Lipschitz maps and nets in Euclidean space

Transcription

1 Lipschitz maps ad ets i Euclidea space Curtis T. McMulle 1 April, Itroductio I this paper we discuss the followig three questios. 1. Give a real-valued fuctio f L (R ) with if f(x) > 0, is there a bi-lipschitz homeomorphism φ : R R such that the Jacobia determiat det Dφ = f? 2. Give f L (R ), is there a Lipschitz or quasicoformal vector field with div v = f? 3. Give a separated et Y R, is there a bi-lipschitz map φ : Y Z? Whe = 1 all three questios have a easy positive aswer. I this paper we show that for > 1 the aswer to all three questios is o. We also fid all three questios have positive solutios if the Lipschitz coditio is relaxed to a Hölder coditio. Defiitios. A map φ is bi-lipschitz if there is a costat K such that 1 K < φ(x) φ(x ) x x < K for x x. A set Y R is a et if there is a R such that d(x,y ) < R for every x R ; it is separated if there is a ǫ > 0 such that y y > ǫ > 0 for every pair y y i Y. History. I 1965, J. Moser showed that ay two positive, C volume forms o a compact maifold with the same total mass are related by a diffeomorphism [Mos]. Extesios of this result to other smoothess classes such as C k,α were give i [Rei1] ad [DM]; see also [RY1], [RY2], ad [Ye]. Research partially supported by the NSF Mathematics Subject Classificatio: Primary 26B35, Secodary 20F32. 1

2 Questios (1) ad (2) remaied ope. Questio (3) was posed i Gromov s 1993 book [Gr, p.23], ad popularized by Toledo s review [Tol]. Recetly couterexamples to (1) ad (3) were discovered idepedetly by Burago ad Kleier [BK], ad the author. Here we show the liearized questio (2) ca be settled usig a 1962 result of Orstei ( 2). The couterexample to (2) suggests the right type of f to make a couterexample to (1), as we sketch i 3. This f L is similar to the oe costructed i [BK], to which we refer for a detailed resolutio of (1). I 4 we show questios (1) ad (3) are equivalet, completig the discussio of Lipschitz mappigs. Fially i 5 we show questios (1-3) have positive aswers i the Hölder category. 2 Vector fields We begi with the ifiitesimal form of the problem of costructig a map with prescribed volume distortio. That is, we study the equatio div v = v i x i = f o R, where f is a real-valued fuctio ad div v is the divergece of the vector field v. We will show: Theorem 2.1 For ay > 1 there is a f L (R ) which is ot the divergece of ay Lipschitz, or eve quasicoformal, vector field. Defiitios. Let D = ( / x i ); the the matrix of partial derivatives of a vector field v is give by the outer product ad div v = tr(dv). Similarly, lettig (Dv) ij = v i x j, (D 2 ) ij = 2 x i x j, we have f = tr(d 2 f). A vector field v is quasicoformal if the distributio Sv lies i L, where the coformal strai Sv = 1 2 (Dv + (Dv) ) 1 (tr Dv)I 2

3 is the symmetric, trace-free part of Dv. Explicitly, (Sv) ij = 1 ( vi + v ) j 1 v k. 2 x j x i x k k Ay Lipschitz vector field is quasicoformal. Quasicoformal vector fields with div v L are more geeral tha Lipschitz vector fields, but they provide good models for ifiitesimal bi- Lipschitz maps. For example, v(z) = iz log z is ot Lipschitz, but it geerates a Lipschitz isotopy of the plae (shearig alog circles). Theorem 2.1 states that eve this broader class of quasicoformal vector fields is isufficiet to solve div v = f. (Further discussio of quasicoformal flows ca be foud i [Rei2] ad [Mc2, Appedix A].) Sigular itegral operators. Before provig Theorem 2.1, we metio how it fits ito the geeral theory of sigular itegral operators ad PDE. Suppose f C 0 (R ) ad f = 0. The most straightforward solutio to div v = f is give by v = Du, the gradiet of the solutio to Laplace s equatio u = f. The regularity of v is thus determied by the behavior of the operator Tf = Dv = D 2 1 f. For example v is Lipschitz iff Dv = Tf L. The operator T is a sigular itegral operator of Calderó-Zygmud type, whose kerel is obtaied by differetiatig a fudametal solutio to Laplace s equatio. By the geeral theory of such operators, T seds L p ito L p for 1 < p <, but it does ot preserve L or L 1. I the case at had, where f is i L, oe ca say at most that Dv = Tf BMO with Dv BMO C f (see [St, IV.4.1]). Just as vector fields with Dv L are Lipschitz, those with Dv BMO satisfy the Zygmud coditio v(x + y) + v(x y) 2v(x) v Z = sup x,y R,y 0 y < (see [Mc2, Thm. A.2]). It follows that v has a xlog x modulus of cotiuity, so while v is geerally ot Lipschitz it is Hölder of every expoet α < 1. O the other had, a solutio to div v = f is oly determied up to a volume-preservig vector field w, so aother solutio v + w might be Lipschitz eve if v is ot. 3

4 To hadle the kerel of the divergece operator, oe is lead to argue by duality. Theorem 2.1 the reduces to a problem i L 1, which is settled by the followig: Theorem 2.2 (Orstei) For ay set of liearly idepedet degree m differetial operators o R, P i = α =m a α i α x α, i = 0,...,k, ad ay C > 0, there exists a g C 0 (R ) such that P 0 g 1 > C k P i g 1. 1 See [Or]; we are grateful to E. Stei for this referece. Proof of Theorem 2.1. The proof is by cotradictio. Suppose for every f L (R ) there exists a quasicoformal vector field v such that div v = f. The there is a costat C such that v ca be chose with Sv C f. (2.1) Ideed, let B be the Baach space of quasicoformal vector fields with bouded divergece, equipped with the (pseudo-)orm v B = Sv + div v ; the the divergece map div : B L (R ) is surjective, so (2.1) follows by the ope mappig theorem. We claim (2.1) implies, for ay compactly supported smooth fuctio g, that g 1 1 C Eg 1. Here E deotes the trace-zero part of D 2 ; it satisfies (D 2 g) ij = (Eg) ij + 1 ( g)i ij, (2.2) where I ij = δ ij is the idetity matrix. The mai poit of the proof is the idetity: tr(e(sv)) = E ij (Sv) ji = 1 div v. (2.3) 4

5 To check (2.3), ote that tr((d 2 )(Dv)) = i,j 3 v i x 2 j x i = div v, while so by (2.2) we have 1 tr(( I)(Dv)) = 1 div v; tr(e(dv)) = 1 div v. But E is trace-zero ad symmetric, so tr(e(dv)) = tr(e(sv )) ad we have (2.3). Now give ay g C0 (R ), choose f L such that f = 1 ad g 1 = f g = g f. Choose a quasicoformal vector field with div v = f ad satisfyig (2.1), so Sv C. The g 1 = g div v = g tr(e(sv)) 1 by (2.3). Itegratig by parts gives g tr(e(sv)) = tr((eg)(sv)), so we have g 1 1 Eg 1 Sv 1 C Eg 1. But E ad are liearly idepedet differetial operators, so this iequality cotradicts Orstei s theorem. 3 Maps I this sectio we sketch the costructio of a couterexample to (1). A similar couterexample is give i [BK, Theorem 1.2]. The L 1 couterexamples give by Orstei i [Or] are also similar i spirit. 5

6 For simplicity we will work i R 2. Let T S deote the square of side 1/3 withi the uit square S. Choose f > 0 to be costat o T ad S T, with S f = 1 ad T f = Cover the edges of S ad T with much smaller squares S i, ad redefie f S i as f h i, where h i : S i S is a liear map. See Figure 1; the regios where f > 1 are black. Figure 1. No-realizable desity. Now repeat the costructio alog the edges of each S i, ad iterate j times to obtai f j. As the costructio is iterated, arrage that the ratio betwee the sizes of the squares at levels j ad j + 1 teds to ifiity. The f(x) = lim j f j (x) exists almost everywhere ad is bouded above ad below. We claim f caot be realized as the Jacobia determiat of a bi- Lipschitz homeomorphism. To see this, let K = sup φ(a) φ(b) / a b, where the sup is over just the edges [a,b] of all squares at all levels j. For simplicity, suppose K is achieved o a horizotal edge [a,b] of a square S at level j. Let S i deote the squares at level j + 1 ruig alog [a,b], ad let R = S i be the log, thi rectagular they form. By the triagle iequality, the horizotal edges of R are mapped to almost straight lies stretched by K. Sice area φ(r) = area(r), the height of R is compressed by 1/K. The horizotal edges of most S i are also stretched by K, so the perimeter of some S i is icreased by a factor of at least K/2. 6

7 But most of the area of φ(s i ) is filled by φ(t i ), the image of the black subsquare T i S i. Sice the perimeter of T i is 1/3 that of S i, it is stretched by a factor of about 3K/2 uder φ, cotradictig the defiitio of K. A detailed proof ca be give alog lies similar to those preseted i [BK], to which the reader is referred for a more complete discussio. This couterexample to (1) was motivated for us by the area-modulus iequality area(s) area(t) (3.1) 1 + 4π mod(a) where A is the aulus betwee two disks T S C [Mc1, Lemma 2.17]. This iequality relates coformal distortio to distortio of relative areas. Sice (3.1) comes from the isoperimetric iequality, for a rigorous proof oe is lead to cosider stretchig alog the edges ad stability of geodesics as above. 4 Nets I this sectio we show questios (1) ad (3) are equivalet. I particular, a couterexample to (1) implies a couterexample to (3). Theorem 4.1 The followig two statemets are equivalet: A. Every measurable f > 0 o R with f ad 1/f bouded ca be realized as the Jacobia determiat of a bi-lipschitz map. B. Every separated et Y R is bi-lipschitz to Z. Proof of Theorem 4.1. (B) = (A). Choose a et Y such that uder rescalig, the measure that assigs a δ-mass to each poit of Y accumulates weakly o the measure µ = f(x)dx. By (B) there is a bi-lipschitz map φ : Y Z. Uder suitable rescalig, φ coverges to a bi-lipschitz homeomorphism Φ : R R with Jacobia f. Compare [BK, Lemma 2.1]. (A) = (B). Let Y R be a separated et. Let C y : y Y be the tilig of R determied by the Vorooi cells C y = {x : x y < x y for all y y i Y }. Sice Y is a et, we have supdiam C y <, ad if vol C y > 0 because Y is separated. Let f(x) = 1. (4.1) vol C y y : x C y 7

8 The f ad 1/f are bouded a.e., so (A) provides a bi-lipschitz homeomorphism φ : R R with Jacobia determiat f. Lettig D y = φ(c y ), we have vol φ(d y ) = 1. For z Z let E z deote the uit cube cetered at z. Cosider the relatio R Y Z give by the set of pairs (y,z) such that D y meets E z. Sice diam D y ad diame z are bouded, the distace φ(y) z is also bouded for all (y,z) R. Now thik of the relatio R as a multi-valued map from Y to Z. The for ay fiite set A Y, we have R(A) A. Ideed, the cubes labeled by R(A) cover the cells D y labeled by A, so the iequality follows from the fact that vol D y = vol E z = 1. Similarly, R 1 (B) B for ay fiite set B Z. By the trasfiite form of Hall s marriage theorem [Mir, Thm ], R cotais the graph of a ijective map ψ 1 : Y Z. Similarly, R 1 cotais the graph of a ijective map ψ 2 : Z Y. By the Schröder-Berstei theorem [Hal, 22], R cotais the graph of a bijectio ψ : Y Z. Sice sup ψ(y) φ(y) <, the map ψ : Y Z is bi-lipschitz, provig (B). The proof of (A) = (B) shows that for ay separated et Y, the quality of a bijectio φ : Y Z ca be cotrolled by the quality of a solutio to det Dφ = f, where f is determied by the Vorooi cells as i (4.1). This fact is exploited i the ext sectio. 5 Hölder maps To coclude we show questios (1-3) have positive aswers if we relax the Lipschitz coditio to a Hölder coditio. Defiitio. We say φ : R R is a homogeeous Hölder map if there are costats K 0 ad 0 < α 1 such that for x, y R we have φ(x) φ(y) KR 1 α x y α. (5.1) If φ(x) satisfies (5.1), the so does rφ(x/r) for every r > 0; it is this sese that the Hölder coditio above is homogeeous. If φ ad φ 1 both satisfy (5.1) the we say φ is a homogeeous bi-hölder homeomorphism. Whe α = 1 we obtai the class of bi-lipschitz maps. Note that for ay homogeeous bi-hölder homeomorphism, we have φ(y) y whe y is large. To see this, set x = 0 ad R = y i (5.1). 8

9 We say a map φ : Y Y betwee subsets of R is a homogeeous bi-hölder bijectio if φ ad φ 1 satisfy (5.1) o their respective domais. Theorem 5.1 Fix 1. The: 1. For ay f L (R ) with if f(x) > 0, there is a homogeeous bi- Hölder homeomorphism φ : R R such that vol(φ(e)) = f(x) dx (5.2) for all bouded ope sets E R. 2. For ay f L (R ), there is a vector field v with Zygmud compoets such that div v = f. 3. For ay separated et Y R, there is a homogeeous bi-hölder bijectio ψ : Y Z. Lemma 5.2 Ay radial fuctio f(r) L (R ) with if f > 0 ca be realized as the Jacobia determiat of a radial bi-lipschitz homeomorphism φ(r,θ) = (ψ(r),θ). Proof. Defie ψ : [0, ) [0, ) by The we have ψ(r) = r 0 E detdφ = ψ (r)ψ(r) 1 r 1 s 1 f(s)ds. = f(r). The upper ad lower bouds o f imply ψ(r) r, so by the formula above we have ψ (r) 1. Thus φ is bi-lipschitz. Proof of Theorem 5.1. (2). This statemet follows from the geeral theory of sigular itegral operators, as sketched i 2. Note that a vector field v with Zygmud compoets has x log x modulus of cotiuity ad geerates a flow whose time-oe map is Hölder [Rei2, Prop. 4]. (1). This result is due to Rivière ad Ye. Cosider the tilig of R {0} by the dyadic auli A i = {x : 2 i x 2 i+1 }, i Z. 9

10 After a prelimiary radial Lipschitz map, whose existece is isured by Lemma 5.2, we ca assume A i f = A i 1 for each i. By [RY2, Thm. 2], there exists a homeomorphism φ 0 : A 0 A 0 such that (i) E f(x)dx = vol(φ 0(E)) for ay ope set E A 0 ; (ii) φ 0 (x) = x o A 0 ; ad (iii) K 1 x y 1/α φ(x) φ(y) K x y α, where α > 0, K > 1 deped oly o f + 1/f (compare [RY2, (2.14)]). Sice A i is simply A 0 rescaled by a factor of 2 i, we ca apply this result to obtai homeomorphisms φ i : A i A i satisfyig the volume distortio equatio (5.2) for E A i. The Hölder bouds i (iii) rescale to give the homogeeous bouds (5.1) for φ i ad φ 1 i, so the φ i piece together to produce the desired homogeeous bi-hölder map φ : R R. (3). Let Y R be a separated et. Let C y be the Vorooi cells for Y, ad let E z deote the uit cube cetered at z Z. Defie f(x) = 1/vol(C y ) for x C y as i (4.1). By (1) there exists a homogeeous bi-hölder map φ : R R sedig f(x)dx to the stadard measure o R. Lettig D y = φ(c y ) we have vol D y = 1 ad diamd y = O(1 + y 1 α ), where α is the expoet i (5.1). As i the proof of Theorem 4.1, Hall s marriage theorem provides a bijectio ψ : Y Z such that D y E z wheever ψ(y) = z. Therefore φ(y) ψ(y) C(1 + y 1 α ) (5.3) for some costat C. We claim ψ : Y Z is a homogeeous bi-hölder map. Ideed, give distict poits x,y Y with x, y R, by (5.1) ad (5.3) we have ψ(x) ψ(y) φ(x) φ(y) + φ(x) ψ(x) + φ(y) ψ(y) KR 1 α x y α + 2C(1 + R 1 α ) = O(R 1 α x y α ) sice x y > ǫ > 0 by separatio of Y. This shows ψ satisfies the homogeeous Hölder coditio. To verify the same coditio for ψ 1, we apply the same reasoig to the iverse image cubes F z = φ 1 (E z ). The Hölder coditio o φ 1 gives diam(f z ) = O(1 + z 1 α ), ad sice F z C ψ 1 (z) we have ψ 1 (z) φ 1 (z) C (1 + z 1 α ). 10

11 Thus for distict z,w Z with z, w R we have ψ 1 (z) ψ 1 (w) KR 1 α z w α + 2C (1 + R 1 α ) = O(R 1 α z w α ) sice z w 1. Therefore ψ 1 also satisfies (5.1) ad we are doe. Refereces [BK] D. Burago ad B. Kleier. Separated ets i Euclidea space ad Jacobias of bilipschitz maps. To appear, Geometric ad Fuctioal Aalysis. [DM] B. Dacoroga ad J. Moser. O a partial differetial equatio ivolvig the Jacobia determiat. A. Ist. H. Poicaré Aal. Noliéaire 7(1990), [Gr] M. Gromov. Asymptotic ivariats of ifiite groups. I G. A. Niblo ad M. A. Roller, editors, Geometric Group Theory, volume 2. Cambridge Uiversity Press, [Hal] P. Halmos. Naive Set Theory. Spriger-Verlag, [Mc1] C. McMulle. Complex Dyamics ad Reormalizatio, volume 135 of Aals of Math. Studies. Priceto Uiversity Press, [Mc2] C. McMulle. Reormalizatio ad 3-Maifolds which Fiber over the Circle, volume 142 of Aals of Math. Studies. Priceto Uiversity Press, [Mir] L. Mirsky. Trasversal Theory. Academic Press, [Mos] J. Moser. O the volume elemets o a maifold. Tras. Amer. Math. Soc. 120(1965), [Or] D. Orstei. A o-iequality for differetial operators i the L 1 orm. Arch. Ratioal Mech. Aal. 11(1962), [Rei1] H. M. Reima. Harmoische Fuktio ud Jacobi- Determiate vo Diffeomorphisme. Commet. Math. Helv. 47(1972), [Rei2] H. M. Reima. Ordiary differetial equatios ad quasicoformal mappigs. Ivet. math. 33(1976),

12 [RY1] T. Rivière ad D. Ye. Ue résolutio de l equatio à forme volume prescrite. C. R. Acad. Sci. Paris Sér. I Math. 318(1994), [RY2] T. Rivière ad D. Ye. Resolutios of the prescribed volume form equatio. Noliear Differetial Equatios Appl. 3(1996), [St] E. M. Stei. Harmoic Aalysis. Priceto Uiversity Press, [Tol] D. Toledo. Geometric group theory, Vol. 2: Asymptotic ivariats of ifiite groups by M. Gromov. Bull. Amer. Math. Soc. 33(1996), [Ye] D. Ye. Prescribig the Jacobia determiat i Sobolev spaces. A. Ist. H. Poicaré Aal. Noliéaire 11(1994), Mathematics Departmet, Uiversity of Califoria, Berkeley CA