for each u such that Du L Φ. We have Du f α Du f ) A 4.3) Dufx)) Dfx) α Dufx)) Dfx) ) dx. A From Subsection 2.2 we know that there is p > such that f W,p A, R n ). Let us fix < s < p and we will divide integral into two integrals over sets U = {x A : Dufx)) Dfx) s } and V = {x A : Dufx)) Dfx) s }. We use the definition of -uasiconformal mappings, 4.) and the area formula 2.8) to bound the left-hand side of 4.3) by Dufx)) Dfx) α Dufx)) Dfx) ) dx+ U + Dufx)) Dfx) α Dufx)) Dfx) ) dx V Dufx)) K J f x) α ) Dufx)) s s dx + Dfx) s α ) Dfx) s dx U V CK + Duy) α Duy) ) ) dy + C Dfx) p + ) dx. fa) and the result follows. A

10 0 STANISLAV HENCL AND LUDĚK KLEPRLÍK For n and α < 0 we do not know at the beginning that u f W, loc. Analogously to [8] we need to construct some approximation seuence with the help of the following lemma. Lemma 4.2. Let B R n be an open ball and suppose that u W, 3B). Then for all Lebesgue points x, y B of function u we have ux) uy) Cn) x y M Du x) + M Du y) ), where Mhx) denotes the Hardy-Littlewood maximal operator of h : 3B R Mhx) = L n Bx 0, r) ) hx) dx. sup Bx 0,r) 3B Bx 0,r) Lemma 4.3. Let L Φ A) be an Orlicz space where Φ satisfies 2 and 2 condition and let h L Φ Ω). Then we have lim Φλ)Ln {x : Mhx) > λ} ) = 0. λ Proof. From [4] we know that the maximal operator M is continuous from L Φ to L Φ and hence we get Φλ)L n {Mh > λ} ) = Φλ) ΦMh) 0. {Mh>λ} {Mh>λ} Lemma 4.4. Let Φ be an Young function satisfying 2 and 2 condition and let u W L Φ Bx 0, 3r)). There is a seuence of functions u k with Lipschitz constant Ck and seuence of measurable sets F k such that F k {u = u k }, F k F k+ lim k Ln Bx 0, r) \ F k ) = 0 and u k k u in W L Φ Bx 0, r)). Proof. Let B = Bx 0, r) and for k > 0 we set F k = {x B : M Du ) k and u k} {x : x is Lebesgue point of function u}. It is easy to see that L n B \ F k ) k 0. From Lemma 4.2 we obtain, that the mapping u is Lipschitz continuous on F k. By the classical McShane extension theorem there exists a function u k : R n R with Lipschitz constant Ck such that u k = u on F k and u k. Since u k is Lipschitz function, there exists a derivative almost everywhere and we can estimate u k Ck. Additionally for this extension we have u k = Du almost everywhere on F k. Indeed, if x is a density point of F k, where the derivative of u k exists and the approximative differential of u is eual to the weak derivative, then it is not difficult to show that u k x) = Dux). First we prove, that the functions u k converge to Du in L Φ B): Φ u k Du ) = Φ u k Du ) C Φ u k ) + C Φ Du ) B B\F k B\F k B\F k CL n B \ F k )Φ k) + C Φ Du ) k 0, B\F k

11 COMPOSITION AND ORLICZ-SOBOLEV SPACES where we have used Lemma 4.3 which together with Chebychev s ineuality easily implies Φk)L n B \ F k ) k 0. Now we want to show that u k converges to u in L Φ B). Φ u k u ) = Φ u k u ) C Φ u k ) + C Φ u ) B B\F k B\F k B\F k C Φ k) + C Φ u ) k 0, B\F k B\F k where we have again used the estimate Φk)L n B \ F k ) k 0. Theorem 4.5. Let Ω, Ω 2 R n be domains, < n and let f W, loc Ω, Ω 2 ) be a -uasiconformal homeomorphism. Suppose that Φ is a Young function such that Φt) = t αt) where 4.4) i) α is non-increasing, ii) lim αt)t δ = for every δ > 0, t iii) αt β ) Cβ)αt) for every β and t. Then the operator T f is continuous from W L Φ loc Ω 2) into W L Φ loc Ω ), i.e. for every open set A Ω we have 4.5) DT f u L Φ A) C Du L Φ fa)) for every u W L Φ Ω 2 ). Proof. Let A Ω 2 be an arbitrary. Suppose that we already know that u f W, loc and that 4.6) Du f) = Du) f ) Df holds. Then we can estimate the modular by Du f α Du f ) A 4.7) ) Du fx) Dfx) α ) ) Du fx) Dfx) dx. A From Subsection 2.2 we know that there is r > 0 such that Df r 0 < s < r and define U = V = { x A : Dufx)) { x A : Dufx)) Dfx) +s Dfx) +s } }. and L loc. Let us fix

12 2 STANISLAV HENCL AND LUDĚK KLEPRLÍK We use the definition of -uasiconformal mappings, 4.4) iii), ii) and the area formula 2.8) to bound the left-hand side of 4.7) by ) Du fx) Dfx) α ) ) Du fx) Dfx) dx+ 4.8) U + V U CK Du fx) ) Dfx) α Du fx) ) Dfx) ) dx Du fx) ) K J f x) α Du fx) ) ) +s dx + + Duy) α Duy) )) dy + C fa) A V Df α ) Df s s ) Df +. r and the result follows once we verify u f W, loc and 4.6). If < < n we obtain from Subsection 2.2 that Df > C and thus we can use the definition of -uasiconformal mapping to obtain that f is also -uasiconformal. We know that u W, loc Ω 2) and hence Theorem 2. implies u f W, loc Ω ) and 4.6). It remains to treat the case = n. It is not difficult to see that our Φ satisfies 2 and 2 condition. To obtain 4.6) we will approximate u using the previous lemma. Let x 0 Ω and fix a ball B and r > 0 such that 3B Ω 2 and f Bx 0, r) ) B. We have to show that u f W, Bx 0, r) ). By applying Lemma 4.4 we find a seuence u k of functions with Lipschitz constant Ck and a seuence of measurable sets F k B such that u k = u on F k, F k F k+ and lim k L n F k ) = L n B). Set g j = u j f for each j N. Since u j are Lipschitz functions, we obtain from Theorem 2., that g j W, Bx 0, r) ). We will show that g j is a Cauchy seuence in L Φ Bx 0, r), R n). Let v be a Lipschitz function. Then 4.6) holds and thanks to 4.7) and 4.8) we get Dv f) L Φ Ω ) C Dv L Φ Ω 2 ). If we apply this estimate to the function v = u j u k, we easily get, that the seuence Du j f) = Dg j is Cauchy in L Φ Bx 0, r), R n). Hence there exists a mapping h L Φ Bx 0, r), R n) such that 4.9) Dg j j h in L Φ Bx 0, r), R n). By [3] we know that our mapping f satisfies Luzin N ) condition, i.e. f maps sets of zero measure onto sets of zero measure. Since L n B \ F j ) converge to 0, we obtain, that sets A j := Bx 0, r) f F j ) satisfy lim j Ln A j ) =L n Bx 0, r) f ) ) F j = L Bx n 0, r) \ f ) ) B \ F j = L n Bx 0, r) ). j= Hence we can find j 0 such that L n A j0 ) Ln Bx 2 0, r) ). From the definition of g j we have g j x) = u fx) for all x A j0, and hence g j x) g i x) = 0 on A j0 for all j=

13 COMPOSITION AND ORLICZ-SOBOLEV SPACES 3 i, j j 0. Denote g = g i g j. It follows from the Poincaré ineuality, g Aj0 = 0 and L n A j0 ) Ln Bx 2 0, r) ) that g i g j = g = gx) g Aj0 dx Bx 0,r) Bx 0,r) Bx 0,r) Cn)r Cn, r) Bx 0,r) gx) g Bx0,r) dx + L n Bx 0, r) ) g Aj0 g Bx0,r) Bx 0,r) Bx 0,r) g + Ln Bx 0, r) ) gx) g L n Bx0,r) dx A j0 ) A j0 g = Cn, r) g j g i. Bx 0,r) Since { g j } is a Cauchy seuence in L Bx 0, r), R n) we obtain that {g j } is a Cauchy seuence in L Bx 0, r) ). And due to convergence of g j to u f in points of j= A j, i.e. almost everywhere, we have g j u f in L Bx 0, r) ). The definition of the weak derivative gives us g j x)φx) = Bx 0,r) Bx 0,r) g j x) φx) for each function φ C Bx 0, r)) with compact support. Since g j h in L Bx 0, r), R n) and g j u f in L Bx 0, r) ), by passing j to infinity we get 4.0) hx)φx) = u fx) φx). Bx 0,r) Bx 0,r) This means, that h is the weak gradient of u f on Bx 0, r) and hence u f W, loc. It remains to show that the familiar formula 4.6) holds a.e. in Ω. Our f satisfies Luzin N ) condition and hence u k fx) ) is well defined a.e. in Bx0, r). Passing to a subseuence still denoted as u k ) we may assume that u k x) Dux) on B\N, where N is a Borel measurable set of zero measure. It easily follows that 4.) u k fx) ) Dfx) k Du fx) ) Dfx) on Bx 0, r) \ f N), i.e. almost everywhere. From 4.9) and 4.0) we know that Du k f) = u k f Df converges to Du f). The condition 4.6) a.e. now follows easily. Remark 4.6. For > n it is not necessary to assume iii) in 4.). In this case we know that Df < C and hence we may estimate 4.3) by 4.2) Dufx)) Dfx) α C Dufx)) ) dx and we can finish similarly to the estimate on U. Analogously for < < n it is not necessary to assume iii) in 4.4). We know that Df > C and α is nonincreasing so we can again estimate by 4.2). Proof of Theorem.. The result follows from Theorem 4. and Theorem 4.5.

14 4 STANISLAV HENCL AND LUDĚK KLEPRLÍK 5. Necessity of -Quasiconformal mappings In this section we will use ideas of Gold stein, Gurov and Romanov [5]. They proved that a homeomorphism F : Ω R n which induces a bounded operator from W, Ω 2 ) to W, Ω ) is a -uasiconformal mapping see [5] for details and [6] for history of similar problems). Lemma 5.. Suppose that a homeomorphism f : Ω Ω 2 induces the operator T f : W L Φ Ω 2 ) W L Φ Ω ), then f is in W L Φ loc Ω ). If we moreover assume that L Φ loc Ω ) is embedded into L p loc Ω ) for some p > n then f is differentiable a.e. Proof. Fix R > 0. Mapping f is a homeomorphism and therefore the set A R := {x Ω : fx) B0, R)} = f B0, R)) is open. Fix i n. Plainly there is a Lipschitz function u : Ω 2 R such that { x i for x Ω 2, x < R ux) = 0 for x Ω 2, x > R +. Hence u W, loc Ω 2) W L Φ loc Ω 2) implies T f u) = u f W L Φ loc Ω ). If fx) < R then u f = f i x) and thus f i x) W L Φ A R ). It is well known that each homeomorphism in the Sobolev space W,p loc is differentiable a.e. if p > n see e.g.[7]). From the embedding of L Φ loc into Lp loc we thus obtain that f is differentiable a.e. In the proof of Theorem 5.3 we will need the following elementary lemma [2, Lemma 3.5]. Lemma 5.2. Let f : Ω R n be a continuous mapping and G R k. Suppose that {K y } y G is a family of pairwise disjoint compact sets such that K y fω). Then L n f K y )) = 0 for all y G except possibly a countable subset of G. Theorem 5.3. Let and suppose that Φ is a Young function such that 5.) lim inf s Φ s) Φ Ks) CK for every K > 0. Suppose that a homeomorphism f : Ω Ω 2 induces the bounded operator T f : W L Φ Ω 2 ) W L Φ Ω ), then 5.2) Df j x 0 ) C f vx 0 ) for almost all x 0 Ω. If we moreover assume that f is differentiable a.e. then f is -uasiconformal. Proof. By Theorem 2.3 we know that f vx) < a.e. Fix ε > 0 and a point x 0 Ω such that f vx 0 ) <. There is r 0 such that for all r 0, r 0 ) we have 5.3) fbx 0, 2r)) f vx 0 ) + ε) Bx 0, 2r) = f vx 0 ) + ε)2 n Bx 0, r). Set M = f vx 0 ) + ε)2 n. Let us call a cube Q h-regular if all its edges are parallel to the coordinate axes, the length of the edge is h and every vertex has the form

15 COMPOSITION AND ORLICZ-SOBOLEV SPACES 5 [k h, k 2 h,..., k n h] where k, k 2... k n are integers. Fix r < r 0 and choose h > 0 such that h < 2 n dist f Sx 0, 2r) ), f Sx 0, r) )). Let A be the union of all h-regular cubes Q such that Q fbx 0, r)). It is evident that fbx 0, r)) A fbx 0, 2r)). Fix j {,..., n} and let us focus on the j-th coordinate. Denote the hyperplanes x j = th by L t. The hyperplanes L m m is an integer) divide R n into the layers Z m = {x R n : mh < x j < m + )h}. Put A m = Z m A. For every A m we construct three functions : ψ m, = x j mh, ψ m,2 = m + )h x j, ψ m,3 = h 2 distp jx), P j A m )). Here P j : R n R n j functions is the orthogonal projection of R n onto R n j. Consider the ψ m = max{0, min{ψ m,, ψ m,2, ψ m,3 }} and ψ = m Put E = {x G : ψx) is not differentiable at the point x}. definition of ψ that ) suppψ) fbx 0, 2r)); 2) ψ is Lipschitz with constant ; 3) ψ W L Φ fω)); 4) ψ is differentiable almost everywhere; ψ m. It follows from the 5) ψx) = ±x j + const in all components of the set fbx 0, r)) \ E. The set E fbx 0, r)) belongs to a finite union of hyperplanes L t, L t2,..., L ts where 2t i is an integer. By Lemma 5.2 for almost all small translations τ y parallel to the axis x j we have f τ y i= L i 2 ) fbx 0, r))) = 0. Thus we can assume without loss of generality that 5.4) f E fbx 0, r))) = 0. Otherwise it is possible to change the j-th coordinate of the point [0, 0,..., 0] at the beginning of the construction of ψ. By the assumption of the theorem T f ψ) = ψ f W L Φ Ω ). It follows from 5) and 5.4) that ψ f)x) = ±f j x) + const for almost all x Bx 0, r). This fact and the continuity of T f give us Df j L Φ Bx 0,r)) = Dψ f) L Φ Bx 0,r)) C Dψ L Φ fbx 0,2r))) C L Φ fbx 0,2r))),

16 6 STANISLAV HENCL AND LUDĚK KLEPRLÍK because ψ is Lipschitz with constant and supported in fbx 0, 2r)). Hence we can use 2.3), 5.3) and 5.) to obtain lim inf r 0 + Theorem.3 now gives us Df j LΦ Bx 0,r)) L Φ Bx 0,r)) C lim inf r 0 + LΦ fbx 0,2r))) L Φ Bx 0,r)) Bx 0,r) ) Φ C lim inf r 0 + Φ fbx 0,2r)) ) Bx 0,r) ) Φ C lim inf r 0 + Φ ) CM M Bx 0,r). Df j x 0 ) C M = Cf v x 0 ) + ε) for almost all x 0 Ω and by letting ε 0 we obtain 5.2). If we know that f is differentiable a.e. we may use Theorem 2.3 to conclude that f is -uasiconformal. Proof of Theorem.2. We know that Φt) t log α e + t) for large t and thus for large values of s. Therefore we obtain lim s Φ s) Φ Ks) CK Φ s) s log α e + s) loge + s) lim s loge + Ks) ) α = CK and the statement now follows easily from Theorem 5.3. The reuirement that f is differentiable a.e. for > n is verified by Theorem 5. and for n we have assumed it. 6. Construction of examples In the theory of n-uasiconformal mappings or their generalization one often uses a radial stretching fx) = x ρ x ) as a counterexample. This f maps spheres of x radius r to spheres of radius ρr) or cubes to cubes if. denotes maximum norm). However these maps are too symmetric and thus not critical for -uasiconformal maps. Instead we need to use mappings that map rectangles to rectangles and are inspired by some construction from [0, Section 5]. 6.. Canonical transformation. If c R n, a, b > 0, we use the notation Qc, a, b) := [c a, c + a] [c n a, c n + a] [c n b, c n + b]. for the interval with center at c and halfedges a in the first n coordinates and b in the last coordinate. If Q = Qc, a, b), the affine mapping ϕ Q y) = c + ay,..., c n + ay n, c n + by n ) is called the canonical parametrization of the interval Q. Let P, P be concentric intervals, P = Qc, a, b), P = Qc, a, b ), where 0 < a < a and 0 < b < b. We set ϕ P,P t, y) = t)ϕ P y) + tϕ P y), t [0, ], y [, ]n.

17 COMPOSITION AND ORLICZ-SOBOLEV SPACES 7 This mapping is called the canonical parametrization of the rectangular annulus P \ P, where P is the interior of P. Now, we consider two rectangular annuli, P \P, and P \ P, where P = Qc, a, b), P = Qc, a, b ), P = Q c, ã, b) and P = Q c, ã, b ), The mapping h = ϕ P, P ϕ P,P ) is called the canonical transformation of P \ P onto P \ P. A B P B P A h P P Fig.. The canonical transformation of P \ P onto P \ P for n = 2 We will need the estimate of the derivate of h on P \ P. Let us assume that 6.) a C 0 a, ã C 0 ã, b C 0 b and b C 0 b. It can be computed see [0, Section 5] for details) that {ã 6.2) Dhx) max a, b b } and J b h x) ã ) n b b b a b b for a.e. x in parts A of Fig. and similarly we get that 6.3) Dhx) max {n 2)ã a, ã ã a a, b } and J b h x) ã ) n 2 ã ã b a a a b for a.e. x in parts B of Fig.. All the constants involved depend only on the dimension n and the constant C 0 from 6.). Moreover we can easily estimate the volume of these sets as 6.4) L n A) a ) n b b) and L n B) a ) n 2 a a)b. Example 6.. Let < n and α > 0. Then there is a -uasiconformal homeomorphism f W,, ) n,, ) n ) such that f / W L log α L loc. It follows that the composition with the identity mapping ux) = x satisfies u f / W L log α L loc. Proof. If = then we set fx, x 2,..., x n ) = g x ) sgn x /g), x 2,..., x n ), where gs) = s 0 t log + α 2 t 2 It is easy to see that f is -uasiconformal and Df / L log α L loc. Suppose now that < < n. Set 6.5) β =, γ = n dt. a δ = α 2, ζ = + α δ, η = n δ ) n + ζ

18 8 STANISLAV HENCL AND LUDĚK KLEPRLÍK With the help of 6.5) it is not difficult to verify that 6.6) Let us set a k = k + ) γ log ζ e + k), b k = β ) γn ) = 0, δ ) + ζ η)n ) = 0, δ + ζn ) α =. k + ) β, ã k = log η e + k) and b k = log δ e + k). Our mapping f will be defined as the corresponding canonical transformation from P k := Q0, a k, b k ) \ Q0, a k+, b k+ ) onto Q0, ã k, b k ) \ Q0, ã k+, b k+ ) for every k N 0. It is easy to check that f is a homeomorphism, absolutely continuous on almost all lines parallel to coordinate axes and differentiable a.e. To get our conclusion it is now enough to show that the corresponding integrals of the derivative are finite or infinite. Clearly and k+) ω log η e+k) k+2) ω log η e+k+) k+) ω+ log η e+k) log ω e+k) for every ω > 0. From 6.5) we obtain that γ < β and hence we can k+) log ω+ e+k) use 6.2) and 6.3) to estimate k + )β k + ) γ ) n 6.7) Dfx) log δ e + k) and Jf k + ) β log η ζ e + k) log δ+ e + k) log ω e+k+) for a.e. x P k. Note that the value of constant C 0 in 6.) does not depend on k and thus the constants in and above are independent of k. Now we may use 6.6) to obtain 6.8) K x) = Dfx) J f x) From 6.4) we know that 6.9) L n P k ) k + ) β ) γn ) log δ )+ζ η)n ) e + k) =. k + ) n )γ+β+ log ζn ) e + k) and thus we use 6.7) and 6.6) to obtain Df dx Df p k + ) β dx Q0,a,b ) k=0 P k k + ) k=0 n )γ+β+ log ζn ) e + k) log δ e + k) k + ) log +α e + k) <. k=0 It follows that f W, and by 6.8) we know that f is -uasiconformal mapping. Note that in 6.7) we have not only on both parts A and B in Fig.) but also on a set of measure comparable to 6.9) on part A). Hence analogously as above we may use 6.6) to get Df log α e + Df ) log α e + k) k N 0 k + ) log δ+ζn ) e + k) = k + ) loge + k) =. k N 0 Q0,a,b )

19 COMPOSITION AND ORLICZ-SOBOLEV SPACES 9 Example 6.2. Let > n and α <. Then there are -uasiconformal homeomorphism f W,, ) n,, ) n ) and u W L log α L, ) n,, ) n ) such that u f / W L log α L loc. Proof. Set 6.0) A =, B = n +α 2, H = α 2 ), E =, γ =, δ = n and θ = + n. With the help of 6.0) and < α it is not difficult to verify that 6.) Let us set a k = log A e + k), b k = n )γ )δ = 0, H ) An ) + B ) = 0, log B e + k), ã k = An ) B ) + E < 0, E H ) α >. k + ) γ and b k = k + ) δ log H e + k). Our mapping f will be defined as the corresponding canonical transformation from P k := Q0, a k, b k ) \ Q0, a k+, b k+ ) onto P k := Q0, ã k, b k ) \ Q0, ã k+, b k+ ) for every k N 0. It is easy to check that f is a homeomorphism, absolutely continuous on almost all lines parallel to coordinate axes and differentiable a.e. From 6.0) we obtain γ > δ and hence analogously to the previous example we can use 6.2) and 6.3) to estimate 6.2) Dfx) logb+ H e + k) k + ) δ and Jf for a.e. x P k. Now we may use 6.) to obtain 6.3) K x) = Dfx) J f x) By 6.4) we get 6.4) L n P k ) log A e + k) ) n log B+ H e + k) k + ) γ k + ) δ log H ) An )+B ) e + k) k + ) δ ) γn ) =. k + ) log +n )A+B e + k) and thus we use 6.2) and 6.) to obtain Df dx k + ) log +n )A+B e + k) Q0,a,b ) k=0 log B+ H) e + k) k + ) δ <. It follows that f W, and by 6.3) we know that f is -uasiconformal mapping. Further let us set ã k = k + ) and θ bk = log E e + k).

20 20 STANISLAV HENCL AND LUDĚK KLEPRLÍK Our mapping u will be defined as the corresponding canonical transformation from P k = Q0, ã k, b k ) \ Q0, ã k+, b k+ ) onto Pk := Q0, ã k, bk ) \ Q0, ã k+, bk+ ) for every k N 0. Since γ θ < δ we can use 6.2), 6.3) and 6.4) to obtain Dux) k + ) δ log E H e + k) on P k and L n P k ) k + ) +n )γ+δ log H e + k). With the help of 6.) this implies Du log α k + ) δ log α e + k) e + Du ) dx Q0,a,b ) k + ) +n )γ+δ log H e + k) log E H) e + k) k=0 k + ) log H+E H) α e + k) <. k=0 It is easy to see that u f is a canonical transformation from P k onto Pk and therefore we can use 6.2) and 6.3) to estimate Du f)x) log B E e + k) a.e. on Pk. Together with 6.4) this implies Du f) log α e + Du f) ) dx Q0,a,b ) k=0 k + ) log +n )A+B e + k) logb E) e + k) log α e + log k). From 6.) we easily see that this sum diverges. References [] Astala K., Iwaniec T. and Martin G., Elliptic partial differential euations and uasiconformal mappings in the plane, Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, [2] Federer H., Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 53 Springer-Verlag, New York, 969 Second edition 996). [3] Farroni F. and Giova R., Quasiconformal mappings and exponentially integrable functions, Studia Mathematica ), [4] Gallardo D., Weighted weak type integral ineualities for the Hardy-Littlewood maximal operator, Israel Journal of Math. 67 no. 989), [5] Gold stein V., Gurov L. and Romanov A., Homeomorphisms that induce monomorphisms of Sobolev spaces, Israel Journal of Math ), [6] Gold stein V., Reshetnyak Yu. G., Quasiconformal Mappings and Sobolev Spaces, Kluwer Academic Publishers. [7] Hencl S., Absolutely continuous functions of several variables and uasiconformal mappings, Z. Anal. Anwendungen 22 no ), [8] Hencl S. and Koskela P., Mappings of finite distortion: composition operator, Ann. Acad. Sci. Fenn. Math ), [9] Hencl S. and Koskela P., Composition of uasiconformal mappings and functions in fractional Sobolev spaces, preprint MATH-KMA-200/348,

21 COMPOSITION AND ORLICZ-SOBOLEV SPACES 2 [0] S. Hencl, P. Koskela and J. Malý, Regularity of the inverse of a Sobolev homeomorphism in space, Proc. Roy. Soc. Edinburgh Sect. 36A no ), [] Hencl S. and Malý J., Jacobians of Sobolev homeomorphisms, Calc. Var. Partial Differential Euations ), [2] Iwaniec T. and Martin G., Geometric function theory and nonlinear analysis, Oxford Mathematical Monographs, Clarendon Press, Oxford 200. [3] Kleprlík L., The zero set of the Jacobian and composition of mappings, preprint MATH-KMA- 200/336, [4] Koskela P., Yang D., Zhou Y., Pointwise Characterization of Besov and Triebel-Lizorkin spaces and uasiconformal mappings, Adv. Math. 226 no. 4 20), [5] Koskela P., Lectures on Quasiconformal and Quasisymmetric Mappings, pkoskela/. [6] Malý J., Absolutely continuous functions of several variables, J. Math. Anal. Appl ), [7] Onninen J., Differentiability of monotone Sobolev functions, Real Anal. Exchange 26 no ), no. 2, [8] Rao M. M., Ren Z. D., Theory of Orlicz spaces, Pure and applied mathematics. [9] Reimann H. M., Functions of bounded mean oscillation and uasiconformal mappings, Comment. Math. Helv ), [20] Rao M. M., Ren Z. D., Theory of Orlicz Spaces, Marcel Dekker, Inc.. [2] Rickman S., Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete 3) [Results in Mathematics and Related Areas 3)], 26. Springer-Verlag, Berlin, 993. [22] Tukia P. and Väisäla J., Quasiconformal extension from dimension n to n +, Ann. of Math. 2) 5 no ), [23] Ziemer W. P., Weakly differentiable functions. Graduate texts in Mathematics, 20, Springer- Verlag. Department of Mathematical Analysis, Charles University, Sokolovská 83, Prague 8, Czech Republic address: hencl@karlin.mff.cuni.cz; kleprlik@karlin.mff.cuni.cz