This is the accepted version of the journal article: Baisón, A. L.; Clop, Albert; Orobitg i Huguet, Joan. «Distributional solutions of the Beltrami equation». Journal of mathematical analysis and applications, Vol. 470, Issue 2 (February 2019), p. 1081-1094. DOI 10.1016/j.jmaa.2018.10.050 This version is available at https://ddd.uab.cat/record/288112 under the terms of the license Distributional solutions of the Beltrami equation A. L. Baisón, A. Clop, J. Orobitg Abstract We study the distributional solutions to the (generalized) Beltrami equation under Sobolev assumptions on the Beltrami coefficients. In this setting, we prove that these distributional solutions are true quasiregular maps and they are smoother than expected, that is, they have second order derivatives in L1+ε loc , for some ε > 0. 2010 Mathematics Subject Classification. 30C62 (35J46) Key words and phrases. Quasiconformal mapping, Beltrami’s equation, Distributional solution, Beltrami operators. 1 Introduction In this work we consider the R-linear Beltrami equation (1) ∂¯ f (z) − µ(z) ∂f (z) − ν(z) ∂f (z) = 0, a.e. z ∈ C where µ and ν, called the Beltrami coefficients of f , are L∞ (C; C) functions such that k|µ|+|ν|k∞ = k < 1, which means that the equation (1) is uniformly elliptic. The derivatives ∂f and ∂f are understood in the distributional sense. Usually, when ν = 0, the equation (1) is called the complex Beltrami equation. When µ = 0 it is called the conjugate Beltrami equation. In the general form one says (1) as the generalized Beltrami equation. 1,2 If a function f : Ω → C belonging to the Sobolev space Wloc satisfies (1), then its distribu- tional derivatives satisfy the distortion inequality, (2) |∂¯ f (z)| ≤ k |∂f (z)| a.e. z ∈ Ω. 1+k Such mappings f are called K-quasiregular, K = 1−k . Bijective K-quasiregular maps are called K-quasiconformal. Quasiregular maps are automatically Hölder continuous, and either 1,p constant or open and discrete. Weakly K-quasiregular maps are functions f ∈ Wloc for some p ≥ 1 and such that (2) is satisfied. In the work [4] from Astala, Iwaniec and Saksman, it was 1 2 A.L. Baisón, A. Clop and J. Orobitg 1,p 2K shown that if a weakly K-quasiregular mapping belongs to Wloc for some p > 1 + k = K+1 1,q 2K actually it belongs to Wloc for all q < K−1 and, therefore, it is a true K-quasiregular map. 1 −1 −1 2K The example f (z) = (z|z| K ) shows that for any p < K+1 there exists a weakly K- 1,p quasiregular map in Wloc which is not K-quasiregular. Later, Petermichl and Volberg [13] 2K proved that the weaker assumption p ≥ K+1 is already sufficient for weakly K-quasiregular maps to be K-quasiregular. (See the monograph [3] for a complete and detailed information). If instead of looking at the distortion inequality (2) one looks at the regularity of Beltrami coefficients, one can say more about quasiregular maps. This was already noticed by Iwaniec [11] for the complex Beltrami equation (see [12] or [8] for a proof for the generalized Beltrami equation). Theorem 1 ([11]). Let µ, ν ∈ VMO be compactly supported, and such that k|µ| + |ν|k∞ = 1,r 1,s k < 1. If f ∈ Wloc for some r ∈ (1, ∞) and further f solves (1) then f ∈ Wloc for every s ∈ (1, ∞). In particular, f is quasiregular. In this sense, the work [10] is a good reference to summarize this aspect about the regularity of the Beltrami coefficients. Recall that VMO denotes the space of vanishing mean oscillation functions, that is, the closure of compactly supported smooth functions with the BMO norm. Obviously the above result is remarkable when r is close to 1. It turns out that one might 1,r go even beyond the space Wloc , r > 1, and deal with solutions f ∈ Lrloc in the sense of distributions. Of course, this requires an extra degree of smoothness on µ and ν, so that the distribution ∂¯ f − µ ∂f − ν ∂f is well defined. Such question for ∂¯ f − µ ∂f was treated in detail in [9]. Later, in [7] interest arose in the counterparts for the distributional conjugate Beltrami operator ∂f − ν ∂f with real valued coefficient ν ∈ W 1,p , p > 2. The well known Weyl’s Lemma asserts that if T is any (Schwartz) distribution such that ∂¯ T (ϕ) = 0 for each test function ϕ ∈ D (by D we mean the algebra of compactly supported C ∞ functions), then T agrees with a holomorphic function. In other words, distributional solutions to Cauchy-Riemann equation are actually strong solutions. When trying to extend this kind of result to the generalized Beltrami equation, one first must define the distribution  (3) ∂ − µ∂ − νC∂ T = ∂T − µ∂T − ν∂T . Here and henceforth, C denotes the complex conjugation operator Cf = f . The expression (3) needs not make sense, because bounded functions in general do not multiply distributions nicely. However, if the multiplier is asked to exhibit some regularity, and the distribution T is an integrable function, then something may be done. Namely, given a function f and a Distributional Beltrami Equation 3 test function ϕ, one can write  ∂ − µ∂ − νC∂ f , ϕ = ∂f − µ ∂f − ν ∂f , ϕ = = − h f , ∂ϕ i + f ,∂ (µϕ) + f , ∂ (ν ϕ) (4) = − h f , ∂ϕ i + f , ϕ∂µ + f , ϕ∂ν + f , µ∂ϕ + f , ν∂ϕ , whenever each term makes sense. For instance, this is the case if µ, ν ∈ W 1,p (C) are compactly supported and f ∈ Lqloc (C) with 1 p + 1 q = 1. Hence we can call ∂f − µ∂f − ν∂f the Beltrami distributional derivative of f . Analogously, we say that f is a distributional solution of ∂¯ f (z) = µ(z) ∂f (z) + ν(z) ∂f (z) + h, when D E ∂¯ f (z) − µ(z) ∂f (z) − ν(z) ∂f (z), ϕ = h h, ϕi for all ϕ ∈ D. Observe that the conditions µ, ν ∈ W 1,p (C) compactly supported, h ∈ L1loc (C) and f ∈ r r−1 Lloc (C) with r ≤ p are sufficient for (4) to make sense. Our first result extends [7, Corollary 3.3] to the case p = 2, and meets its C-linear counterpart in [9, Theorem 7]. Theorem 2. Let 1 < r < p ≤ ∞ and let µ, ν ∈ W 1,p (C) be compactly supported such that r K−1 r−1 k|µ| + |ν|k∞ = k = K+1 < 1. Assume that f ∈ Lloc (C) is a distributional solution of (1), that is, ∂¯ f (z) = µ(z) ∂f (z) + ν(z) ∂f (z). 2,p 1. If p > 2, then f ∈ Wloc (C). 2,q 2. If p = 2, then f ∈ Wloc (C), for all q < 2. 2K 1 1 2,s 3. When µ ≡ 0, if K+1 < p < 2 and r > p + K−1 2K , then f ∈ Wloc (C) for all 1 s > 1 p + K−1 2K 1 1 K−1 2,s 1 1 K−1 4. When K < 2, K < p < 2 and r > p + 2K , then f ∈ Wloc (C) for all s > p + 2K In all previous cases, f is quasiregular. We emphasize this fact in the following corollary. The result is new for cases ν 6= 0 and p ≤ 2. Corollary 3. Let 1 < p ≤ ∞ and let µ, ν ∈ W 1,p (C) be compactly supported such that 1,2 k|µ| + |ν|k∞ = k < 1. Assume that f ∈ Wloc (C) satisfies ∂¯ f (z) = µ(z) ∂f (z) + ν(z) ∂f (z), a.e. z ∈ C. 2,p 1. If p > 2, then f ∈ Wloc (C). 2,q 2. If p = 2, then f ∈ Wloc (C), for all q < 2. 2K 2,s 1 1 K−1 3. When µ ≡ 0 and K+1 < p < 2, then f ∈ Wloc (C) for all s > p + 2K . 4 A.L. Baisón, A. Clop and J. Orobitg 2,s 1 1 K−1 4. When K < 2 and K < p < 2, then f ∈ Wloc (C) for all s > p + 2K . We note that if f is quasiconformal, following an argument similar to that in [9, Proposition 4], we can remove the restriction K < 2. That is, K−1 Proposition 4. Let µ, ν ∈ W 1,p (C) be compactly supported such that k|µ| + |ν|k∞ = K+1 2K 1,2 for some K ≥ 1 and K+1 < p < 2. Assume that f ∈ Wloc (C) is a homeomorphism satisfying (1). Then, 2,s 1 1 K −1 f ∈ Wloc (C), for all > + . s p 2K At the end of this note we’ll provide a proof. Let’s mention that Proposition 4 is useful to study (1) when µ, ν ∈ W s,p (C) with 1 < s < 2 and sp < 2 (see [14]). 1 As stated in [9, p.205], the radial stretching f (z) = z|z| K −1 if |z| ≤ 1 and f (z) = z if |z| > 1 shows the sharperness of the above proposition because f has Beltrami coefficient in W 1,p for 2K every p < 2 and, however, the second derivatives of f live in no better space than L 2K−1 ,∞ . 2 Preliminaries ´ When f is a locally integrable function and ϕ is a test function, usually one sets hf, ϕi = f ϕ̄, ´ but here we will prefer the real inner product h f , g i = Re f g. We note that under this structure, the distributional derivatives behave as ˆ ∂f , ϕ = −Re f ∂ϕ = − h f , ∂ϕ i ˆ h ∂f , ϕ i = −Re f ∂ϕ = − f , ∂ϕ . Meanwhile the pointwise multiplication for proper functions µ satisfies ˆ ˆ h µ f , ϕ i = Re µ f ϕ = Re f µϕ = hf , µϕi . Naturally, this is consistent with the integral expression that we have used in the definition of the Beltrami distributional derivative. This does not prevent us of using C-valued test functions. In particular, if a C-valued function f ∈ L1loc is such that ˆ ∂f , ϕ = −Re f ∂ϕ = 0, for each C-valued ϕ ∈ D, then f is holomorphic as stated by the classical Weyl’s Lemma. The reason why we use this notion of duality is that the conjugation operator becomes R-self-adjoint ˆ ˆ h Cf , g i = Re f g = Re f g = h f , Cg i . Related to the Beltrami equations appear the Cauchy transform and the Beurling transform. By Cf one denotes the solid Cauchy transform of f , ˆ 1 f (w) Cf (z) = dA(w), π C z−w Distributional Beltrami Equation 5 while by Bf one means the Beurling transform of the function f , a principal value convolution operator, ˆ ˆ −1 f (w) −1 f (w) Bf (z) = lim 2 dA(w) = pv dA(w). π ε→0 |w−z|≥ε (w − z) π (w − z)2 The basic relations are ∂C = B and B∂ = ∂. By B ∗ we mean the singular integral operator obtained by simply conjugating the kernel of B, that is, ˆ −1 f (w) B ∗ f (z) = pv dA(w). π (w̄ − z̄)2 As Calderón-Zygmund operators, B and B ∗ are bounded on Lp (C), 1 < p < ∞. Even more, they are bounded on W s,p (C), because Sobolev spaces W s,p = Gs ∗ Lp are Bessel potential spaces. Note that B ∗ is the L2 -inverse of B (BB ∗ = B ∗ B = Id), moreover it is also the adjoint of B hBf, gi = hf, B ∗ gi and CB(f ) = B(f ) = B ∗ C(f ). From now, and abusing of notation, we’ll denote CB by B. For smooth functions with compact support f , we clearly have C(∂f ) = ∂C(f ) = f, ∂B(f ) = B(∂f ) and ∂B(f ) = B(∂f ) = ∂C(∂f ) = ∂f, ∂B (f ) = B ∗ (∂f ) = ∂C ∗ (∂f ) = ∂f, ∗ and these identities extend for Sobolev functions in W 1,p , p > 1. 0 When h ∈ Lp , p0 = p/(p − 1), 1 < p < ∞, the distributions ∂h and ∂h act continuosly on g ∈ W 1,p h∂h, gi = −hh, ∂gi, h∂h, gi = −hh, ∂gi. In this setting, B(∂h) can be defined as the distribution ∂(Bh), that is, hB(∂h), gi = h∂h, B ∗ gi = −hh, ∂B ∗ gi = −hh, B ∗ (∂g)i = −hBh, ∂gi = h∂(Bh), gi 0 and also ∂Bh = B∂h. So, when h ∈ Lp we have B(∂h) = ∂h and B ∗ (∂h) = ∂h in the distributional sense. After these considerations, we write ∂f (z) − µ(z) ∂f (z) − ν(z) ∂f (z) = h in the form (Id − µ B − ν B)(∂f ) = h, where Id denotes the identity operator. Therefore, we realize that the study of (1) is linked to the invertibility of the Beltrami operator Id − µ B − ν B in an appropriate function space. This is the natural argument that one uses in the remarkable papers from Ahlfors [1], Iwaniec [11] and Astala, Iwaniec and Saksman [4], as well as the most recent works on the topic (see [10], [12], [8]). In addition, in this note we will follow ideas from Clop et al. [9] and Baratchart et al. [7]. 6 A.L. Baisón, A. Clop and J. Orobitg 3 Distributional solutions are true solutions We will begin proving that distributional solutions of (1) under the hypothesis from Theorem 2 have first derivatives locally integrable. However, to get Theorem 2 we don’t use the Stoilow Theorem, which was fundamental in the approach of [9]. Theorem 5. Let 1 < r < p ≤ ∞ and let µ, ν ∈ W 1,p (C) be compactly supported such that r r K−1 r−1 r−1 k|µ| + |ν|k∞ = K+1 for some K ≥ 1. Fix h ∈ Lloc (C). Assume that f ∈ Lloc (C) is a distributional solution of (5) ∂¯ f (z) − µ(z) ∂f (z) − ν(z) ∂f (z) = h. Then we have r 1, 1. f ∈ Wlocr−1 (C), if p ≥ 2. 1,2 2K 1 1 K−1 2. f ∈ Wloc (C), if K+1 < p < 2 and r > p + 2K . Proof. The scheme of the proof is as follows. First of all we localize the solution f to get another generalized Beltrami equation where all the involved terms will have compact support. Then we will check that a bounded combination of this located function have first derivatives. Finally, we transfer the integrability of the first derivatives to the original solution f . Consider an smooth function with compact support ψ ∈ Cc∞ (C) with real values and define r F = ψ f ∈ Lcr−1 (C). Obviously, the global regularity on F will imply the local regularity on f , as we wish. Using (5), µ, ν ∈ Wc1,p (C) and ψ is real valued, one easily checks that F is a distributional solution of (6) ∂F = µ ∂F + ν ∂F + H ,  r where H = f − νf ∂ψ − µ f ∂ψ + hψ belongs to Lcr−1 (C). Now we define G = ( Id − µ B − ν C ) F . The distributional derivative ∂G of G has sense r because G ∈ Lcr−1 (C) (the compact support of G comes from the compact support of F , µ and ν). Then for any ϕ ∈ Cc∞ (C) we have ∂G , ϕ = − h G , ∂ϕ i = − F − ν F − µ BF , ∂ϕ = − h F , ∂ϕ i + ν F , ∂ϕ + h µ BF , ∂ϕ i = ∂F , ϕ − ∂F , ν ϕ − F ∂ν , ϕ + h BF , ∂ ( µ ϕ ) i − ∂µ BF , ϕ = ∂F , ϕ − ∂F , ν ϕ − F ∂ν , ϕ − ∂∂CF , µ ϕ − ∂µ BF , ϕ = ∂F , ϕ − ∂F , ν ϕ − F ∂ν , ϕ − h ∂F , µ ϕ i − ∂µ BF , ϕ = H − F ∂ν − ( BF ) ∂µ , ϕ . Distributional Beltrami Equation 7 Here we have used ∂CF = F in the fifth line and (6) in the last one. Consequently G is a distributional solution of (7) ∂G = H − F ∂ν − ( BF ) ∂µ pr 0 and clearly ∂G ∈ Lsc (C) with s = p+r 0 and r0 = r r−1 . Applying the Cauchy transform to the equation (7) we get (because G has compact support)  G = C H − F ∂ν − ( BF ) ∂µ and then ∂G = B H − F ∂ν − ( BF ) ∂µ ∈ Ls ( C ) .  Therefore, G ∈ Wc1,s ( C ). G − νG Remark. If µ ≡ 0 then G = (Id − ν C)F , that is, F = ∈ Wc1,s ( C ). This reinforces 1 − |ν|2 the fact already observed in [7] that the study of the regularity in the conjugate Beltrami equation is easier than in the complex Beltrami equation. We want to transfer the regularity from G to F , and therefore to f . We’ll work with some identities with the distributional derivatives of F and finally we’ll conclude that ∂F is an integrable function. For all ϕ ∈ Cc∞ ( C ) we have h ∂G , ϕ i = − G , ∂ϕ = − F + ν F + µ ( BF ) , ∂ϕ = h ∂F , ϕ i + F , ∂ (ν ϕ) + BF , ∂ ( µ ϕ ) − F ∂ν , ϕ − h ( BF ) ∂µ , ϕ i . Now, we replace h F , ∂ ( ν ϕ ) i using that ∂F = ∂F , moreover ∂F satisfies (6) and the expression h∂ F , φ i = h ∂F , µ φ i + h ∂F , ν φ i + h H , φ i holds for any φ ∈ Wc1,p (C) ∩ L∞ (C). After this change we reach that D E 2 h ∂G , ϕ i = h ∂F , ϕ i − ∂F , | ν | ϕ − ∂F , µ ν ϕ + BF , ∂ ( µ ϕ ) − νH, ϕ − F ∂ν + ( BF ) ∂µ , ϕ . Simplifying and rearranging terms, ∂ F , 1 − |ν|2 ϕ  + BF , ∂ ( µ ϕ ) − ∂F , µ ν ϕ = = ∂G + ν H + F ∂ν + ( BF ) ∂µ , ϕ . Furthermore, this equality extends to any ϕ in Wc1,p (C)∩L∞ (C). In particular, for functions of type ϕ = Ψ 1−|ν|2 with Ψ ∈ Cc∞ (C) arbitrary. Thus,     µ µν h∂ F , Ψ i − ∂ BF , Ψ − ∂F , Ψ 1 − |ν|2 1 − |ν|2 8 A.L. Baisón, A. Clop and J. Orobitg * + ∂G + ν H + F ∂ν + ( BF ) ∂µ = 2 ,Ψ . 1 − |ν | Recall that ∂BF = B∂F and B ∗ is the adjoint operator of B. Rewriting the above equality we get * ! + ∗ µ µν (8) ∂F , Id − B 2 −C 2 Ψ 1 − |ν | 1 − |ν | * + ∂G + ν H + F ∂ν + ( BF ) ∂µ = 2 ,Ψ . 1 − |ν | Now the invertibility of the operator µ µν T := Id − B ∗ 2 −C 2 1 − |ν | 1 − |ν | and its adjoint µ µν T ∗ := Id − 2 B − 2 C 1 − |ν | 1 − |ν | come into play (see Lemma 6 below). 2K 1 1 K−1 Second part of Theorem 5. Now, since K+1 < p < 2 and r > p + 2K we have 2K pr0 s 2K 1, we can ensure that B∂F = ∂F and write r with H ∈ Ltc (C) for all t ≤  (9) Id − µ B − ν B ∂F = H , . r−1 r Since r < p < 2 one has > 2. For t = 2 we use the invertibility of the Beltrami r−1 operator, −1 ∂F = Id − µ B − ν B H Distributional Beltrami Equation 9 1,2 to get F ∈ Wc1,2 (C), and so f ∈ Wloc (C) as we wished. First part of Theorem 5. In this case the operators T and T ∗ are invertible on Lq (C) for all q ∈ (1, ∞). Repeating the previous argument we arrive to (9) and then −1 r ∂F = Id − µ B − ν B H ∈ L r−1 (C), r 1, implying that f ∈ Wlocr−1 (C). Lemma 6. Let µ, ν ∈ L∞ (C) be compactly supported such that k|µ| + |ν|k∞ = k < 1. Let T and T ∗ the two operators previously defined. 2K 2K (a) Both operators are invertible on Lt (C) when 1 + k = K+1 2. By Theorem 5 we know F ∈ Wc (C) and satisfies (6) ∂F = µ ∂F + ν ∂F + H , where  (10) H= f − νf ∂ψ − µ f ∂ψ r r 1, belongs to Lcr−1 (C). Following this scheme we also get f ∈ Wlocr−1 (C). Therefore, by the 2r r−2 Sobolev embedding, we have f ∈ Lloc (C) if r > 2, f ∈ Lqloc (C) for all q < ∞ if r = 2 and 2r f ∈ L∞ r−2 loc (C) if r < 2. Consequently, H ∈ Lc (C) when r > 2, H ∈ Lqc (C) for all q < ∞ if r = 2 and H ∈ L∞ c (C) when r < 2. From the identity −1 (11) ∂F = Id − µ B − ν B H and by the invertibility of the Beltrami operator we have F ∈ Wc1,t (C) for some t > 2 regardless of the value of r. Then, F ∈ L∞ ∞ ∞ c (C), f ∈ Lloc (C) and so H ∈ Lc (C). Re-using the invertibility of the operator, this time we get F ∈ Wc1,q (C) for all q < ∞. Immediately 1,q we have f ∈ Wloc (C) for all q < ∞. With this, it is easy to verify that H ∈ Wc1,p (C). Now, since p > 2, we know (see [10, Section 3] and the argument in [8, Lemma 10]) that the generalized Beltrami operator Id − µ B − ν B : W 1,p ( C ) → W 1,p ( C ) is bounded and boundedly invertible. From (11) ∂F ∈ Wc1,p (C) (and ∂F = B(∂F ) too), 2,p that is, f ∈ Wloc (C). Part 2. When p = 2 the proof follows the same argument as in Part 1. Just keep in mind that in this case (see Lemma 7 below) the operator Id − µ B − ν B : W 1,q ( C ) → W 1,q ( C ) 2,q is bounded and boundedly invertible for all q ∈ (1, 2) (but not for q = 2). Thus, f ∈ Wloc (C) for all q < 2 as we claimed. 2K For a moment, we consider Part 3 and Part 4 together. Recall that K+1 < p < 2 and 1 1 K−1 1,2 r > p + 2K . By Theorem 5 we know F ∈ Wc1,2 (C) and f ∈ Wloc (C). Then, the function H given by (10) belongs to Ltc (C) for all t < ∞. The invertibility of Id − µ B − ν B and (11) give ∂F ∈ Lqc (C) (and so, ∂F too) for all 2 ≤ q < 2K K−1 . In particular we obtain f ∈ L∞ loc (C) and with that we achieve H ∈ Wc1,p (C) ∩ L∞ . Distributional Beltrami Equation 11 Define G := ∂F . Our goal is to check that the distribution ∂G is in fact an integrable function. Consider Ψ ∈ Cc∞ (C) arbitrary, then it is true that ∂G , Ψ = − h G , ∂Ψ i = − h ∂F , ∂Ψ i = − B(∂F ) , ∂Ψ = − ∂F , B ∗ (∂Ψ) = − ∂F , ∂Ψ = − ∂F , µ ∂Ψ − ∂F , ν ∂Ψ − H , ∂Ψ = − G , µ ∂Ψ − G , ν ∂Ψ − H , ∂Ψ = h ∂G , µ Ψ i + ∂G , ν Ψ + h ∂H , Ψ i + G ∂µ + G ∂ν , Ψ . That is, (12) ∂G , Ψ = h ∂G , µ Ψ i + ∂G , ν Ψ + G ∂µ + G ∂ν + ∂H , Ψ Note that µΨ, νΨ ∈ Wc1,p (C). Thanks to this we can ensure that (13) h ∂G , µ Ψ i = − G , ∂ ( µ Ψ ) = − h G , ∂B ∗ ( µ Ψ ) i = ∂G , B ∗ ( µ Ψ ) and (14) ∂G , ν Ψ = C∂G , ν Ψ = ∂G , C ( ν Ψ ) . Plugging (13) and (14) into (12), we have (15) ∂G , ( Id − B ∗ µ − Cν ) Ψ = G ∂µ + G ∂ν + ∂H , Ψ . 1 1 K−1 By construction, G ∂µ, G ∂ν, ∂H ∈ Lsc (C) for any 1 > s > p + 2K . Therefore, (15) has 0 s 0 s sense for all Ψ ∈ L (C), s = s−1 . Part 3. Since µ ≡ 0, (15) becomes ∂G , ( Id − Cν ) Ψ = G ∂ν + ∂H , Ψ . The operator Id − Cν and its adjoint Id − νC are obviously invertible on Lt for any t. ϕ+ν ϕ̄ Take Ψ := (Id − Cν)−1 ϕ = 1−|ν|2 and so ∂G , ϕ = G ∂ν + ∂H , (Id − Cν)−1 ϕ = (Id − νC)−1 (G ∂ν + ∂H) , ϕ 0 2,s ¯ for all ϕ ∈ Ls . In conclusion ∂∂F = ∂G ∈ Lsc and f ∈ Wloc (C) if 1 > 1 + K−1 s p 2K . Part 4. The operator Id − B ∗ µ − Cν and its adjoint Id − µB − νC are invertible on Lt if 2K 2K K+1 0 such that |f (z) − a z| ≤ |z| as |z| → ∞. In particular, this tells us that the value ∂f (∞) is well defined, and uniquely determines a well defined branch of the complex logarithm log ∂f , precisely as done in [5]. We mollify µ and ν and obtain a sequence µn , νn ∈ C ∞ of compactly supported coefficients, K−1 |µn | + |νn | ≤ |µ| + |ν| ≤ K+1 such that lim kµn − µkW 1,p (C) + kνn − νkW 1,p (C) = 0. n→∞ To each pair µn , νn we can associate a unique quasiconformal map fn such that ∂fn − a = C B(∂fn ), and in particular |fn (z) − a z| ≤ |z| as |z| → ∞, since ∂fn has compact support. It can be shown that in this situation the sequences {∂fn }n and {∂fn − a}n are bounded in 2K Lr (C) for each 2 < r < K−1 , and indeed, modulo subsequences, 2K lim kDfn − Df kLr (C) = 0, 2 + . s 2K p It then follows that {∂∂fn }n has a weak accumulation point in Lsloc (C), which obviously can only be ∂∂f , so that ∂∂f ∈ Lsloc (C). A similar argument shows that ∂∂f belongs to Lsloc (C) (actually Ls (C), since ∂f has compact support). In order to see that ∂∂f ∈ Ls (C), just notice that from ∂f − a = B(∂f ) one gets ∂f = B ∗ (∂f − a). Thus, ∂∂f = B ∗ (∂∂f ). Acknowledgements. The authors were partially supported by research grants 2017SGR395 (Generalitat de Catalunya), MTM2016-75390-P (Spanish Government). A.C. was partially supported by MTM2016-81703-ERC (MINECO, Spain). References [1] L. Ahlfors, Lectures on quasiconformal mappings. Second ed., University Lecture Series, 38, American Mathematical Society, Providence, RI, 2006. [2] K. 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Cruz, Weighted estimates for Beltrami equations, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 1, 91–113. [9] A. Clop, D. Faraco, J. Mateu, J. Orobitg, and X. Zhong, Beltrami equations with coefficient in the Sobolev Space W 1,p , Publ. Mat. 53 (2009), 197-230. [10] V. Cruz, J. Mateu and J. Orobitg. Beltrami Equation with Coefficient in Sobolev and Besov Spaces. Canad. J. Math. 65 (2013), 1217-1235. 16 A.L. Baisón, A. Clop and J. Orobitg [11] T. Iwaniec, Lp -theory of quasiregular mappings, Quasiconformal space mappings, vol- ume 1508 of Lecture Notes in Math., pp 39–64. Springer, Berlin, 1992. [12] A. Koski, Singular integrals and Beltrami type operators in the plane and beyond. Mas- ter Thesis, Department of Mathematics, University of Helsinki, 2011. [13] S. Petermichl, A. Volberg,Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281–305. [14] M. Prats, Beltrami equations in the plane and Sobolev regularity, arXiv:1606.07751. 2016 Antonio L. Baisón Olmo Departamento de Ciencias Básicas. Universidad Autónoma Metropolitana - Unidad Azcapotzalco. Av. San Pablo No. 180 Col. Reynosa Tamaulipas C.P. 02200 Delegación Azcapotzalco Distrito Federal (México) A. Clop, J. Orobitg Departament de Matemàtiques Universitat Autònoma de Barcelona 08193-Bellaterra (Catalonia) E-mail addresses: albo@correo.azc.uam.mx albertcp@mat.uab.cat orobitg@mat.uab.cat