On reduced Beltrami equations and linear families of quasiregular mappings
aa r X i v : . [ m a t h . C V ] A ug On reduced Beltrami equations andlinear families of quasiregular mappings
Jarmo Jääskeläinen
Abstract
This paper studies linear classes of planar quasiregular mappings. We givea positive answer to a conjecture of K. Astala, T. Iwaniec, and G. Martin(2009) on reduced Beltrami equations. Moreover, we use it to prove aWronsky-type theorem for general linear Beltrami systems. This is a keyto show that the associated Beltrami equation of a linear quasiregularfamily is unique.
Distinctive for the reduced Beltrami equation (1.1) ∂f∂ ¯ z = λ ( z ) Im (cid:18) ∂f∂z (cid:19) , | λ ( z ) | k < , for almost every z ∈ Ω , is that its solutions create an R -linear family of quasireg-ular mappings. The reduced equation arises naturally in a great variety of topics;for instance, in the study of linear families of quasiregular mappings, and also inthe Stoïlow factorization and the G -closure problems for the general Beltramiequation(1.2) ∂f∂ ¯ z = µ ( z ) ∂f∂z + ν ( z ) ∂f∂z , | µ ( z ) | + | ν ( z ) | k < , for almost every z ∈ Ω .It is clear from the definition that a differential constraint in (1.1) is strongerthan the one in the classical Beltrami equation. Hence f is K -quasiregular with K = k − k . In this paper we assume an appropriate regularity, i.e., f ∈ W , (Ω) for a domain Ω ⊂ C .Studies of the reduced Beltrami equation (1.1) indicate that for its solutions, reduced quasiregular mappings , the null Lagrangian J ( z, f ) = Im (cid:18) ∂f∂z (cid:19) has many properties similar to the Jacobian determinant of a Sobolev func-tion. We prove the following key result in this direction, answering in positiveConjecture 6.3.1 in [4]. heorem 1.1. Suppose f : Ω → C , f ∈ W , (Ω) , is a solution to the reducedBeltrami equation (1.1) . Then either ∂ z f is a constant or else Im (cid:18) ∂f∂z (cid:19) = 0 almost everywhere in Ω .Thus if Im( ∂ z f ) vanishes on a set of positive measure, then f ( z ) = az + b ,where a ∈ R and b ∈ C . In Geometric Function Theory the above fact has a similar important roleas that of the Jacobian determinant of a general quasiregular map, J ( z, f ) = | ∂ z f | − | ∂ ¯ z f | = 0 . The null Lagrangian Im( ∂ z f ) appears naturally as thedenominator in algebraic fractions; for example, in the G -compactness studiesof K -quasiregular families.For special properties of the reduced Beltrami equation (1.1), we refer thereader to the recent monograph [4]. An early application of the reduced equationcan be found in [7]. The reduced equation has generated a considerable new-found interest, see [1], [4], [5], [12], [14] [15], [17], and [18].First steps in giving the positive answer to the conjecture were made by F.Giannetti, T. Iwaniec, L. Kovalev, G. Moscariello, and C. Sbordone in [12]. Theyproved the statement for global homeomorphisms, that is, homeomorphisms ofthe plane C , when k < in (1.1). Next, G. Alessandrini and V. Nesi showed in [1]the assertion for global homeomorphisms. A direct, and substantially simplified,proof of this result can be found in [5] or Theorem 6.4.1 in [4].The case of plane homeomorphisms was used, for example, to study linearfamilies of quasiconformal mappings. Moreover, by combining it with the ideasand results developed in [8] and [12], one can show that the family of Beltramidifferential operators is G -compact, see Section 16.6 in [4].In spite of the close analogy with the global homeomorphic case, our proofrequires more involved methods and rigorous analysis. Difference is that weconsider an arbitrary domain Ω ⊂ C instead of the whole plane C in the reducedBeltrami equation (1.1); the earlier proofs use the property that a solution is aglobal homeomorphism C → C . The key element of our proof is the weak reverseHölder inequality for the solutions to so-called adjoint equations. We use it witha smoothness at a point to derive our result.The reduced Beltrami equation (1.1) is closely related to linear families ofquasiregular mappings. The key ingredient is the following Wronsky-type theo-rem . It shows that the singular set of a two-dimensional linear family of quasireg-ular mappings has measure zero. This was conjectured for homeomorhisms in[8] and proven for them in [1], [5]. We establish a more general theorem.
Theorem 1.2.
Suppose Φ , Ψ ∈ W , (Ω) are solutions to (1.2) . Solutions Ψ and Φ are R -linearly independent if and only if complex gradients ∂ z Φ and ∂ z Ψ arepointwise independent almost everywhere, i.e., J (Φ , Ψ) := Im (cid:18) ∂ Φ ∂z ∂ Ψ ∂z (cid:19) = 0 almost everywhere in Ω . Above J (Φ , Ψ) plays the role of Wronskian. Note that, if Ψ and Φ are R -linearly dependent, then J (Φ , Ψ) ≡ . 2iven an R -linear subspace F ⊂ W , (Ω) , we say that F is a linear familyof quasiregular mappings , if there is K < ∞ such that for every f ∈ F thefunction f is K -quasiregular in Ω . The family F is generated by the maps f i , i = I , if F = (X i ∈I a i f i : a i ∈ R ) for some R -linearly independent quasiregular mappings f i : Ω → C . We al-ways assume that the linear family is generated by countable set of functions. Itquickly follows that in case of linear families that consist of quasiconformal map-pings, dim F , see [8]. Recall that a linear family of quasiregular mappings isnot always two-dimensional, e.g., -quasiregular family spanned by f i ( z ) = z i , i = 1 , , .In general, quasiregularity is not preserved under linear combinations, simpleexample is f ( z ) = k ¯ z + z , g ( z ) = k ¯ z − z . However, if we have mappings thathappen to be solutions to the same general Beltrami equation (1.2), then theirlinear combinations are quasiregular. Conversely, [8] associates to a linear two-dimensional family F of quasiregular mappings a general Beltrami equation ofthe type (1.2) satisfied by every g ∈ F , see also Remark 16.6.7 in [4]. We showthat the associated equation is unique. Theorem 1.3.
For any linear family F of quasiregular mappings, there existsa corresponding general linear Beltrami equation ∂f∂ ¯ z = µ ( z ) ∂f∂z + ν ( z ) ∂f∂z , | µ ( z ) | + | ν ( z ) | k < , almost everywhere in Ω , satisfied by every element f ∈ F .Moreover, the associated equation is unique. We begin with the adjoint equation approach, similarly as in [1] and [5]. Westudy the solution f ∈ W , (Ω) to the reduced Beltrami equation(2.1) ∂ ¯ z f ( z ) = λ ( z ) Im (cid:0) ∂ z f ( z ) (cid:1) , | λ ( z ) | k < , for almost every z ∈ Ω . Let us write f ( z ) = u ( z ) + iv ( z ) , where u and v arereal-valued; similarly notate λ ( z ) = α ( z ) + iβ ( z ) .Using the definition of complex partial derivatives (i.e., ∂ z = ∂ x − i∂ y and ∂ ¯ z = ∂ x + i∂ y ) and taking the imaginary part of the reduced equation gives u y + v x = β ( v x − u y ) , that is, u y = β − β + 1 v x . Thus(2.2) (cid:0) ∂ z f ( z ) (cid:1) = v x − u y = 2 β + 1 v x = 2 β − u y . Since | β ( z ) | | λ ( z ) | k < , the coefficients / ( β ( z ) ± in (2.2) are uniformlybounded from below. Hence Im( ∂ z f ) and u y have the same zeros.3or the reduced Beltrami equation (2.1), the derivative u y is a weak solutionto the adjoint equation determined by a non-divergence type operator. Moreprecisely, consider an operator(2.3) L = X i,j =1 σ ij ( z ) ∂ ∂x i x j , where σ ij = σ ji are measurable and the matrix σ ( z ) = (cid:20) σ ( z ) σ ( z ) σ ( z ) σ ( z ) (cid:21) is uniformly elliptic, K | ξ | h σ ( z ) ξ, ξ i = σ ( z ) ξ + 2 σ ( z ) ξ ξ + σ ( z ) ξ ≤ K | ξ | for all ξ ∈ C and z ∈ Ω . Above K is the ellipticity constant . The mapping ω ∈ L (Ω) is a weak solution to the adjoint equation L ∗ ( ω ) = 0 if(2.4) Z Ω ωL ( ϕ ) dm = 0 , for every ϕ ∈ C ∞ (Ω) .To identify u y as a weak solution to an adjoint equation of the type (2.4), werecall that the components of solutions f = u + iv to general Beltrami systems(1.2) satisfy a divergence type second-order equation, see Section 16.1.5 in [4];note that one gets the divergence type equations for solutions to general Beltramisystems defined in any domain Ω , but getting from the divergence type equationsto (1.2) one needs the domain to be simply connected.In the case of the reduced Beltrami equation (2.1), the component u satisfies(2.5) div A ∇ u = 0 , A ( z ) := (cid:20) a ( z )0 a ( z ) (cid:21) , where the matrix elements are(2.6) a = 2 Re( λ )1 − Im( λ ) = 2 α − β , a = 1 + Im( λ )1 − Im( λ ) = 1 + β − β . Specifically (2.5) means that for every ϕ ∈ C ∞ (Ω) (2.7) Z Ω ∇ ϕ · A ∇ u = Z Ω ϕ x ( u x + a u y ) + ϕ y a u y . But since derivatives of smooth test functions are again test functions, we canreplace ϕ by ϕ y ∈ C ∞ (Ω) in (2.7). Now, a straightforward calculation showsthat u y is a weak solution to the adjoint equation L ∗ ( u y ) = 0 , where(2.8) L = ∂ ∂x + a ∂ ∂x∂y + a ∂ ∂y and a , a are given by (2.6). We note that the original matrix A ( z ) is not sym-metric. However, the operator L in (2.8) can be represented by the symmetricmatrix σ ( z ) = (cid:20) a ( z ) / a ( z ) / a ( z ) (cid:21) and, as | λ ( z ) | k < , from (2.6) we see that σ is uniformly elliptic.4 Weak reverse Hölder inequality
Theorem 3.1.
Let ω ∈ L (Ω) be a real-valued weak solution to the adjointequation L ∗ ( ω ) = 0 of the type (2.4) . Then a weak reverse Hölder inequalityholds for ω ; namely, (3.1) (cid:18) r Z B ω dm (cid:19) / cr Z B | ω | dm, for every disk B := D ( a, r ) such that B := D ( a, r ) ⊂ Ω . The constant c depends only on the ellipticity constant K . There is a stronger result for non-negative solutions: a reverse Hölder in-equality holds, see [11] or Theorem 6.4.2 in [4].We start with a well-known interpolation inequality, for instance, [6], [19],and [20].
Lemma 3.2.
Let g ∈ W ,p ( U ) be real-valued, where U ⊂ C is a bounded smoothdomain and p > . Then (3.2) k∇ g k L p ( U ) c (cid:0) k D g k L p ( U ) + k g k L p ( U ) (cid:1) , where c depends only on U .Proof. By a basic interpolation between Sobolev spaces, see § 4.3.1, Theorem 1with § 2.4.2, (11) in [20], (cid:0) W ,p ( U ) , W ,p ( U ) (cid:1) / = W ,p ( U ) , and thus k g k W ,p ( U ) c ( U ) k g k / W ,p ( U ) k g k / W ,p ( U ) . Then we use the inequality ab ε a + 12 ε b , a, b > , ε > , to derive k∇ g k L p ( U ) k g k W ,p ( U ) εc ( U ) (cid:0) k g k L p ( U ) + k∇ g k L p ( U ) + k D g k L p ( U ) (cid:1) + c ( U, ε ) k g k L p ( U ) . Choosing ε sufficiently small the term with ∇ g is absorbed to the left-hand side,and we arrive at the estimate (3.2).Next, let us recall some of the key estimates in this connection. There isalways a unique solution to the following Dirichlet problem in a bounded domain D with a thick boundary, see Chapter 17 in [4],(3.3) L ( g ) = h, h ∈ L ( D ) , g ∈ W , ( D ) with g = 0 on ∂ D , where the operator L is of the type (2.3). By the Alexandrov-Bakel ′ man-Puccimaximum principle, see Theorem 17.3.1 in [4] or Theorem 9.1 in [13],(3.4) k g k L ∞ ( D ) c diam( D ) k h k L ( D ) and c depends only on the ellipticity constant K .5 emma 3.3. Let g be a real-valued solution to the Dirichlet problem (3.3) .If L ( g ) = 0 in a subdomain V ⊂ D , then for every relatively compact smoothsubdomain V ′ ⊂ V k∇ g k L ∞ ( V ′ ) c k g k L ∞ ( D ) , where c depends on p ∈ (cid:16) , KK − (cid:17) , V , and V ′ .Proof. We use the interior regularity from [3]; alternatively, see Chapter 17 in[4]. By Lemma 4.1 in [3], the complex gradient g z is quasiregular in V . Further,by Corollary 5.1 in [3], D g ∈ L p loc ( V ) , p < KK − . Moreover, the corollary implies for every relatively compact subdomain V ′ ⊂ V the uniform estimate(3.5) k D g k L p ( V ′ ) c k g k L ( V ) , where c depends on p , V , and V ′ . Actually, for the potential function, g ∈ W ,p loc ( V ) for all < p < KK − . Thus by the Sobolev embedding, or more strictlyby Morrey’s inequality, we achieve the following estimate for the Hölder norm k∇ g k C ,γ ( V ′ ) c ( p, V ′ ) k∇ g k W ,p ( V ′ ) c ( p, V ′ ) (cid:0) k∇ g k L p ( V ′ ) + k D g k L p ( V ′ ) (cid:1) , where γ = 1 − /p . Now we use the interpolation inequality (3.2) to write k∇ g k L p ( V ′ ) c ( V ′ ) (cid:0) k D g k L p ( V ′ ) + k g k L p ( V ′ ) (cid:1) . Hence, by combining the previous estimates with (3.5), we have shown k∇ g k L ∞ ( V ′ ) k∇ g k C ,γ ( V ′ ) c ( p, V, V ′ ) k g k L ∞ ( D ) . In the following proof, we will also use the consequence of the Leibniz rule,(3.6) L ( ϕg ) = 2 h σ ∇ ϕ, ∇ g i + ϕL ( g ) + gL ( ϕ ) , ϕ, g ∈ W , ( D ) , where L is an operator defined as in (2.3). Proof of Theorem 3.1.
Without loss of generality we can assume a = 0 . Weshow the claim for the unit disk D and then use a rescaling argument.For the unit disk case, it is enough to prove(3.7) Z D ω dm c ( K ) k ω k L ( D ) Z D | ω | dm. We solve the Dirichlet problem (3.3) for D = 2 D and h = ωχ D ∈ L (2 D ) . Asin (3.3) we notate the W , -solution by g .Let < δ < / and ϕ ∈ C ∞ (cid:0) (3 / δ D (cid:1) satisfy ϕ ≡ on δ D with (cid:12)(cid:12) ∂ α ϕ/∂x α (cid:12)(cid:12) c α for | α | . Now ϕg ∈ W , (2 D ) . Since ω is an adjoint solution in D , L ∗ ( ω ) = 0 , by approximating with smooth functions we find that Z D ωL ( ϕg ) dm = 0 . Z D ω = Z D ωL ( g ) ϕ = − Z D ω h σ ∇ ϕ, ∇ g i − Z D ωgL ( ϕ ) Z D | ω ||h σ ∇ ϕ, ∇ g i| + Z D | ω || g || L ( ϕ ) | c ( K ) Z (3 / δ D \ δ D | ω ||∇ ϕ ||∇ g | + k g k L ∞ (2 D ) Z D | ω || L ( ϕ ) | c ( K ) Z (3 / δ D \ δ D | ω ||∇ g | + c ( K ) k g k L ∞ (2 D ) Z D | ω | . Further, by using Lemma 3.3 to the sets V := 2 D \ D ⊂ D =: D and V ′ := (3 / δ D \ δ D , we have Z D ω c ( K ) k g k L ∞ (2 D ) Z D | ω | . The inequality (3.7) follows by the consequence of the Alexandrov-Bakel ′ man-Pucci maximum principle (3.4).We are left with the rescaling. Assume B ⊂ Ω . We set ω r ( z ) = ω ( rz ) and L r ( ϕ )( z ) = L ( ϕ )( rz ) . By definition, L r is a uniformly elliptic non-divergence type operator with thesame ellipticity constant K as the operator L . Further, ω r is an adjoint solutionin Ω r := { z ∈ Ω : rz ∈ Ω } for L r and D ⊂ Ω r . Thus the above proof shows(3.8) (cid:18)Z D ω r ( z ) dm ( z ) (cid:19) / c ( K ) Z D | ω r ( z ) | dm ( z ) . Next we use the change of variables to get r Z D ω r ( z ) dm ( z ) = Z B ω ( z ) dm ( z ) , r Z D | ω r ( z ) | dm ( z ) = Z B | ω ( z ) | dm ( z ) . Combining the above calculation with (3.8) gives (cid:18) r Z B ω ( z ) dm ( z ) (cid:19) / c ( K ) r Z B | ω ( z ) | dm ( z ) . Remark 3.4.
It is well-known, see Gehring’s lemma, for example, from Section4.3 in [9] or Chapter 14 in [16], that a weak reverse Hölder inequality improvesintegrability: if a weak reverse Hölder inequality (3.1) holds for ω , then there is p > such that (cid:18) r Z B | ω | p dm (cid:19) /p c ( K ) r Z B | ω | dm. Zeros of infinite order
A weak reverse Hölder inequality implies that almost every zero is of infiniteorder; that is,
Theorem 4.1.
Let ω satisfy a weak reverse Hölder inequality (3.1) . Then, foralmost every zero z of ω and for every positive integer N , there is r ( z , N ) > such that Z D ( z ,r ) | ω | dm r N r N Z D ( z , r ) | ω | dm = O ( r N ) , < r r ( z , N ) . Proof.
We use the iteration argument from pp. 299–300 in [9] and Theorem14.5.1 in [16].Set E = { z ∈ Ω : ω ( z ) = 0 } . Assume | E | > . Let z be a point of densityof E . We fix a positive integer N . Since z is a density point, we find that, for r := r ( z , N ) sufficiently small, | D ( z , δr ) \ E | ( δr ) c N holds for all < δ , where c is the constant from the weak reverse Hölderinequality. Thus Z D ( z ,δr ) | ω | = Z D ( z ,δr ) \ E | ω | | D ( z , δr ) \ E | / (cid:18)Z D ( z ,δr ) | ω | (cid:19) / | D ( z , δr ) \ E | / cδr Z D ( z , δr ) | ω | N Z D ( z , δr ) | ω | . Iterating yields for k = 1 , , . . . Z D ( z , − k r ) | ω | ( k +1) N Z D ( z , r ) | ω | . For each < r r there exists k such that − k r r < − k +1 r . Hence Z D ( z ,r ) | ω | Z D ( z , − k +1 r ) | ω | kN Z D ( z , r ) | ω | = r N kN r N Z D ( z , r ) | ω | r N r N Z D ( z , r ) | ω | = O ( r N ) . Proof of Theorem 1.1
We are ready to answer in positive Conjecture 6.3.1 in [4].
Proof of Theorem 1.1.
Let f = u + iv be a solution to the reduced Beltramiequation ∂ ¯ z f ( z ) = λ ( z ) Im (cid:0) ∂ z f ( z ) (cid:1) , | λ ( z ) | k < , for almost every z ∈ Ω . By (2.2) E := { z ∈ Ω : Im (cid:0) ∂ z f ( z ) (cid:1) = 0 } = { z ∈ Ω : u y ( z ) = 0 } . Assume | E | > . We have shown above that u y is a solution to the adjointequation and thus a weak reverse Hölder inequality holds for u y . Further, byTheorem 4.1, Z D ( z ,r ) | ∂ ¯ z f | k Z D ( z ,r ) | Im (cid:0) ∂ z f (cid:1) | k − k Z D ( z ,r ) | u y | = O ( r N ) , (5.1)for almost every z ∈ E and for all positive integers N , when r > is smallenough. In the second inequality we use (2.2) again. A. Series representation
We will prove that for almost every z ∈ E and for all positive integers n ,(5.2) f ( w ) = c + c ( w − z ) + E ( w ) near the point z ,where c ∈ C , c ∈ R are constants depending only on f and z and(5.3) Z D ( z ,r ) | D E| dm = O ( r n +1 ) holds for small enough r > . We deduce the statement of our theorem fromthis by quasiregularity.Fix a positive integer n . Choose z ∈ E and r ∈ (0 , such that D ( z , r ) ⊂ Ω and (5.1) holds for N = n + 2 and < r r . By the weak reverse Hölderinequality (3.1) and the improved integrability, see Remark 3.4, (5.1) impliesfor some p > (5.4) Z D ( z ,r ) | ∂ ¯ z f | p dm = O ( r Np − p +2 ) , p p ,when < r r . Alternatively one could use Astala’s higher integrability, see[2] or Theorem 13.2.3 in [4].Suppose w ∈ D ( z , r ) . We begin by showing that(5.5) f ( w ) = n − X j =0 c j ( w − z ) j + E ( w ) , Z D ( z ,r ) | D E| dm = O ( r n +1 ) , where < r r and c j ∈ C are constants depending only on f and z .Smoothness at a point has been studied, for example, in [10] and we use afew similar ideas. 9 tep 1. Generalized Cauchy formula First, the generalized Cauchy formula gives f ( w ) = 12 πi Z ∂ D ( z ,r ) f ( z ) z − w dz + 1 π Z D ( z ,r ) ∂ ¯ z f ( z ) w − z dm ( z ) . Since the first term is analytic in the disk D ( z , r ) , using the Taylor expansionabout z it can be written in the form n − X j =0 a j ( w − z ) j + R n ( w ) , R n ( w ) = O ( | w − z | n ) . For the second term, π Z D ( z ,r ) ∂ ¯ z f ( z ) w − z dm ( z ) = − n − X j =0 ( w − z ) j π Z D ( z ,r ) ∂ ¯ z f ( z )( z − z ) j +1 dm ( z )+ ( w − z ) n π Z D ( z ,r ) ∂ ¯ z f ( z )( z − z ) n ( w − z ) dm ( z )=: n − X j =0 b j ( w − z ) j + T ( w ) , as soon as we show the convergence of the coefficient integrals(5.6) | b j | π Z D ( z ,r ) | ∂ ¯ z f ( z ) || z − z | j +1 dm ( z ) , j = 1 , , . . . , n − . Observe that after we have the convergence of integrals, f ( w ) is a sum of aholomorphic part and T . Step 2. Convergence of the integrals
Dividing in annuli A k := D ( z , − k +1 r ) \ D ( z , − k r ) , for j = 1 , , . . . n − , Z D ( z ,r ) | ∂ ¯ z f ( z ) || z − z | j +1 dm ( z ) = ∞ X k =1 Z A k | ∂ ¯ z f ( z ) || z − z | j +1 dm ( z ) ∞ X k =1 − k r ) j +1 Z D ( z , − k +1 r ) | ∂ ¯ z f | dm c n +2 ∞ X k =1 k , by (5.1). Indeed, our choice N = n + 2 gives for k = 1 , , . . . , r N r j +10 ( − k +1) N − k ( j +1) n +2 k , j = 1 , , . . . n − . Z D ( z ,r ) | ∂ ¯ z f ( z ) | p | z − z | np dm ( z ) c ( K, p , n ) ∞ X k =1 k , p p , by (5.4). Step 3. Remainder term
Set c j = a j − b j for j = 0 , . . . , n − . Thus, as the remainder term in (5.5), wehave E = R n + T , where R n is holomorphic with R n ( w ) = O ( | z − w | n ) and T ( w ) = ( w − z ) n π Z D ( z ,r ) ∂ ¯ z f ( z )( z − z ) n ( w − z ) dm ( z ) . To prove (5.5), we are left to show the estimate for the derivative D E .Recall | D E| = | ∂ ¯ z E| + | ∂ z E| . By definition, the derivative of the holomorphicpart DR n has the correct convergence rate. After combining ∂ ¯ z f = ∂ ¯ z E = ∂ ¯ z T with (5.1), we see that only the estimation of ∂ z T remains.The integral term in T is the Cauchy transform C of F ( z ) := χ D ( z ,r ) ( z ) ∂ ¯ z f ( z )( z − z ) n , C ( F )( w ) := 1 π Z C χ D ( z ,r ) ( z ) ∂ ¯ z f ( z )( z − z ) n ( w − z ) dm ( z ) . The above is well-defined, since F ∈ L ( C ) by (5.7). Now, for almost every w ,(5.8) ∂ z T ( w ) = n ( w − z ) n − C ( F )( w ) + ( w − z ) n S ( F )( w ) , where the Beurling transform S is given by the principal value integral S ( F )( w ) := − π Z C χ D ( z ,r ) ( z ) ∂ ¯ z f ( z )( z − z ) n ( w − z ) dm ( z ) . We start with the first term in (5.8). By inequality (5.7), there is a Hölderconjugate pair < q < < p < ∞ such that F ∈ L p ( C ) ∩ L q ( C ) , since F has acompact support. Thus C ( F ) ∈ C ( ˆ C ) , and moreover, kC ( F ) k L ∞ ( C ) √ − q (cid:0) k F k L p ( C ) + k F k L q ( C ) (cid:1) , see, for example, Theorem 4.3.11 in [4]. We have Z D ( z ,r ) | n ( w − z ) n − C ( F )( w ) | dm ( w ) cr n +1 . For the second term in (5.8) the Hölder inequality implies Z D ( z ,r ) | w − z | n |S ( F )( w ) | dm ( w ) (cid:18)Z D ( z ,r ) | w − z | n dm ( w ) (cid:19) / kS ( F ) k L ( C ) cr n +1 k F k L ( C ) . We have proven the estimate (5.5).11 tep 4. No higher-order terms
Observe for j = 2 , . . . , n − Z D ( z ,r ) (cid:12)(cid:12) Im (cid:0) j c j ( w − z ) j − (cid:1)(cid:12)(cid:12) dm ( w )= j r j +1 j + 1 Z π (cid:12)(cid:12) β j cos (cid:0) ( j − θ (cid:1) + α j sin (cid:0) ( j − θ (cid:1)(cid:12)(cid:12) dθ = cr j +1 , where we notate c j = α j + iβ j . Further, straight from (5.5) Im (cid:0) ∂ z f ( w ) (cid:1) = Im c + n − X j =2 Im (cid:0) j c j ( w − z ) j − (cid:1) + Im (cid:0) ∂ z E ( w ) (cid:1) . Now, the estimates for the convergence rate in (5.1) and (5.5) imply Im c = 0 and c j = 0 , j = 2 , . . . , n − .We have shown the series representation (5.2) with the estimate (5.3). B. Conclusion by quasiregularity
We use similar methods as in pp. 299–300 [9] and Section 16.10 in [16]. In thereferences these ideas are used to prove that the Jacobian of a nonconstantquasiregular mapping is nonvanishing almost everywhere.The constant c in (5.2) is real and hence g ( w ) := f ( w ) − c − c ( w − z ) solves the same reduced Beltrami equation as f . Therefore, g is quasiregularwith the following property for every positive integer n Z D ( z ,r ) | Dg | dm = Z D ( z ,r ) | D E| dm = O ( r n +1 ) , < r r . Since the weak reduced Hölder inequality holds for | Dg | by quasiregularity, weachieve for every positive integer N (cid:18)Z D ( z ,r ) | Dg | dm (cid:19) / O ( r N +1 ) , when r is small enough . Quasiregularity and a version of Morrey’s inequality implies the Hölder con-tinuity of the form | g ( z ) − g ( w ) | c (cid:18) | z − w | r (cid:19) α ( K ) (cid:18)Z D ( z ,r ) | Dg | dm (cid:19) / , w ∈ D ( z , r/ , and < α ( K ) < , see, for instance, Theorem 5.2 in [9]. Thus(5.9) sup | z − w | < r/ | g ( z ) − g ( w ) | = O ( r N +1 ) . This proves our statement: g is quasiregular and hence the classical Stoïlowfactorization holds; that is, g = h ◦ G , where h is holomorphic and G a qua-siconformal homeomorphism. If g is nonconstant, the quasisymmetry of G and h ( z ) = O ( | z − G ( z ) | m ) , m > , imply that there exists γ > such that cr γ sup | z − w | < r/ | g ( z ) − g ( w ) | . This would be a contradiction with (5.9). Thus g is a constant; and f ( z ) = c + c ( z − w ) , where c ∈ C , c ∈ R , proving our claim.12 Linear families of quasiregular mappings. Proofsof Theorems 1.3 and 1.2
Proof of Theorem 1.2.
Suppose Φ , Ψ ∈ W , (Ω) are solutions to the generalBeltrami equation (1.2). Moreover, assume Φ , Ψ are not affine combinations ofeach other. We show that ∂ z Φ and ∂ z Ψ are linearly independent over the field R , that is, Im (cid:18) ∂ Φ ∂z ∂ Ψ ∂z (cid:19) = 0 almost everywhere in Ω .We can assume Φ is nonconstant. As a nonconstant quasiregular mapping,it follows that Φ is discrete, open, and the branch set consists of isolated points.Thus it is enough to study points outside the branch set. Let z be such a point.There exists a ball B := D ( z , r ) such that Φ | B : B → Φ( B ) is a homeomor-phism, hence quasiconformal. From the Stoïlow factorization of general Beltramiequations, Theorem 6.1.1 in [4], we know that Ψ = F ◦ Φ in B, where F solves the reduced Beltrami equation (1.1) in Φ( B ) with λ ( w ) = − i ν ( z )1 + | ν ( z ) | − | µ ( z ) | , w = Φ( z ) , z ∈ B. Let z ∈ B . Using the chain rule and identities J ( z, f ) h ¯ w ( w ) = − f ¯ z ( z ) , J ( z, f ) h w ( w ) = f z ( z ) , where h = f − and w = f ( z ) , we arrive at J ( z, Φ) F w ( w ) = Ψ z ( z )Φ z ( z ) − Ψ ¯ z ( z )Φ ¯ z ( z )= (1 − | µ | )Ψ z Φ z − | ν | Ψ z Φ z − µν Ψ z Φ z ) , w = Φ( z ) . Thus J ( z, Φ) Im( F w ◦ Φ) = ( − | µ | − | ν | ) Im(Φ z Ψ z ) . Since Φ | B preserves sets of zero measure, the statement follows by Theorem1.1. Proof of Theorem 1.3. Step 1. Two-dimensional family . It is known that forany linear two-dimensional family F of quasiregular mappings Ω → C thereexists a corresponding general Beltrami equation(6.1) ∂f∂ ¯ z = µ ( z ) ∂f∂z + ν ( z ) ∂f∂z , | µ ( z ) | + | ν ( z ) | k < , almost everywhere in Ω , satisfied by every element f ∈ F , see the beginningof Section 5.3 in [8] or the proof of Theorem 16.6.6 in [4]. We recall the ideasof the proof for the reader’s convenience, and further, show that the associatedequation is unique. 13ssume Φ , Ψ ∈ W , (Ω) generate a linear family F of K -quasiregular map-pings. The goal is to find coefficients µ and ν such that(6.2) ∂ ¯ z Φ = µ∂ z Φ + ν∂ z Φ and ∂ ¯ z Ψ = µ∂ z Ψ + ν∂ z Ψ , almost everywhere in Ω . In the regular set R F of F , i.e., the set of points z ∈ Ω where the matrix M ( z ) = (cid:20) ∂ z Φ( z ) ∂ z Φ( z ) ∂ z Ψ( z ) ∂ z Ψ( z ) (cid:21) is invertible, the values µ ( z ) and ν ( z ) are uniquely determined by (6.2), that is, µ ( z ) = i Ψ ¯ z ( z )Φ z ( z ) − Ψ z ( z )Φ ¯ z ( z )2 Im (cid:0) Φ z ( z )Ψ z ( z ) (cid:1) , (6.3) ν ( z ) = i Φ ¯ z ( z )Ψ z ( z ) − Φ z ( z )Ψ ¯ z ( z )2 Im (cid:0) Φ z ( z )Ψ z ( z ) (cid:1) . (6.4)Note that changing the generators corresponds to multiplying M ( z ) by an in-vertible constant matrix. Hence the regular set and its complement, the singularset S F = (cid:8) z ∈ Ω : 2 i Im (cid:0) Φ z ( z )Ψ z ( z ) (cid:1) = det M ( z ) = 0 (cid:9) , depend only on the family F and not the choice of generators.It can be proven that for almost every z ∈ S F the vector (cid:0) Φ ¯ z ( z ) , Ψ ¯ z ( z ) (cid:1) lies in the range of the linear operator M ( z ) : C → C . It follows that on thesingular set one may define ν ( z ) = 0 . Here the assumption that the family F consists entirely of quasiregular mappings is needed. By quasiregularity, one hasfor every α, β ∈ R (6.5) | α ∂ ¯ z Φ( z ) + β ∂ ¯ z Ψ( z ) | k | α ∂ z Φ( z ) + β ∂ z Ψ( z ) | , for a.e. z ∈ Ω . There is a technical difficulty: the set where the inequality (6.5) holds depends,in general, on α and β . A short argument shows that (6.5) holds on the sameset of full measure for all reals, see Lemma 12.1 in [12] or p. 465 in [4].Finally, ellipticity bounds in (6.1) follow for the singular set S F by definitionof µ and ν , since Φ and Ψ are K -quasiregular. For the regular set one tests theinequality (6.5) by real-valued measurable functions θ ( z ) instead of parameters α and β .As seen above, the existence of the general Beltrami equation (6.1) followsfrom local properties. For the uniqueness we need also global qualities. HereTheorem 1.2 comes into play. The coefficients are uniquely determined on theregular set R F by (6.3) and (6.4). Moreover, Theorem 1.2 shows that the sin-gular set S F has measure zero, thus proving the uniqueness. Step 2. General linear family . It is a straightforward calculation after the two-dimensional case to achieve the same for general linear families.Since the family consists of K -quasiregular mappings, we have the inequality(6.6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈I a i ∂ ¯ z f i ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈I a i ∂ z f i ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , for a.e. z ∈ Ω , k = K − K + 1 . a i . Noting that thereare only countable many generators by assumption, the same argument as be-fore works, see Lemma 12.1 in [12] or p. 465 in [4]. Hence we have a set E ⊂ Ω of full measure such that (6.6) holds for all real coefficients a i .Note that two generating mappings, for example, f and f , define a two-dimensional linear family of quasiregular mappings, and thus functions of thetwo-dimensional family satisfy the unique general linear Beltrami equation (1.2).Further, by Theorem 1.2, we have a set of full measure E ′ ⊂ E such that ∂ z f ( z ) and ∂ z f ( z ) are R -linearly independent on E ′ .Our goal is to find H : Ω × C → C that satisfies(H1) For w , w ∈ C , |H ( z, w ) − H ( z, w ) | k | w − w | , for almost every z ∈ Ω .(H2) H ( z, ≡ .Moreover, we want every mapping f ∈ F to solve the Beltrami equation(6.7) ∂ ¯ z f ( z ) = H ( z, ∂ z f ( z )) . We define H for z ∈ E ′ by (6.7) going through all mappings f ∈ F . Thefunction H is not over-determined. Indeed, on E ′ the inequality (6.5) holds,and hence if ∂ z f ( z ) = ∂ z g ( z ) for some f, g ∈ F , then ∂ ¯ z f ( z ) = ∂ ¯ z g ( z ) . More-over, since ∂ z f ( z ) and ∂ z f ( z ) are R -linearly independent on E ′ , by our aboveremark, H ( z, w ) is defined for all w ∈ C .The definition of H and quasiregularity of mappings imply k -Lipschitz prop-erty on the second variable, that is, condition (H1). Also (H2) and that theequation (6.7) holds for all f ∈ F follow straight from definition.Finally, we see that the linearity of the family F is inherited by H , thatis, w
7→ H ( z, w ) is R -linear. Thus we have H ( z, w ) = µ ( z ) w + ν ( z ) ¯ w , where | µ ( z ) | + | ν ( z ) | k < . Since µ and ν are uniquely defined and measurable fortwo-dimensional linear families, our claim follows. Acknowledgements
The author thanks Kari Astala and Tadeusz Iwaniec for stimulating discussionson the subject of this paper.
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Department of Mathematics and Statistics,P.O. Box 68, FI-00014 University of Helsinki, Finland [email protected]@helsinki.fi