Beltrami equations in the plane and Sobolev regularity

New results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation $\bar{\partial} f = \mu \partial f + \nu \overline{\partial f}$ for discontinuous Beltrami coefficients $\mu$ and $\nu$ are obtained, using Kato-Ponce commutators, obtaining that $\overline \partial f$ belongs to a Sobolev space with the same smoothness as the coefficients but some loss in the integrability parameter. A conjecture on the cases where the limitations of the method do not work is raised.

Theorem 1.1 ([AIS01, Theorem 3]). Given µ, ν P L 8 with }|µ|`|ν|} L 8 " κ ă 1, the operator Id´µB´νB is invertible on L p pCq for 1 pκ ă 1 p ă 1 When the coefficients satisfy some extra assumption, we can improve the previous results. We discuss this results in Section 2.3, after a brief introduction of the function spaces we consider. Nevertheless, we sketch here the general facts. Given 0 ă s ă 8 and 1 ă p ă 8, we say that the Sobolev space W s,p pR d q (in the sense of Bessel potential spaces, see [Tri83, Section 2.2.2]) is critical if s´d p " 0. If s´d p ą 0 it is called supercritical and if, instead, s´d p ă 0, then it is called subcritical. Note that so-called differential dimension s´d p coincides with the homogeneity exponent of the semi-norms of these spaces. The functions in supercritical spaces are continuous, while the functions on critical spaces are just in the space of vanishing mean oscillation functions (V M O), i.e., the closure of C 8 c in BM O. For the subcritical spaces we have less self-improvement (see Section 2.1 below).
Roughly speaking, when the coefficients µ and ν are in a supercritical Sobolev space, thenBf inherits the regularity of the Beltrami coefficient. In the critical situation, there is a small loss, and in the subcritical case, there is a bigger gap, in the spirit of (1.3), where µ is in every L p space butBf is only p-integrable for a certain range.
The supercritical case is well understood (see [AIM09,Chapter 15] for Hölder spaces and [CFM`09] and [CMO13] for Sobolev, as well as Besov and Triebel-Lizorkin spaces). The critical case is studied in the latter two papers as well as in [BCO17] and [BCG`16], while the literature on the subcritical cases is less complete (see [CFM`09] and [CFR10]). This note is devoted to unify the approaches for the critical and subcritical situations, in the quest to find a complete sharp theory.
Sharp bounds in this theory may lead to a better understanding of the stability of the Calderón inverse problem, as shown in [CFR10]. There, the authors prove that if one knows all the possible pairs of Dirichlet and Neumann data of the solutions to the conductivity equation for conductivities satisfying certain a priori subcritical Sobolev conditions, then the recovery is stable. A crucial step there is to solve a Beltrami equation as (1.2) above: after showing Sobolev regularity of the principal solution to a certain family of equations, the authors show an asymptotic decay of the so-called Complex Geometric Optics Solution. Greater Sobolev regularity of these solutions is translated into higher decay of the solutions and better stability estimates.
We show the following result.
If s " 2 p , thenB f P W s,q for every 1 q ą 1 p . (1.5) If s ă 2 p and 1 p ă 1 (1.6) However, the restriction 1 p ă 1 p 1 κ´1 pκ seems rather unnatural (see Section 2.3). Due to this fact, there is only room for p in the conditions for (1.6) if s ă 2 1´κ 1`κ , which is equivalent to κ ă 2´s 2`s . Therefore it is natural to ask whether it can be removed or not (see Conjecture 2.4 below).
The paper is organized in the following way. Section 2 is devoted to making the background of this article clear. In Section 2.1 the definitions and basic properties of the Triebel-Lizorkin and related spaces are given. Section 2.2 specifies some properties of compactly supported Triebel-Lizorkin functions, in order to make a clear picture of the problem and to provide the reader a guide to understand the full scale of Sobolev regularity obtained for the principal mappings in Theorem 1.2. In Section 2.3 there is a discussion on the existing results using the concepts introduced in the former sections. Finally, Section 3 contains the proof of Theorem 1.2.

Definitions and well-known properties of function spaces
First we recall some results on Triebel-Lizorkin spaces.
We will use the classical notation p f for the Fourier transform of a given Schwartz function, and q f will denote its inverse. It is well known that the Fourier transform can be extended to the whole space of tempered distributions by duality and it induces an isometry in L 2 (see for example [Gra08, Chapter 2]).
Definition 2.1. Let s P R, 0 ă p ă 8, 0 ă q ď 8. For any tempered distribution f P S 1 pR d q we define its non-homogeneous Triebel-Lizorkin quasi-norm and we call F s p,q Ă S 1 to the set of tempered distributions such that this quasi-norm is finite. These quasi-norms (norms when p, q ě 1) are equivalent for different choices of tψ j u j"0 (see [Tri83, Section 2.3]). Changing the order of integration and summation above we get the nonhomogeneous Besov quasi-norm which makes sense for 0 ă p ď 8. For q " 2 and 1 ă p ă 8 the Triebel-Lizorkin spaces coincide with the so-called Bessel-potential spaces. In addition, if s P N they coincide with the usual Sobolev spaces of functions in L p with weak derivatives up to order s in L p , and they coincide with L p for s " 0 ([Tri83, Section 2.5.6]). In the present text, we use the convention W s,p :" F s p,2 for s ě 0 and 1 ă p ă 8. and for s " 0 we write h p :" F 0 p,2 for 0 ă p ă 8, that is, the non-homogeneous hardy space (which coincides with L p for 1 ă p ă 8 . For instance, if´8 ă s ă 8, 0 ă p ă 8, 0 ă q 0 , q 1 ď 8 and ε ą 0, we have that (2.2) (see Figure 2.1). Besov spaces present a similar structure. Regarding classical spaces, whenever 0 ă p ă 8 and 0 ă q ă 8, the following holds true:

Compactly supported functions
Let us assume that µ P W s,p is compactly supported with p ą 1. Since µ P L ppqs (note that one can do the same if µ P F s p,q when s´d p ą´d as we noted above), which coincides with the Hardy space h p , also µ P h r for 0 ă r ă p (see [Tri83, Section 2.2.2]). By interpolation, we have that µ P W σ,r as well for σ ă s and 1 ă r ď p. Combined with the embeddings described in (2.1) and (2.2), this gives us an almost complete picture of the spaces where µ does belong. It remains to see what happens in the endpoint σ " s and in the remaining spaces of the scale F s p,q . If s is a natural number, then the derivatives of order s are compactly supported as well and it follows that µ P W s,r for every 0 ă r ď p, but in case s is not a natural number, some extra work needs to be done. Since the author does not know any mention in the literature about this case, it is studied in this section.
We will argue using expressions in terms of differences. Let us write ∆ 1 is compactly supported in the disk D R`M for t ď 1, and so is (with the usual modifications for q " 8).
By hypothesis we have d mintp0,p1,qu´d ă s. By [Tri06, Theorem 1.116], writing p j :" maxt1, p j u for every locally integrable function g we have that Now, the norm in (2.3) and the Hölder inequality grant that Remark 2.3. Note that the one can obtain analogous results for Besov spaces using also norms in terms of differences, with the more general assumption 0 ă q ď 8, d d`s ă p 1 ď p 0 ď 8. The norm in [Tri83, (2.5.12.4)] by replacing ş R d by ş |h|ď1 , for instance, will do the job.
Finally let us compile all the information regarding functions such as the Beltrami coefficients involved in Theorem 1.2 (see Figure 2.2 (b)). In this case, we can interpolate norms with L 8 by [RS96, Theorem 2.2.5]: Let 0 ă s 2 ă s ă d p , d d`s ă p 1 ď p ă 8 and 0 ă p 2 ă 8, 0 ă q ď q 1 ď 8 with d d`s ă q 1 and 0 ă q 2 ď 8. For every compactly supported function µ P F s p,q X L 8 we have that as long as s 2 p 2 ď sp. (2.4)

Discussion on the former regularity results
Kari Astala showed in [Ast94] that every quasiconformal mapping f with Beltrami coefficient µ P L 8 compactly supported in the unit disk with }µ} L 8 " κ ă 1 satisfies that L pp2qs 2 (a) Embeddings for compactly supported functions.  (see Figure 2.3(a)). Moreover, the operator H " Id´µB´νB is invertible in Lebesgue spaces with exponent on the critical range pp 1 κ , p κ q as shown in [AIS01]. When the regularity of the coefficients is greater, we can expect the principal solution to (1.2) to have greater regularity as well. The first result in that direction was given by Iwaniec, who could prove the compactness of the commutator rµ, Bs in every L p space when µ is a Beltrami coefficient in V M O. By means of a Fredholm theory argument this implies the invertibility of H in L p when ν " 0, and the general case is shown by the same argument (see Figure 2.3(b)).
In recent years there has been a great improvement in results. A remarkable one, given by • If 1 p ă 1 2 , thenBf P W 1,p .
Notice how W 1,p being either supercritical, critical or subcritical determines the regularity behavior of f . In the subcritical case, the setting ν ‰ 0 was studied in [Bai16, Corolario 3.8]. The author shows that when 1 2 ă 1 p ă 1 p 1 κ´1 pκ , thenBf P W 1,q for every 1 q ą 1 p`1 pκ . Here we observe the same phenomenon as in Theorem 1.2, that is, we need that κ ă 1 3 in order to have room for p in the hypothesis.
In the supercritical case there is no loss of regularity: if µ P X, thenBf P X as well. Indeed this is the case in the Hölder spaces of fractional order (see [AIM09, Chapter 15], Figure 2.3(c) above) and in the whole Triebel-Lizorkin and Besov scales, as shown by Cruz, Mateu and Orobitg, see [CMO13, Theorem 1], Figure 2.3(f). Here, the authors had to prove the invertibility of I´µB on F s p,q which they did using Fredholm theory following Iwaniec's scheme, since the boundedness of B on Triebel-Lizorkin spaces can be deduced from [FTW88, Corollary 3.33], while in Besov spaces it is a consequence of Fourier multipliers theory and interpolation (see [Tri83, Section 2.6], for instance). In the supercritical context but restricted to domains, µ P W s,p pΩq, there are also some positive results in [CMO13] and in [Pra15].    When µ belongs to a critical space X Ă V M O, we cannot expect thatBf is in X, but the loss is minimal. The first result in that spirit is Iwaniec's above. This theorem has been extended to other critical spaces in [CFM`09] for s " 1 as commented above, and in [BCO17, Theorem 1] which settles the case 1 2 ď s ă 1, finding that µ P W s,2{s ùñ logpBf q P W s,2{s .
This result implies (1.5) for 1 2 ď s ă 1. Thus, the progress in Theorem 1.2 for the critical setting is to cover the whole range 0 ă s ă 2. Finally, small improvements on the spaces lead to better results, as shown in [CMO13], where the authors prove that replacing W s,2{s by the Riesz potential of a Lorentz space I s pL 2{s,1 q, then there is no loss. In this case, the functions are continuous, so we can classify these spaces as supercritical.
In the subcritical situation, in addition to the results on classical spaces in [Ast94, Corollary 1.2], [CFM`09, Proposition 4] and [Bai16, Corolario 3.8], the fractional smoothness case 0 ă s ă 1 was considered in [CFR10,Theorem 4.3]. In this case, the authors show thatBf P W Θs,2 for every Θ ă 1´2 pκ . Note that in this result there is a loss in "smoothness", that is, in the s parameter. We will show Theorem 1.2 using the same techniques as the authors of that result with some extra care to avoid this loss. We recover their result in Corollary 1.3. However, when s " 1 and ν " 0 [CFM`09, Proposition 4] is stronger than Theorem 1.2, since the restriction 1 p ă 1 Figures 2.3(d) and 2.4(c)). Thus, it is natural to ask whether the following conjecture holds.  The Beurling transform commutes with fractional differentiation of order smaller or equal than s on W s,p . Indeed, since the Beurling transform is bounded in W s,p (see [Tri83, Section 2.6.6], for instance), we have that both D s˝B and B˝D s map W s,p into L p . In the S class, which is dense, we have that D s Bf "´|ξ| sξ ξ p f p¨q¯q" BD s f , so the equality extends to W s,p . Next we define the commutator of the multiplication by a test function with the fractional differentiation.
Definition 3.2. For every pair µ, f P C 8 c , we define rµ, D s sf :" µD s f´D s pµf q.
Next we recover Kato-Ponce Leibniz' rule as presented by Hofmann. mapping L p to L q for every 1 ď q ă p ď 8, whenever 1 r :" 1 q´1 p P p0, 1q. Moreover, }rµ, D s sf } L q À p,q }D s µ} L r }f } L p .
We will also use the Young inequality. It states that for measurable functions f and g, we have that }f˚g} L q ď }f } L r }g} L p (3.1) for 1 ď p, q, r ď 8 with 1 q " 1 p`1 r´1 (see [Ste70,Appendix A2]). Proof of Theorem 1.2. We will study in one stroke the critical and the subcritical case. To do so, we will use the convention p κ " 8 in the critical situation, that is, whenever sp " 2, while we use the standard notation p κ " 1`1 κ in the subcritical situation (sp ă 2). Thus, I´µB is invertible in L q for p 1 κ ă q ă p κ (see Theorem 1.1 for the subcritical setting and [Iwa92, Section 1] for the critical one).
On the other hand, being the convolution with ψ n an approximation of the identity, D s µ n " D s µ˚ψ n converges to D s µ in L p if D s µ P L p for p ă 8, and the same can be said about tν n u. and h n :" pI´µ n B´ν nB q´1pµ n`νn q. Note that h n is C 8 c , and µ n , ν n P C 8 c as well, while Bh n P C 8 X W m,p for every m P N and 1 ă p ă 8. Thus, we can take fractional derivatives of order s to get D s h n " D s pµ n Bh n`νn Bh n`µn`νn q " µ n D s Bh n´r µ n , D s spBh n q`ν n D s Bh n´r ν n , D s spBh n q`D s µ n`D s ν n , that is, since the Beurling transform commutes with the fractional derivatives for C 8 c functions, D s h n´µn BpD s h n q´ν n BpD s h n q "´rµ n , D s spBh n q´rν n , D s spBh n q`D s µ n`D s ν n . (3.6) Next, by (3.5) we have that and, by (3.2), for n big enough }D s µ n } L r`} D s ν n } L r ď C κ,p,r,}µ} W s,p ,}ν} W s,p for every 1 r ě 1 p . (3.8) Given 1 q ą 1 p`1 pκ , we can write 1 q " 1 ℓ`1 r satisfying restrictions (3.7) and (3.8), so Theorem 3.3 with (3.5) and (3.7) yields › › rµ n , D s sBh n`r ν n , D s sBh n Thus, we have a uniform control on the L q norm of the right hand side of (3.6). In case 1 p`1 p κ ă 1 p 1 κ we can find 1 q in between where we can invert the operator I´µ n B´ν nB and, using (3.6), we can write }D s h n } L q " › › pI´µ n B´ν nB q´1prµ n , D s spBh n q`rν n , D s spBh n q´D s µ n´D s ν n q › › L q À › › rµ n , D s spBh n q`rν n , D s spBh n q´D s µ n´D s ν n › › L q which is uniformly bounded by (1.4), (3.9) and (3.8). Combining these estimates with (3.5), we get }h n } W s,q À C κ,p,q,}µ} W s,p ,}ν} W s,p . (3.10) The Banach-Alaoglu Theorem shows that there exists a subsequence h n k converging to r h P W s,p as tempered distributions, with } r h} W s,q À C κ,p,q,}µ} W s,p ,}ν} W s,p . It only remains to transfer this information to h. First, note that (3.11) Indeed, from (3.3) and (3.4), we have that |h´h n | " |µBh´µ n Bh n`νB h´ν nB h n`µ´µn`ν´νn | ď |µ||Bph´h n q|`|µ´µ n ||Bh n |`|ν||Bph´h n q|`|ν´ν n ||Bh n |`|µ´µ n |`|ν´ν n |.
Since κ ă 1 and the Beurling transform is an isometry in L 2 , the left-hand side can absorb the first term in the right-hand side. To deal with the second and the third terms, choose 2 ă r q ă p κ , and let 1 r`1 r q " 1 2 . Then, by (3.4) and Theorem 1.1, the norm of h n in L r q is uniformly bounded, and using the L p version of (3.2) (i.e., Young's inequality) as well we get }pµ´µ n qBh n } L 2 ď }µ´µ n } L r }Bh n } L r q ď C κ,r q,n ÝÝÝÑ nÑ8 0.
Using again Young's inequality for the remaining terms, we see that showing (3.11). Since this implies convergence as tempered distributions and S 1 is a Hausdorff space, we get that h " r h. This finishes the case s ă 1. It remains to show that (1.6) holds when 1 p ă 1 p 1 κ´1 pκ and s ą 1. If this is the case, then for every 1 ă q ă 8 we have that }h} W s,q « }h} L q`}Bh} W s´1,q by the lifting property (see [Tri83, Section 2.3.8]) and the boundedness of the Beurling transform in Sobolev spaces. We will assume that ν " ν n " 0 to keep a compact notation, leaving the necessary modifications to the reader. Fix n ě 1. By assumption, µ P W s,p X L 8 . By (2.4) we also have that µ, µ n P W 1,sp X W s´1, sp s´1 uniformly in n. (3.12) As a consequence, Bh, Bh n P L r uniformly in n for every given 1 r ą 1 sp`1 p κ (3.13) by [CFM`09, Proposition 4] and [Bai16, Corolario 3.8], and, as we have proven above, D s´1 h, D s´1 h n P L r uniformly in n for every given 1 r ą s´1 sp`1 p κ . (3.14) Thus, we only need to deal with the homogeneous norm, that is, › › D s´1 Bh › › L q . From (1.2) we deduce that Bh n " µ n BBh n`B µ n Bh n`B µ n and differentiating we get D s´1 Bh n " µ n D s´1 BBh n´r µ n , D s´1 spBBh n q`Bµ n D s´1 Bh n´r Bµ n , D s´1 spBh n q`D s´1 Bµ n leading to › › pI´µ n BqpD s´1 Bh n q › › L q ď › › rµ n , D s´1 spBBh n q › › L q`› › Bµ n D s´1 Bh n › › L q`› › rBµ n , D s´1 spBh n q › › L q › › D s´1 Bµ n › › L q " 1`2`3`4 .
(3.15) Note that in the previous case, (3.6) had a simpler form, but the essential ideas to control the norm of the right-hand side are the same. We will find 1 pκ ă 1 q ă 1 p 1 κ such that j ď C κ,q for j P t1, 2, 3, 4u.
First of all, by Theorem 3.3 we have that 1 " › › rµ n , D s´1 spBBh n q › › L q ď › › D s´1 µ n › › L sp s´1 }BBh n } L r 1 for 1 q " s´1 sp`1 r1 . The first term is uniformly bounded by (3.12), and the last one is controlled by (3.13) as long as 1 r1 ą 1 sp`1 pκ . It is possible to find such a value for r 2 as long as 1 q ą s´1 sp`1 sp`1 p κ " 1 p`1 p κ .
On the other hand, using the commutation of fractional derivatives and the Beurling transform and Hölder's inequality, we have that for 1 q " 1 sp`1 r2 . The first term is uniformly bounded by (3.12), and the last one is controlled by (3.14) as long as 1 r2 ą s´1 sp`1 pκ . It is possible to find such a value for r 2 as long as 1 q ą 1 sp`s´1 sp`1 p κ " 1 p`1 p κ .
The latter two terms are bounded as before. By Theorem 3.3 we have that 3 " › › rBµ n , D s´1 spBh n q › › L q ď › › D s´1 Bµ n › › L p }Bh n } L r 3 for 1 q " 1 p`1 r3 , the last term being uniformly controlled for 1 r3 ą 1 pκ by Theorem 1.1. It is possible to find such a value for r 3 as long as 1 q ą 1 p`1 p κ .
This facts, together with (3.15) and Theorem 1.1, show that (3.10) holds for s ą 1 with exactly the same restrictions as when s ă 1, that is, when 1 p`1 pκ ă 1 q ă 1 p 1 κ . The theorem follows by the Banach-Alaoglu Theorem again.