(2) * ; v(x,y)=r-j [p(x + ty)~ p(x-ty)]dt (r > 0)

Transcription

1 proceedings of the american mathematical society Volume 100, Number 4, August 1987 ON THE DILATATION ESTIMATES FOR BEURLING-AHLFORS QUASICONFORMAL EXTENSION DELIN TAN ABSTRACT. Let ß(x) be a p-quasisymmetric function. the dilatation K(z) of Beurling-Ahlfors extension with r = 1 satisfies the inequalitites K < 1p - 7(p - l)/6(p + 1) K < 2p O(lfp) for sufficiently large p. 1. Introduction. Let p(x) be p-quasisymmetric, 1 < p < oo, i.e. p(x) is a continuous incresasing function mapping the real line onto itself satisfying (1) 1 < p(x + t) - p(x) p ~ p(z) p(x i) for all x t ^ 0. Beurling Ahlfors [1] using the formulas u(x-,y) = 7,l 1 f1 [p(x + ty) - p(x-ty)]dt, (2) * ; v(x,y)=r-j [p(x + ty)~ p(x-ty)]dt (r > 0) constructed the function u>(2) = u(x,y) + iv(x,y) (z = x + iy), which is a q.c. mapping from the upper half-plane onto itself having p(x) as its boundary value. It is of interest to estimate the dilatation K(z) of w(z). Beurling Ahlfors [1] first proved for some r (3) K < p2. Reed [2] improved the inequality (3) as follows: (4) K < 8p. Li Zhong [3] then obtained (5) K < 4 2p. Ahlfors [4] also proved that the Beurling-Ahlfors extension function with r = 1 is quasi-isometric, i.e. there exists a constant A such that (6) -d(zi,z2) < d(w(zi),w(z2)) <Ad(zi,z2) for any zi,z2 in the upper half-plane, where d(-, ) denotes the non-euclidean distance. Ahlfors obtained (7)_ A<4p2(p+1). Received by the editors November 15, 1986, in revised form, May 21, Mathematics Subject Classification (1985 Revision). Primary 30C60; Secondary 26D American Mathematical Society /87 $ $.25 per page

2 656 DELIN TAN Chen Ji-xiu refined Reed's method to obtain (8) K < 2 58p (9) A < 3p. In this paper we have refined the Beurling-Ahlfors technique obtained the following theorem. THEOREM 1. Let p(x) be a p-quasisymmetric function. the dilatation K(z) of the Beurling-Ahlfors extension with r = 1 satisfies the inequalities (11) K < 2p (1/p) for sufficiently large p. THEOREM 2. Let p(x) be a p-quasisymmetric function let w(z) be the Beurling-Ahlfors extension of p(x) for r = 1. (12) 2~dOi,*2)< d(w(zi),w(z2))<2pd(zi,z2) for any Z\, z2 in the upper half-plane. Remark 1. When p = 1 p(0) = 0, we easily see that w(z) = c(x+yi/2) (c > 0). It is evident that K(z) = 2 lim dm*m*2)) = 2. ImÄ%>0 d^z2) Therefore the coefficient 2 of p either in Theorem 1 or 2 cannot be replaced by any smaller number. REMARK 2. This work was done during the summer of It is independent of Lehtinen's paper. Lehtinen [5] obtained (13) K < 2p. 2. LEMMA. Let p(x) be a p-quasisymmetric function normalized by p(0) = 0 p(l) = 1. (14) (l + 2p)c: + ßv>l+ß, (15) ß(l + 2p)r) + t>l + ß, where ß = -p(-l), = 1 - /0 p(t) dt, n = 1 + /3_1 f_1 p(t) dt. PROOF. Taking x = t > 0 in (1) we have (1 + p)p(x) > p(2x). Thus (1 + p) j p(x)dx> I p(2x)dx=- I p(x)dx+- / p(x)dx. Jo Jo 2 Jo 2 Jo

3 THE DILATATION OF BEURLING-AHLFORS EXTENSION 657 Therefore (16) (l + 2p) p(x)dx> p(x)dx. Substituting 1 p(l x) for p(x), we get (1 + 2p) / p(x) dx > 1 pt(x) dx. This yields (14). Similarly, substituting 1 + p(x l)/ß for p(x) yields (15). 3. Proof of Theorem 1. Because of linear invariance we only need to estimate K(z) for x = 0, y = 1, p(x) normalized by p(0) = 0, p(l) = 1. Thus the dilatation K = K(i) with r = 1 satisfies [1] the equation (17) K+K t: + n[ß l + e2)+/?(l + r/2) = F(tl,n,ß), where ß < p, 1/(1 + p) < Ç,n < p/(l + p). Furthermore, we can suppose ß > 1, otherwise consider w( z)/ß. Let G be a closed domain bounded by a polygon ABC DE. The side AB lies on the line of (1 + 2p) + ßn = 1 + ß\ the other sides BC, CD, DE, AE lie on the lines of i; = 1/(1 + p), V - p/(l + p), = p/(l + p), _?? = 1/(1+p), respectively. It is sufficient to look at the maximum of F(, n, ß) in G. By calculating df dn we have ß(t:+n)2-(ß+i/ß)(i+e) d2f 2(ß+i/ß)(i + e) ( + r?)2 dn2 (t+vy Since d2f/dr 2 > 0,max of F in G is in CDuAEl) AB. Since d2f/di2 > 0, the max is in DE U BC U AB, so the max is in AB U {C, D, E}. Since df/dn < 0 in BC df/dt] < 0 in AE, the max is in AB U { >}. When (, n) G AB, then ßt] ß - (1 + 2p)^ F(c», _ nl+ß + ß2-(l + ß)(l + 2pU+(l + 2p + 2p2)e (18) '«^'W i+^_(i + 2p-^

4 658 DELIN TAN But W(,ß) is a convex function of either ß or, therefore the max of W(,ß) must occur when = 1/(1 + p), ß = p, or 1 + P + pß (l + p)(l + 2p) Hence we only need to consider the following cases: Case 1: ylf pom >. = rj = p/(l + p), (19) (20) 1\ l + (p/l + p)2 l + p l + p j \ ß 2p/(l + p) 1 (p-l)(2p3+4p2- -2p+2p" 2p2(p+l) 2p + l) Case 2: At point B with ß = p. = 1/(1 + p), r = p/(l + p), 1 p \ _ YT~p,TT~p,P) p p + T+7 (1 + P)2 1 (p-l)(4p2 + 5p + 3) P+2p 2p(p+l)2 Case 3: 4i pom«a. J = (1 + p + /?p)/(l + p)(l + 2p), 1 < ß < p, n V(l+ P). (21) 1 + P + /?P 1 fl (l + p)(l + 2p)'l + p'p _ß+l/ß + ß/(l + P)2 + [(1 + P + ßp)2/(l + P)(1 + 2p)]2/ß (2 + 3p + ßp)/(l + 2p)(l + p) <(,,^,r,n, ß+l/ß + ß/(l+p)2 + [(l+p + p2)/(l + p)(l+2p)}2/ß S(i + Pj(i + ^j p + /?p = (l + p)(l + 2p)Y(ß,p). Denote X(p) = [(1 + p + p2)/(l + p)(l + 2p)]2. (2 + 3p + /3p)2 = (2 + 3p) (i+p)2 When ß increases from 1 to p, the sign of dy/dß (22) max Y(ß,p) i<ß<p (l + p)(l + 2p)y(l,p) = ^p+l + = 2p + =max{y(l,p),y(p,p)}, [1 + A(P)] 2p 1 ß < ^2+^\ changes only once. Hence p 16(1+ 2p) 16(1+ 2p)2 1 (p-l)(7p4 + 13p3+4p2-2p-l) 2p 2p(l + p)(l + 2p)2 (23) 3 4 ;i + p)(i + 2p)y(p,p) = 2p-3 + i + ^- ïtp-ït-p + i = 2p + 1 (p-l)(12p4 + 26p3+23p2+9p + 2) 2p 2p(p+l)2(p + 2)(2p + l)

5 THE DILATATION OF BEURLING-AHLFORS EXTENSION 659 From (19), (20), (22), (23) we have (25) ii+l<2p+-l-2 + of- K 2p \p for sufficiently large p. Inequalities (10) (11) follow. 4. Proof of Theorem 2. Because the non-euclidean distance is also a linear invariant we only need to prove (26) 1 < 2p- dw(i) < 2p v(i) dz I for p(0) = 0 p(l) = 1. Similarly, we suppose ß > 1. From (2) From [4] Hence (27) (28) From Theorem 1 (29)»W = \ I MO - K-t)\ dt = (1 + ß) - ( + ßrj). 1 + ß.,-. P(i+ß) < v(i) < 2(1 + p) --V-/-- 2(1 +p)' l*».wia = g[(l + Í2) + /?2(1 + ^2) + 2ß(t + r,)]- MO* < ^+/?i/1+7r^U4^ (i + p)2 'rv~ ' (i + p)2 Wj(z') V[l wz(i) v(i) (l + /?)2(2p2+2p+l)-2/? 8(1+ p)2 < < K(0l2> (i + /?2). 1+pJ {l + ß)2(2p + 2p+l)-2ß 4(1+ p)5 8(1+ p)2 2p2 + 2p + 1-2p2 + 2p + 1 (1 + /?)2 2/3 :i + /?)2J 2p (I+P)5 >i(i+^.4ii±c>(1+^ K <2p p2(l+ß)2~ p - 1 2p2 + p + 1 P+l P+l K 2p2+p+l A + 1-2p2 + 2p + 2 ' 4p2

6 660 DELIN TAN For dw wzdz + wz dz, we have (30) (31) dw(i) V(i)dz dw(i) v(i) dz \dw\ < \w \dw\ > \w < AK2 (K + l) w. v[i) 2A \dz\ -- = v x \wzdz\, K + 2 \dz\ \wz dz\. Ä+11 2 r 2p2 + p + 1 2p <2 2p2 + 2p + 1-2p2 + 2p + 2 (I + Pf 2(p - l)(2p5 + 12p4 + 15p3 + 13p2 + 5p + 1) = 4p2-2(l+p)2(l + p + p2)2 <4p2, > 4 (A + l) wz(i) > (2p + l)2 il± t>j_ 4p2-4p2' that completes the proof. Finally I want to express my heartfelt thanks to Professor Clifford Earle for his careful reading many suggestions which made this paper better than the original one. References 1. A. Beurling L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), J. Reed, Quasiconformal mappings with given boundary values, Duke Math. J. 33 (1966), Li Zhong, On Beurling-Ahlfors extension, Acta Math. Sinica 3 (1983), L. Ahlfors, Lecture on the quasiconformal mappings, Van Nostr, Princeton, N.J., M. Lehtinen, Ann. Acad. Sei. Fenn. Ser. A I Math. 8 (1983), department of mathematics, state university of new york at stony Brook, Stony Brook, New York 11794