On harmonic and QC maps

Transcription

1 On harmonic and QC maps Vesna Manojlović University of Belgrade Helsinki Analysis Seminar FILE: vesnahelsinkiharmqc tex , Vesna Manojlović On harmonic and QC maps 1/25

2 Goal A survey of hqc maps, with emphasis on moduli of continuity. Martio s question If u is harmonic in the unit disk D, and f is the boundary function on T = D, find necessary and sufficient conditions that lim z ζ u r (z) and lim z ζ u θ (z) exists at each ζ T. Vesna Manojlović On harmonic and QC maps 2/25

3 Theorem (Pavlović) Let u be a harmonic homeomorphism of D. Then the following conditions are equivalent: (a) u is qc; (b) u is bi-lipschitz in the euclidean metric; (c) f is bi-lipschitz and the Hilbert transformation of its derivative is in L. Vesna Manojlović On harmonic and QC maps 3/25

4 Partyka and Sakan gave explicit estimations of the bi-lipschitz constants for u expressed by means of the maximal dilatation K of u and u 1 (0). Additionally if u(0) = 0, they used Mori s theorem as in Pavlović, to get asymptotically sharp estimates as K 1. A mapping f : (X, d X ) (Y, d Y ) is bi-lipschitz if it is bijective and both f and f 1 are Lipschitz continuous. Vesna Manojlović On harmonic and QC maps 4/25

5 Theorem (M) Suppose D and D are proper domains in R 2. If u : D D is K-qc and harmonic, then it is bilipschitz with respect to quasihyperbolic metrics on D and D. The proof is based on the theorem of Astala and Gehring. Mateljević and Vuorinen extended this result to domains in R n under the hypothesis that K < 2 n 1 and u is of class C 1,1. Vesna Manojlović On harmonic and QC maps 5/25

6 Theorem (M-Pavlović) If f is a complex-valued k-quasiregular harmonic function defined on a region G C, and q = 4k/(k + 1) 2, then f q is subharmonic. The exponent q is optimal. Vesna Manojlović On harmonic and QC maps 6/25

7 For a continuous function f : D C harmonic in D we define two moduli of continuity: ω(f, δ) = sup{ f (e iθ ) f (e it ) : e iθ e it δ, t, θ R}, for all δ 0, and ω(f, δ) = sup{ f (z) f (w) : z w δ, z, w D}, for all δ 0. Clearly ω(f, δ) ω(f, δ), but the reverse inequality need not hold. Vesna Manojlović On harmonic and QC maps 7/25

8 However, as was proved by Rubel, Shields and Taylor and Tamrazov, if f is a holomorphic function, then ω(f, δ) Cω(f, δ), where C is independent of f and δ. Here is an extension that result to quasiregular harmonic functions. Theorem (M-Pavlović) Let f be a k-quasiregular harmonic complex-valued function which has a continuous extension on D, then there is a constant C depending only on k such that ω(f, δ) Cω(f, δ). Vesna Manojlović On harmonic and QC maps 8/25

9 Question 2. If u is a hqc mapping on the unit ball and u S n 1 has some continuity property, does it follow that u has the same property on the ball? If this property is Lipschitz continuity, the answer is affirmative. If this property is the modulus of continuity, the answer is also affirmative. Vesna Manojlović On harmonic and QC maps 9/25

10 The following problem is considered: can one control the modulus of continuity ω u of a harmonic quasiregular (briefly, hqr) mapping u in B n by the modulus of continuity ω f of its restriction to the boundary S n 1, i.e. is it true that ω u Cω f? Vesna Manojlović On harmonic and QC maps 10/25

11 In fact this problem has been studied extensively for harmonic functions and mappings without assumption of quasiregularity. For the unit ball the answer is positive in the case ω(δ) = δ α, 0 < α < 1 (Hölder continuity) and negative in the case ω(δ) = Lδ (Lipschitz continuity). Vesna Manojlović On harmonic and QC maps 11/25

12 In fact, for bounded plane domains the answer is always negative for Lipschitz continuity Aikawa. However, it is proved by Hinkkanen, that for general plane domains one has logarithmic loss of control : ω u (δ) Cω f (δ) log(1/δ). Vesna Manojlović On harmonic and QC maps 12/25

13 Theorem(Arsenović-Božin-M) There is a constant q = q(k, n) (0, 1) such that u q is subharmonic in Ω R n whenever u : Ω R n is a K-quasiregular harmonic map. Vesna Manojlović On harmonic and QC maps 13/25

14 Theorem(Arsenović-Božin-M) If u : B n R n is a continuous map which is K-quasiregular and harmonic in B n, then ω u (δ) Cω f (δ) for δ > 0, where f = u S n 1 and C is a constant depending only on K, ω f and n. Vesna Manojlović On harmonic and QC maps 14/25

15 We note that every every continuous map u : B n Ω which is hqc in B n, where Ω is bounded and has C 2 boundary, is Lipschitz continuous (Kalaj). Also, every holomorphic quasiregular mapping on a domain Ω C n (n > 1) with C 2 boundary is Lipschitz continuous (Poletsky). The same paper contains an example of a holomorphic quasiregular map in a domain Ω C 2 (with non-smooth boundary) which is not Lipschitz. Vesna Manojlović On harmonic and QC maps 15/25

16 In view of the above, one is tempted to make the following conjecture: every hqc map u : B n Ω is Lipschitz continuous. However, this is false, as we show by an example for n = 3 given by V. Božin. Vesna Manojlović On harmonic and QC maps 16/25

17 Counterexample We use the following notation: X = (x, y, z), Π + = {(x, y, z) : z > 0}. We construct a mapping f : Π + R 3 such that 1. f is continuous on Π f is not Lipschitz on L = {(0, 0, z) : 0 z 1}. 3. f is hqc on Π +. Vesna Manojlović On harmonic and QC maps 17/25

18 Clearly, for general quasiconformal mappings u : Ω 1 Ω 2 one can not expect that the modulus of continuity behaves as in the above theorem, even for Ω 1 = B n. However, for bounded Ω 1, Hölder continuity of u Ω1 implies Hölder continuity of u in Ω 1, but with possibly different Hölder exponent, see Näkki-Palka and Martio-Näkki. Vesna Manojlović On harmonic and QC maps 18/25

19 Theorem (Martio-Näkki) Let D be a bounded domain in R n and let f be a continuous mapping of D into R n which is quasiconformal in D. Suppose that, for some M > 0 and 0 < α 1, f (x) f (y) M x y α (1) whenever x and y lie on D. Then f (x) f (y) M x y β (2) for all x and y on D, where β = min(α, K 1/(1 n) I ) and M depends only on M, α, n, K(f ) and diam(d). Vesna Manojlović On harmonic and QC maps 19/25

20 The exponent β is the best possible, as an example of a radial quasiconformal map f (x) = x α 1 x, 0 < α < 1, of B n onto itself shows (Väisälä). Also, the assumption of boundedness of the domain is essential. Indeed, one can consider g(x) = x a x, x 1 where a > 0. Then g is quasiconformal in D = R n \ B n, it is identity on D and hence Lipschitz continuous on D. However, g(te 1 ) g(e 1 ) t a+1, t, and therefore g is not globally Lipschitz continuous on D. Vesna Manojlović On harmonic and QC maps 20/25

21 Arsenović extended Martio-Näkki theorem to more general continuity conditions. Namely, he replaced condition (1) with f (x) f (y) ω( x y ), x, y D where ω is a gauge function, satisfying an additional condition. Then the conclusion is that f (x) f (y) Mω ( x y ) for all x and y in D, where ω (δ) = max(δ α, ω(δ)) and α = K 1/(1 n) I (f ). Note that there are no assumptions on the regularity of the boundaries D and f (D). Vesna Manojlović On harmonic and QC maps 21/25

22 P. Koskela posed the following question: Question 3. Is it possible to replace β with α if we assume, in addition to quasiconformality, that f is harmonic? Vesna Manojlović On harmonic and QC maps 22/25

23 The answer is positive, if D is a uniformly perfect set (Järvi-Vuorinen). In fact, one can prove (Arsenović-M-Vuorinen) a more general result, including domains having a thin, in the sense of capacity, portion of the boundary. However, this generality is in a sense illusory, because any hqc mapping extends harmonically and quasiconformally across such portion of the boundary (Caffarelli-Kinderlehrer). Vesna Manojlović On harmonic and QC maps 23/25

24 Open problems Is Hölder continuity on the boundary preserved for hqc mappings in any bounded domain? Characterize moduli of continuity of boundary values of functions in hqc(b n ). Note that these include non-lipschitz ones. Is any hqc mapping bi-lipschitz continuous with respect to quasihyperbolic metric in bounded domains in R n for n > 2? Vesna Manojlović On harmonic and QC maps 24/25

25 M. Arsenović, V. Božin, and V. Manojlović, Moduli of continuity of harmonic quasiregular mappings in B n, Potential Anal. M. Arsenović, V. Manojlović, and M. Vuorinen, Hölder continuity of harmonic quasiconformal mappings, submitted V. Manojlović, Bi-Lipschicity of quasiconformal harmonic mappings in the plane, Filomat 23 (2009), no. 1, O. Martio and R. Näkki, Boundary Hölder continuity and quasiconformal mappings, J. London Math. Soc. (2)44 (1991), M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, Vesna Manojlović On harmonic and QC maps 25/25