We extend monotone quasiconformal mappings from dimension n to n+1 while
preserving both monotonicity and quasiconformality. The extension is given
explicitly by an integral operator. In the case n=1 it yields a refinement of
the Beurling-Ahlfors extension.Comment: 9 pages. Added references, edited concluding remark

Kovalev, Leonid V.

Onninen, Jani

Annales Academiae Scientiarum Fennicae. Mathematica

English

arXiv

Journal.fi

An N-dimensional version of the Beurling-Ahlfors extension

We extend monotone quasiconformal mappings from dimension n to n+1 while preserving both monotonicity and quasiconformality. The extension is given explicitly by an integral operator. In the case n=1 it yields a refinement of the Beurling-Ahlfors extension

Kovalev, Leonid V

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Mathematics - Faculty Scholarship Mathematics 4-27-2009 An N-Dimensional Version of the Beurling-Ahlfors Extension Leonid V. Kovalev Syracuse University Jani Onninen Syracuse University Follow this and additional works at: https://surface.syr.edu/mat  Part of the Mathematics Commons Recommended Citation Kovalev, Leonid V. and Onninen, Jani, "An N-Dimensional Version of the Beurling-Ahlfors Extension" (2009). Mathematics - Faculty Scholarship. 49. https://surface.syr.edu/mat/49 This Article is brought to you for free and open access by the Mathematics at SURFACE. It has been accepted for inclusion in Mathematics - Faculty Scholarship by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:0904.3090v2  [math.CV]  27 Apr 2009AN N-DIMENSIONAL VERSION OFTHE BEURLING-AHLFORS EXTENSIONLEONID V. KOVALEV AND JANI ONNINENAbstract. We extend monotone quasiconformal mappings from di-mension n to n + 1 while preserving both monotonicity and quasicon-formality. The extension is given explicitly by an integral operator. Inthe case n = 1 it yields a refinement of the Beurling-Ahlfors extension.1. IntroductionExtension Problem. Given a mapping f : Rn → Rn of class A , find F : Rn+1 →Rn+1 of class A such that the restriction of F to Rn agrees with f .Let us introduce coordinate notation x = (x1, . . . , xn) and f = (f1, . . . , fn).By setting F i = f i for i = 1, . . . , n and Fn+1 = xn+1 one immediately ob-tains a solution to the extension problem for many classes A such as con-tinuous (A = C0), smooth (A = Ck), homeomorphic, diffeomorphic, and(bi-)Lipschitz mappings.When A = QC, the class of quasiconformal mappings, the extensionproblem is much more difficult. It was solved• for n = 1 by Beurling and Ahlfors [4] in 1956,• for n = 2 by Ahlfors [1] in 1964,• for n ≤ 3 by Carleson [8] in 1974, and• for all n ≥ 1 by Tukia and Va¨isa¨la¨ [16] in 1982.The Tukia-Va¨isa¨la¨ extension uses, among other things, Sullivan’s the-ory [15] of deformations of Lipschitz embeddings. Our goal is to give anexplicit extension for a subclass of QC. Quasiconformal mappings can bedefined as orientation-preserving quasisymmetric mappings [11, 17].Definition 1.1. A homeomorphism f : Rn → Rn is quasisymmetric if thereis a homeomorphism η : [0,∞)→ [0,∞) such that(1.1)|f(x)− f(z)||f(y)− f(z)|≤ η(|x− z||y − z|).for x, y, z ∈ Rn, z 6= y.Date: April 17, 2008.2000 Mathematics Subject Classification. Primary 30C65; Secondary 47H05, 47B34.Kovalev was supported by the NSF grant DMS-0913474.Onninen was supported by the NSF grant DMS-0701059.12 LEONID V. KOVALEV AND JANI ONNINENOne can say that quasisymmetry is a three-point condition. But there aretwo subclasses of QC that are defined by two-point conditions, namely bi-Lipschitz class BL and the class of nonconstant delta-monotone mappings [2,Chapter 3]. Recall that a mapping f : Rn → Rn is monotone if(1.2) 〈f(x)− f(y), x− y〉 ≥ 0 for all x, y ∈ Rn.We called f delta-monotone if there exists δ > 0 such that(1.3) 〈f(x)− f(y), x− y〉 ≥ δ|f(x)− f(y)||x− y| for all x, y ∈ Rn.The class of nonconstant delta-monotone mappings is denoted by DM.When we want to specify the value of δ we write that f is δ-monotone.In contrast to the bi-Lipschitz case, the extension problem for the classDM cannot be solved by means of the trivial extension. For example, themapping f(x) = |x|px, p > −1, belongs to DM but its trivial extensiondoes not (unless p = 0).Main Result. Let n ≥ 2. For any mapping f : Rn → Rn of class DMthere exists F : Rn+1 → Rn+1 of class DM such that the restriction of F toRn agrees with f .Our proof is by an explicit construction that can be viewed as an n-dimensional version of the Beurling-Ahlfors extension. Suppose f ∈ DM.Let Rn+1+ = Rn × [0,∞) and(1.4) φ(x) = (2pi)−n2 e−|x|2/2, x ∈ Rn.We define F : Rn+1+ → Rn+1+ byF i(x, t) =∫Rnf i(x+ ty)φ(y) dy i = 1, . . . , n(1.5)Fn+1(x, t) =∫Rn〈f(x+ ty), y〉φ(y) dy(1.6)where x ∈ Rn, t ≥ 0 (see §4 for the convergence of these integrals). Observethat F (x, 0) = (f(x), 0). Furthermore, Fn+1(x, t) ≥ 0 because∫Rn〈f(x+ ty), y〉φ(y) dy =∫Rn〈f(x+ ty)− f(x), y〉φ(y) dy ≥ 0due to the monotonicity of f . Finally, we extend F to Rn+1 by reflectionF i(x, t) = F i(x,−t) i = 1, . . . , n and Fn+1(x, t) = −Fn+1(x,−t).Theorem 1.2. Let n ≥ 2. If f : Rn → Rn is δ-monotone, then F : Rn+1 →Rn+1 is δ1-monotone where δ1 depends only on δ and n. In addition,F : Hn+1 → Hn+1 is bi-Lipschitz in the hyperbolic metric.Here Hn+1 = Rn×(0,∞) and the hyperbolic metric on Hn+1 is |dx|/xn+1.Theorem 1.2 can be also formulated for n = 1, in which case it becomes arefinement of the Beurling-Ahlfors extension theorem.AN N-DIMENSIONAL VERSION OF THE BEURLING-AHLFORS EXTENSION 3Proposition 1.3. If f : R → R is increasing and quasisymmetric, thenF : R2 → R2 is δ1-monotone where δ1 depends only on η in Definition 1.1.Furthermore, F : H2 → H2 is bi-Lipschitz in the hyperbolic metric.Fefferman, Kenig and Pipher [9, Lemma 4.4] proved that F in Propo-sition 1.3 is quasiconformal. Proposition 1.3 was originally proved in [12]using their result. In this paper we give a direct proof.Theorem 1.2 has an application to mappings with a convex potential [7],i.e., those of the form f = ∇u with u convex. The basic properties andexamples of quasiconformal mappings with a convex potential are givenin [13].Corollary 1.4. Suppose that f : Rn → Rn, n ≥ 2, is a K-quasiconformalmapping with a convex potential. Then f can be extended to aK1-quasicon-formal mapping F : Rn+1 → Rn+1 with a convex potential, where K1 de-pends only on K and n.2. PreliminariesLet e1, . . . , en+1 be the standard basis of Rn+1. All vectors are treatedas column vectors. The transpose of a vector v is denoted by vT . We usethe operator norm ‖·‖ for matrices. A Borel measure µ on Rn is doubling ifthere exists Dµ, called the doubling constant of µ, such thatµ(2B) ≤ Dµ µ(B)for all balls B = B(x, r). Here 2B = B(x, 2r).The geometric definition of classQC given in the introduction is equivalentto the following analytic definition [11, 17].Definition 2.1. A homeomorphism f : Rn → Rn (n ≥ 2) is quasiconformalif f ∈W 1,nloc (Rn,Rn) and there exists a constant K such that the differentialmatrix Df(x) satisfies the distortion inequality‖Df(x)‖n ≤ K detDf(x) a.e. in Rn.Delta-monotone mappings also have an analytic definition.Lemma 2.2. Let Ω be a convex domain in Rn, n ≥ 2. Suppose f ∈W 1,1loc (Ω,Rn) is continuous. The following are equivalent:(i) f is δ-monotone in Ω for some δ > 0; that is, (1.3) holds for allx, y ∈ Ω;(ii) there exists δ > 0 such that for a.e. x ∈ Ω the matrix Df(x) satisfiesvTDf(x)v ≥ δ|Df(x)v||v| for every vector v ∈ Rn;(iii) there exists γ > 0 such that for a.e. x ∈ Ω the matrix Df(x) satisfiesvTDf(x)v ≥ γ‖Df(x)‖|v|2 for every vector v ∈ Rn.The constants δ and γ depend only on each other.4 LEONID V. KOVALEV AND JANI ONNINENProof. The equivalence of (i) and (ii), with the same constant δ, was provedin [12, p. 397]. It is obvious that (iii) implies (ii) with δ = γ. It remainsto establish the converse implication (ii) =⇒ (iii). To this end we need thefollowingClaim: if a real square matrix A satisfiesvTAv ≥ δ|Av||v| for every v ∈ Rnthen(2.1) |Av| ≥ c‖A‖|v| c = c(δ) > 0.Although this claim is known, even with a sharp constant [3], we give a prooffor the sake of completeness. It suffices to estimate |Av| from below underthe assumptions that Av 6= 0 and ‖A‖ = 1 = |v|. Let u be a unit vector inRn such that |Au| = 1. Replacing u by −u if necessary we may assume thatuTAv + vTAu ≤ 0. Let λ =√|Av|. On one hand we have(2.2) (λu+ v)TA(λu+ v) ≤ λ2uTAu+ vTAv ≤ λ2 + λ2 = 2λ2.On the other hand(2.3) (λu+ v)TA(λu+ v) ≥ δ|λAu+Av||λu + v| ≥ δ(λ − λ2)(1− λ).Combining (2.2) and (2.3) we obtain 2λ ≥ δ(1 − λ)2, henceλ ≥ δ−1 + 1−√(δ−1 + 1)2 − 1 > 0.This proves the claim. 3. Delta-monotone mappings and doubling measuresThe following result shows that DM ⊂ QC. In particular, f ∈ DMimplies that f is a continuous Sobolev mapping, and therefore (ii)–(iii) ofLemma 2.2 hold.Proposition 3.1. [12, Theorem 6] Every nonconstant δ-monotone mappingis η-quasisymmetric where η depends only on δ.It is well-known that quasisymmetric mappings are closely related to dou-bling measures [11]. The following lemma is another instance of this relation.Lemma 3.2. For any nonconstant δ-monotone mapping f : Rn → Rn (n ≥2) the measure µ = ‖Df(x)‖ dx is doubling. The doubling constant Dµdepends only on δ and n.Proof. Recall that f is quasisymmetric. Lemma 3.2 in [14] implies the exis-tence of a constant C = C(δ, n) such that(3.1) C−1diam f(B)diamB≤1|B|∫B‖Df‖ dx ≤ Cdiam f(B)diamBfor all balls B ⊂ Rn. Since diam f(2B) ≤ C diam f(B) with C = C(η), thelemma follows. AN N-DIMENSIONAL VERSION OF THE BEURLING-AHLFORS EXTENSION 5Recall that φ : Rn → (0,∞) is the Gaussian kernel (1.4). Let B = B(0, 1)be the open unit ball in Rn.Lemma 3.3. Let µ be a doubling measure in Rn and p ≥ 0. Let Ω be eitherRn or the half space {y : 〈y, ξ〉 ≥ 0} for some ξ ∈ Rn. Then(3.2) C−1µ(B)≤∫Ω|y|pφ(y) dµ(y) ≤ Cµ(B)where the constant C depends only on Dµ, p and n.Proof. We begin by estimating the integral in (3.2) from above as follows∫Rn|y|pφ(y) dµ(y) =∫B|y|pφ(y) dµ(y) +∞∑k=0∫2k<|y|≤2k+1|y|pφ(y) dµ(y),where ∫B|y|pφ(y) dµ(y) ≤ φ(0)µ(B) = (2pi)−n2 µ(B)and ∫2k<|y|≤2k+1|y|pφ(y) dµ(y) ≤ 2p(k+1)(2pi)−n2 e−22k−1µ(B(0, 2k+1))≤ 2p(k+1)(2pi)−n2 e−22k−1Dk+1µ µ(B).Summing over k = 0, 1, 2 . . . we obtain∫Rnφ(y) dµ(y) ≤ C µ(B)where C = C(Dµ, p, n) > 0.We turn to the left side of (3.2). The inequality|y|pφ(y) ≥e−1/22p(2pi)n/2for12≤ |y| ≤ 1implies ∫Ω|y|pφ(y) dµ(y) ≥e−1/22p(2pi)n/2µ(Ω ∩ {1/2 ≤ |y| ≤ 1}).Since µ(Ω ∩ {1/2 ≤ |y| ≤ 1}) ≥ D−3µ µ(B), the left side of (3.2) follows. 4. Proof of main resultsProof of Theorem 1.2. Since f is quasisymmetric by Proposition 3.1, it sat-isfies the growth condition |f(x)| ≤ α|x|p + β for some constants α, β, p, see[11, Theorem 11.3]. Therefore, the integrals (1.5) and (1.6) converge and Fis C∞-smooth in Hn+1. Let γ = γ(δ) > 0 be as in part (iii) of Lemma 2.2.Our first step is to prove that for (x, t) ∈ Hn+1 the matrix B := DF (x, t)satisfies the condition(4.1) wTBw ≥ γ1‖B‖|w|2 for every vector w ∈ Rn+16 LEONID V. KOVALEV AND JANI ONNINENwhere γ1 = γ1(δ, n) > 0. Fix x ∈ Rn and t > 0. We compute the partialderivatives of F at (x, t) ∈ Hn+1 as follows.∂F i∂xj=∫Rnf ij(x+ ty)φ(y)dy, 1 ≤ i, j ≤ n;∂F i∂t=∫Rnn∑j=1f ij(x+ ty)yiφ(y)dy, 1 ≤ i ≤ n;∂Fn+1∂xj=∫Rnn∑i=1f ij(x+ ty)yjφ(y)dy, 1 ≤ j ≤ n;∂Fn+1∂t=∫Rnn∑i=1n∑j=1f ij(x+ ty)yiyjφ(y)dy.To simplify formulas we write A(y) = Df(x + ty) and let B(y) be the(n+ 1)× (n+ 1) matrix written in block form below.(4.2) B(y) = A(y) A(y)yyTA(y) yTA(y)y .With this notation we have(4.3) DF (x, t) =∫RnB(y)φ(y) dy.First we show that the norm of B is dominated by the quantityα :=∫B(0,1)‖A(y)‖ dy.Indeed,‖B‖ ≤∫Rn‖B(y)‖φ(y) dy ≤∫Rn‖A(y)‖(1 + |y|)2φ(y) dy.By Lemma 3.2 the measure µ = ‖A(y)‖ dy is doubling. Applying Lemma 3.3we obtain(4.4) ‖B‖ ≤ Cα, C = C(δ, n).Next we estimate the quadratic form w 7→ wTBw generated by B frombelow. For this we fix a vector w ∈ Rn+1, written as w = v + sen+1 withv ∈ Rn and s ∈ R. It is easy to see thatwTB(y)w = (v + sy)TA(y)(v + sy).AN N-DIMENSIONAL VERSION OF THE BEURLING-AHLFORS EXTENSION 7Let Ω = {y ∈ Rn : 〈v, sy〉 ≥ 0}. ThenwTBw =∫Rn{(v + sy)TA(y)(v + sy)}φ(y) dy≥ γ∫Rn‖A(y)‖|v + sy|2φ(y) dy≥ γ∫Ω‖A(y)‖|v + sy|2φ(y) dy≥ γ|v|2∫Ω‖A(y)‖φ(y) dy + γs2∫Ω‖A(y)‖|y|2φ(y) dy.Applying Lemma 3.3 with µ = ‖A(y)‖ dy we obtain(4.5) wTBw ≥ c αγ(|v|2 + s2) = c αγ|w|2, c = c(δ, n).Combining (4.4) and (4.5) we obtain (4.1) with γ1 = (c/C)γ. By virtueof Lemma 2.2 F is δ1-monotone in the upper half-space Hn+1 where δ1 =δ1(δ, n). By symmetry, F is also δ1-monotone in the lower half-space.To prove that F is δ1-monotone in the entire space Rn+1, we consider twopoints a, b ∈ Rn+1 such that the line segment [a, b] crosses the hyperplaneRn at some point c. We have〈F (a)− F (b), a− b〉 = 〈f(a)− f(c), a− b〉+ 〈F (c) − F (b), a − b〉≥ δ1|F (a)− F (c)||a − b|+ δ1|F (c) − F (b)||a− b|≥ δ1|F (a)− F (b)||a − b|Therefore, F ∈ DM.It remains to show that F : Hn+1 → Hn+1 is bi-Lipschitz in the hyperbolicmetric. Since F ∈ QC and Hn+1 is a geodesic space, it suffices to prove that(4.6) ‖DF (x, t)‖ ≈Fn+1(x, t)t.Here X ≈ Y means that X and Y are comparable, i.e., C−1Y ≤ X ≤CY where C = C(δ, n). It follows from (4.4) and (4.5) that ‖DF (x, t)‖ iscomparable to the integral average of ‖Df‖ over the ball B(x, t). By (3.1)this average is comparable to t−1 diam f(B(x, t)). The quasisymmetry of Fimplies (cf. [11, 11.18])diam f(B(x, t)) ≈ |F (x, t) − F (x, t/2)| ≈ Fn+1(x, t).This proves (4.6). Proof of Proposition 1.3. The proof of Theorem 1.2 also works in the casen = 1 with the following interpretation. Since quasisymmetric mappingson the line need not be absolutely continuous [4], the derivative f ′ mustbe understood in the sense of distributions. In fact, µ := f ′ is a positivedoubling measure with Dµ = Dµ(η) [11, 13.20]. Lemma 3.2 is not needed inthis case. The rest of the proof carries over with γ = 1 and γ1 = γ1(Dµ). 8 LEONID V. KOVALEV AND JANI ONNINENProof of Corollary 1.4. According to [12, Lemma 18], a K-quasiconformalmapping with a convex potential is also δ-monotone with δ = δ(K,n). Let Fbe the δ1-monotone extension of f provided by Theorem 1.2. Since the dif-ferential matrix Df is symmetric, the formulas (4.2) and (4.3) show that DFis symmetric as well. In addition, DF is positive semidefinite by Lemma 2.2.Thus, F = ∇U for some convex function U : Rn+1 → R. 5. Concluding remarksBoth classes QC (quasiconformal) and BL (bi-Lipschitz) are groups undercomposition. However, the class of delta-monotone mappings DM is notclosed under composition (consider the rotation of the complex plane givenby z 7→ eiθz where |θ| < pi/2). Let QCd ⊂ QC be the group generated by BLand DM. In other words, f belongs to QCd if it can be decomposed intobi-Lipschitz and delta-monotone mappings. This should be compared withthe notion of polar factorization of mappings introduced by Brenier [6].Theorem 1.2 together with the trivial extension of bi-Lipschitz mappingsyield a solution to the extension problem for QCd.Corollary 5.1. Let n ≥ 2. For any mapping f : Rn → Rn of class QCdthere exists F : Rn+1 → Rn+1 of class QCd such that the restriction of F toRn agrees with f .It seems likely that QCd is a proper subset of QC. This motivates thefollowing question:Question 5.2. Which quasiconformal mappings are decomposable?Both bi-Lipschitz and delta-monotone mappings take smooth curves intorectifiable curves [2, Theorem 3.11.7]. This is no longer true for their com-position. More precisely, for any 1 < α < 2 one can construct a mappingf : R2 → R2 such that f ∈ QCd and f(R) has Hausdorff dimension at leastα. To this end, one first finds a bi-Lipschitz mapping g : R2 → R2 suchthat g(R) contains a planar Cantor set E of dimension 0 < β < 1 (seeLemma 3.1[5] and the comment after its proof). Second, there is a delta-monotone mapping h : R2 → R2 such that the Hausdorff dimension of h(E)is equal to α (see the construction in [10, Theorem 5]). Finally, let f = h◦g.AcknowledgmentsWe thank Mario Bonk and Jang-Mei Wu for conversations related to thesubject of this paper.References1. L. V. Ahlfors, Extension of quasiconformal mappings from two to three dimensions,Proc. Nat. Acad. Sci. U.S.A. 51 1964 768–771.2. K. Astala, T. Iwaniec, and G. J. Martin, Elliptic partial differential equations andquasiconformal mappings in the plane, Princeton University Press, Princeton, 2009.AN N-DIMENSIONAL VERSION OF THE BEURLING-AHLFORS EXTENSION 93. K. Astala, T. Iwaniec, and G. J. Martin, Monotone maps of Rn are quasiconformal,Methods Appl. Anal. 15 (2008) no. 1, 31–38.4. A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal map-pings, Acta Math. 96 (1956), 125–142.5. C. J. Bishop, Quasiconformal mappings which increase dimension, Ann. Acad. Sci.Fenn. Math. 24 (1999), no. 2, 397–407.6. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions,Comm. Pure Appl. Math. 44 (1991), 375–417.7. L. A. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math.Soc. 5 (1992), no. 1, 99–104.8. L. Carleson, The extension problem for quasiconformal mappings, in “Contributions toanalysis” (ed. by L. V. Ahlfors et al.), Academic Press, New York, 1974, 39–47.9. R. A. Fefferman, C. E. Kenig and J. Pipher, The theory of weights and the Dirichletproblem for elliptic equations, Ann. of Math. (2) 134 (1991), 65–124.10. F. W. Gehring and J. Va¨isa¨la¨, Hausdorff dimension and quasiconformal mappings, J.London Math. Soc. (2) 6 (1973), 504–512.11. J. Heinonen, Lectures on analysis on metric spaces, Universitext. Springer-Verlag,New York, 2001.12. L. V. Kovalev, Quasiconformal geometry of monotone mappings, J. Lond. Math. Soc.(2) 75 (2007), no. 2, 391–408.13. L. V. Kovalev and D. Maldonado, Mappings with convex potentials and the quasicon-formal Jacobian problem, Illinois J. Math. 49 (2005), no. 4, 1039–1060.14. L. V. Kovalev, D. Maldonado, and J.-M. Wu, Doubling measures, monotonicity, andquasiconformality, Math. Z. 257 (2007), no. 3, 525–545.15. D. Sullivan, Hyperbolic geometry and homeomorphisms. Geometric topology, editedby J. C. Cantrell, Academic Press, New York-London, 1979, 543–555.16. P. Tukia and J. Va¨isa¨la¨, Quasiconformal extension from dimension n to n + 1, Ann.of Math. (2) 115 (1982), no. 2, 331–348.17. J. Va¨isa¨la¨, Lectures on n-dimensional quasiconformal mappings, Lecture Notes inMathematics, Vol. 229. Springer-Verlag, Berlin-New York, 1971.Department of Mathematics, Syracuse University, Syracuse, NY 13244, USAE-mail address: lvkovale@syr.eduDepartment of Mathematics, Syracuse University, Syracuse, NY 13244, USAE-mail address: jkonnine@syr.edu

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