Localized Regularity of Planar Maps of Finite Distortion
aa r X i v : . [ m a t h . C V ] J a n LOCALIZED REGULARITY OF PLANAR MAPS OF FINITE DISTORTION
OLLI HIRVINIEMI, ISTV´AN PRAUSE, AND EERO SAKSMAN
Abstract.
In this article we study fine regularity properties for mappings of finite distortion.Our main theorems yield strongly localized regularity results in the borderline case in the classof maps of exponentially integrable distortion. Analogues of such results were known earlier inthe case of quasiconformal mappings. Moreover, we study regularity for maps whose distortionhas higher exponential integrability. Introduction
Let f : Ω → C be a function where Ω ⊂ C is a domain. We say that f is a (homeomorphicand orientation preserving) mapping of finite distortion if following conditions are satisfied.(i) f ∈ W , loc ( C ).(ii) f : Ω → f (Ω) is a homeomorphism with J f ≥ | Df | = K f ( z ) J ( z, f ) for a.e. z ∈ C , where K f is a measurable function that is finitealmost everywhere.In an analogous way one may define mappings of finite distortion on subdomains of R d . Inthis article we only consider mappings of finite distortion on the plane. A planar mapping offinite distortion satisfies a Beltrami equation f z ( z ) = µ f ( z ) f z ( z ) , where µ is a measurable function with | µ ( z ) | < z . One actually has | µ ( z ) | = K f ( z ) − K f ( z )+1 .An important subclass is formed by mappings of finite exponential distortion which have theproperty that for some positive constant p > e pK f ( z ) ∈ L loc . A natural version of the measurable Riemann mapping theorem, Stoilow factorization theorem,and many other basic features of the standard quasiconformal theory generalise to this classses.For a good account of the theory refer the reader to [4, Chapter 20]. Improving on earlier results[9], it was shown in [3] that for a mapping of exponentially integrable distortion satisfying (1)there is the regularity(2) | Df | log β ( e + | Df | ) ∈ L loc for β < p − , and this is not necessarily true for β = p − . Note that in the above results one is interested onlyon the local regularity of mappings of finite distortion with exponentially integrable distortion.Similarly, in our work it is enough to consider only regularity of principal maps near the originsince local regularity results may then be transferred by the Stoilow factorisation theorem to
The work was supported by the Finnish Academy Coe ’Analysis and Dynamics’ and the Finnish Academyprojects 1266182, 1303765, and 1309940. maps that are defined on subdomains. Let us recall that a principal map f : C → C is conformal(i.e. µ f ( z ) = 0) outside the unit disk D and f ( z ) = z + O (1 / | z | ) near infinity.For (standard) K -quasiconformal maps (i.e. K f ( z ) ≤ K < ∞ , where K is a constant), theoptimal area distortion [2] implies that Df ∈ L ploc for p < KK − , but this fails in general in theborderline case p = KK − . There is a substitute [4, Cor. 13.2.5] in the form of containment tothe weak space Df ∈ L KK − , ∞ loc . Another kind of result in the borderline case was given in [5,Theorem 3.5] stating that a K -quasiconformal map satisfies(3) ( K − K f ( z )) | Df | KK − ∈ L loc , which gives strongly localized regularity information on the map, especially R K f ≤ K − ε | Df | KK − < ∞ for all ε >
0. Not quite opptimal version of this result were proven in n dimensions in [9].For further basic results on planar maps of exponentially integrable distortion we refer e.g. to[6, 15, 12, 14, 10] and the references therein.The principal aim of the present note is to establish a strongly localized regularity result formappings of finite distortion analogous to (3). Our main result states the following: Theorem 1.
Assume that f is a planar ( homeomorphic ) mapping of exponentially integrabledistortion on a planar domain satisfying (1) with p = 1 . Then it holds that (4) Z A ε ( e + K f ) | f z | < ∞ for any ε > for any compact subset A ⊂ Ω . As a corollary of the proof of this result we obtain.
Corollary 2.
With f a mapping of integrable distortion satisfying (1) for some p > we have | Df | log( e + | Df | ) p − (log log( | Df | + 10) − (1+3 p + ε ) ∈ L loc for any ε > . Our next result considers mapping that in some sense lie in between mapping of exponentiallyintegrable distortion and standard quasiconformal maps.
Theorem 3.
Assume that f is a planar ( homeomorphic ) mapping of finite distortion on a planardomain satisfying the integrability e ( K f ( z )) α ∈ L loc for some α > . Then it holds that (5) Z D | f z | exp (cid:0) log β ( e + | f z | ) (cid:1) < ∞ . for any β < − /α . In radial case, one can improve Theorem 1:
Theorem 4.
Let f : C → C be a planar and radial homeomorphic mapping of finite distortionwith exponentially integrable distortion satisfying (1) with p = 1 . Then Z A ε ( e + K f ) | f z | < ∞ for any ε > . for any compact subset A ⊂ C . OCALIZED REGULARITY 3
This result naturally leads to the conjecture
Conjecture 5.
One expects that Theorem 1 (resp. Corollary 2) remain true if the logarithmicfactor in the statement is replaced by log( e + K f ) − (1+ ε ) (resp. (log log( | Df | + 10) − (1+ ε ) ).Theorem 4 is sharp up to the possible borderline case. For any 0 < ε <
1, we can choose g ε : C → C to be g ε ( z ) := z | z | (cid:20) log (cid:18) e + 1 | z | (cid:19)(cid:21) − / (cid:20) log log (cid:18) e + 1 | z | (cid:19)(cid:21) − ε/ for | z | < g ε ( z ) := cz elsewhere for some constant c . Then one directly verifies that g ε is a mappingof finite distortion satisfying (1) with p = 1 but Z D − ε ( e + K g ε ) | ( g ε ) z | = ∞ . Section 2 below contains the proof of Theorem 1 assuming the quantitative estimate of Lemma7. Next, Section 3 gives careful quantitative estimates for the decay of the Neumann seriesassociated with the Beltrami equation. Then in Section 4 we are ready to accomplish the proofof Lemma 7, and also to complete the proof of Theorem 3. Finally, Section 5 treats the case ofradial mappings. 2.
Proof of the main theorem
In this section we prove Theorem 1 assuming a lemma whose proof we provide later. We willemploy the standard notation ∂ := ddz = ( ∂ x + i∂ y ) and ∂ := ddz = ( ∂ x − i∂ y ). Proof of Theorem 1.
Let f be a principal mapping of finite distortion with Z D e K f < ∞ , and let µ := µ f be the associated Beltrami coefficient. For δ ∈ (0 , ξ = ξ δ be the truncatedversion of µ , ξ ( z ) = ( µ ( z ) , | µ ( z ) | < − δ (1 − δ ) µ ( z ) | µ ( z ) | , | µ ( z ) | ≥ − δ . The truncation satisfies k ξ k ∞ ≤ − δ , so we can use the theory of quasiconformal mappings onthe solutions of corresponding Beltrami equations.We next fix 0 < ε < / w with 0 ≤ Re w ≤ f w be the unique principalsolution to the Beltrami equation ∂f w ( z ) = ν w ( z ) ∂f w ( z ) , where ν w ( z ) := ξ ( z ) | ξ ( z ) | | ξ ( z ) | w + ε . A main idea in the proof is to consider the functions g w = (1 − | µ | ) (1 − w ) / ∂f w and apply analytic interpolation theorem, or actually a very special case of it that reduces to avector-valued Phargem-Lindel¨of type maximal principle. O. HIRVINIEMI, I. PRAUSE, AND E. SAKSMAN
To accomplish this, note that the dependence w ν w is analytic, we deduce by the Ahlfors-Bers theorem that the dependence w f w (as a L ( D )-valued function) is analytic over theclosed strip, and hence also g depends analytically on w . Especially, the map w g w iscontinuous in the strip and analytic in the interior. Moreover, by a standard application of theNeumann series and the definition of g we see that k g w k L ( D ) ≤ C δ for any δ and for all w inthe closed strip { ≤ Re w ≤ } . Fix h ∈ C ∞ ( D ) with k h k L ( D ) = 1. A fortiori, the function w R D g w ( z ) h ( z ) dm ( z ) is a continuous and bounded analytic function in the closed strip andanalytic in the interior. If we denote f M r := sup Re w = r (cid:12)(cid:12)(cid:12)(cid:12)Z D g w ( z ) h ( z ) dm ( z ) (cid:12)(cid:12)(cid:12)(cid:12) , and M r := sup Re w = r k g w k L ( D ) , then we have by a classical version of the Hadamard’s three lines theorem that f M θ ≤ f M − θ f M θ ≤ M − θ M θ . Since h ∈ C ∞ ( D ) is arbitrary we in fact have for any θ ∈ (0 , M θ ≤ M − θ M θ . In order to continue the proof we need several auxiliary results.
Lemma 6.
For any w with Re w = 0 we have Z D | g w | ≤ Cε − . The constant C is independentof δ . In particular, M ≤ C ε − / .Proof. As f w is a quasiconformal principal mapping, we obtain by the Bieberbach area formula Z D J ( z, f w ) = | f ( D ) | ≤ π. To use this, note first that as J ( z, h ) = | h z | − | h z | , we have by the definition of g for any w with Re w = 0 Z D | g w | = Z D (1 − | µ | ) | ∂f w | = Z D − | µ | − | ν w | J ( z, f w ) ≤ Z D − | µ | − | µ | ε J ( z, f w ) . As x x ε is a concave function whose derivative at x = 1 equals 2 ε , we have x ε ≤ ε ( x − x >
0. This implies that 1 − | µ | − | µ | ε ≤ − | µ | ε (1 − | µ | ) = 12 ε , finishing the proof. (cid:3) Lemma 7.
For any w with Re w = 1 it holds that Z D | g w | ≤ Cε − . The constant C is indepen-dent of δ . In particular, M ≤ C ε − We postpone the proof of this lemma to Section 4 as it needs more preparation, especially oneneeds to carefully check the dependence of constants in certain arguments of [3].
OCALIZED REGULARITY 5
In order to continue the proof we choose θ = 1 − ε in (6) to obtain Z D (1 − | µ ( z ) | ) ε | ∂f − ε ( z ) | dm ( z ) ≤ Cε . Here f − ε has Beltrami coefficient ν . Since the bound is independent of δ , by standard compact-ness of principal mappings of finite distortion (with uniform bound in condition (1)), we obtainimmediately(7) Z D (1 − | µ ( z ) | ) ε | f z | ≤ Cε . Again, the constant C in (7) does not depend on ε .The inequality (7) already provides a non-trivial localization result because we may considersmall values of ε . However, as we have all values ε ∈ (0 , /
2) at our disposal, the result can beimproved on by invoking the following observation:
Lemma 8.
Let h and W be non-negative functions on D , with W ( z ) ≤ for all z . Let also ε ∈ (0 , / , α, C > be positive constants. Assume that for any < ε < ε we have (8) Z D ( W ( z )) ε h ( z ) dm ( z ) ≤ Cε α . Then there is a constant C = C ( ε , α, C ) such that for < η ≤ Z D (cid:16) log (cid:16) e + W ( z ) (cid:17)(cid:17) α + η h ( z ) dm ( z ) ≤ C η . Proof.
The assumption remains valid if W is replaced by min( W, /
2) and the conclusion ob-tained in this case yields the original one, in view of the assumption. We may hence assumethat W ( z ) ∈ [0 , /
2] for all z . From (8) it immediately follows that if 0 < η ≤
1, then Z ε ε α − η Z D ( W ( z )) ε h ( z ) dm ( z ) dε ≤ C Z ε ε η − dε = Cη ε η . On the other hand, we can use Fubini’s theorem to conclude that Z ε ε α − η Z D ( W ( z )) ε h ( z ) dm ( z ) dε = Z D h ( z ) Z ε ε α − η ( W ( z )) ε dε dm ( z )For those z with W ( z ) = 0, the inner integral is 0. Let now 0 < a := W ( z ) ≤ /
2. Then theinner integral is equal to Z ε x α + η − a x dx = 1 (cid:0) log (cid:0) a (cid:1)(cid:1) α + η − Z ε (cid:18) log (cid:18) a (cid:19) x (cid:19) α + η − e − ( log ( a ) x ) dx = 1 (cid:0) log (cid:0) a (cid:1)(cid:1) α + η Z ε log ( a ) s α + η − e − s ds. The last integral factor approaches Γ( α + η ) uniformly on η ∈ [0 ,
1] as a →
0. The positivefunction φ : (0 , / × [0 , → R , φ ( a, η ) := (cid:0) log (cid:0) e + a (cid:1)(cid:1) α + η (cid:0) log (cid:0) a (cid:1)(cid:1) α + η Z log ( a ) ε s α + η − e − s ds, O. HIRVINIEMI, I. PRAUSE, AND E. SAKSMAN extends therefore to a continuous positive function on [0 , / × [0 , c > z we have Z ε ε α − η ( W ( z )) ε dε ≥ c (cid:16) log (cid:16) e + W ( z ) (cid:17)(cid:17) α + η which finishes the proof. (cid:3) Theorem 1 is obtained by applying the previous Lemma in conjunction with inequality (7)using the choices W ( z ) := (1 − | µ ( z ) | ) and α = 4 . (cid:3) Proof of Corollary 2.
Following the argument of the proof and keeping track of the dependenceof constant factors, we obtain under the assumption (1) that instead of the result stated inLemma 7 we obtain for general p that Z D | Df | log( e + | Df | ) p − log − ε (cid:0) e + | Df | (cid:1) ≤ Cε − (1+3 p ) . Then, as before the statement follows by an application of Lemma 8. (cid:3)
We note that one may actually apply Lemma 8 directly on the result stated in Theorem 1and obtain a statement of the form Z A ( e + K f ) log log ε (10 + K f ) | f z | < ∞ for any ε > . An industrious reader may refine this result by iterating the lemma, obtaining estimates forweights with more iterations of logarithms if they so wish.3.
Decay of the Neumann series
For the proof of Lemma 7 we need quantitative versions of several auxiliary results in [3]. Inthis section we establish decay estimates for the Neumann series that suffice both for Theorem1 and for Theorem 3. Our proof follows rather closely the ideas of [8, 3] but keeping track ofthe dependence of the constants is somewhat non-trivial even in the case α = 1 which relates tothat considered in [8, 3].The Beurling operator S is the singular integral S φ ( z ) := − π Z C φ ( τ )( z − τ ) dτ. Recall that in the context of quasiconformal mappings, the classical Beltrami equation in W , loc ( C ) ∂f ( z ) = µ ( z ) ∂f ( z )has a unique principal solution f ( z ) = z + O (1 /z ) – for this and other basic facts on quasicon-formal maps we refer the reader to [1, 4]. We can use the identity ∂f − S ( ∂f ) to write theBeltrami equation equivalently for ω = ∂fω ( z ) = µ ( z )( S ω ( z ) + 1) , which is solved by the Neumann series ω = ( I − µ S ) − µ = µ + µ S µ + µ S µ S µ + · · · . OCALIZED REGULARITY 7
The series converges absolutely when | µ ( z ) | ≤ k < S is a unitaryoperator in L ( C ). This is no longer true if only | µ ( z ) | <
1, but we have the following estimate.
Lemma 9.
Let | µ ( z ) | < almost everywhere, with µ ( z ) ≡ for | z | > . Assume that thedistortion function K ( z ) = | µ ( z ) | −| µ ( z ) | satisfies e K α ∈ L p ( D ) , for some p > and α ≥ In case α > we have for every p > and β ∈ [ p/ , p ) Z C | ( µ S ) n µ | ≤ C exp (cid:18) − β/ /α − /α (cid:0) ( n + β/ − /α − ( β/ − /α (cid:1)(cid:19) , n ∈ N , where by denoting δ := ( p − β ) β ( p + β ) , e C := pp − β (cid:0)R D e pK α (cid:1) ( p − β ) / p , b := ( β/ /α , and B := max (cid:16) b − /α (cid:0) (2 b/δ ) α − − (1 + β/ − /α (cid:1) , (cid:17) we have (9) C := (4 δ − e Ce B + 1) . In case α = 1 one instead has Z C | ( µ S ) n µ | ≤ C (cid:16) n + β/ β/ (cid:17) − β , n ∈ N , where (10) C := 12 β +3 ( p/β − − (5+2 β ) (cid:18)Z D e pK (cid:19) (1 − β/p ) Proof.
We first note that a simple computation shows that the case α = 1 follows from the case α > k µ k ∞ < α → + in estimate (9). Hence we mayassume that α > < β < p . For n ∈ N divide the unit disk into twosets B n = (cid:26) z ∈ D : | µ ( z ) | > − β /α (4 n ) /α + β /α (cid:27) and G n = D \ B n . By Chebychev’s inequality, | B n | ≤ (cid:16) Z D e pK α (cid:17) e − np/β . The terms of the Neumann series ψ n = ( µ S ) n µ and the auxiliary terms g n are obtained induc-tively ψ n = µ S ( ψ n − ) , ψ = µ and g n = χ G n µ S ( g n − ) , g = µ. For g n we can estimate by using the fact that S is L -isometry to see that k g n k L = Z G n | µS ( g n − ) | ≤ (cid:18) − β /α (4 n ) /α + β /α (cid:19) k g n − k L , O. HIRVINIEMI, I. PRAUSE, AND E. SAKSMAN and therefore k g n k L ≤ n Y j =1 (cid:18) − β /α (4 j ) /α + β /α (cid:19) k g k L = exp n X j =1 log (cid:18) − β /α (4 j ) /α + β /α (cid:19)! k µ k L . As log(1 − x ) ≤ − x for x <
1, and k µ k L ≤ √ π , it follows that k g n k L ≤ exp − − /α β /α n X j =1 j /α + ( β/ /α ! π / ≤ exp − − /α β /α n X j =1 j + β/ /α ! π / , where we applied the inequality ( j ) /α + ( β/ /α ≤ − /α ( j + β/ /α . The sum inside theexponential can be estimated by an integral n X j =1 j + β/ /α ≥ Z n +11 dx ( x + β/ /α = ( n + 1 + β/ − /α − (1 + β/ − /α (1 − /α ) , (11)so that(12) k g n k L ≤ exp (cid:18) − − /α β /α ( n + 1 + β/ − /α − (1 + β/ − /α − /α (cid:19) . The difference of ψ n and g n is ψ n − g n = χ G n µ S ( ψ n − − g n − ) + χ B n µ S ( ψ n − ) . For the norms, this gives k ψ n − g n k L ≤ (cid:18) − β /α (4 j ) /α + β /α (cid:19) k ψ n − − g n − ) k L + p R ( n )with R ( n ) = k χ B n µ S ( ψ n − ) k L = Z B n | ( µ S ) n µ | . OCALIZED REGULARITY 9
By induction and estimating like in (11) we deduce k ψ n − g n k L ≤ (cid:18) − β /α (4 n ) /α + β /α (cid:19) k ψ n − − g n − ) k L + p R ( n ) ≤ n X j =1 p R ( j ) n Y k = j +1 (cid:18) − β /α (4 k ) /α + β /α (cid:19) = n X j =1 p R ( j ) exp n X k = j +1 log (cid:18) − β /α (4 k ) /α + β /α (cid:19)! ≤ n X j =1 p R ( j ) exp − − /α β /α n X k = j +1 k /α + ( β/ /α ! ≤ n X j =1 p R ( j ) exp (cid:18) − − /α β /α ( n + 1 + β/ − /α − ( j + 1 + β/ − /α − /α (cid:19) = exp (cid:18) − − /α β /α − /α (cid:0) ( n + 1) − /α − (1 + β/ − /α (cid:1)(cid:19) ×× n X j =1 exp (cid:18) − /α β /α − /α (cid:0) ( j + 1 + β/ − /α − (1 + β/ − /α (cid:1)(cid:19) p R ( j )(13)We next recall that in [3] the Astala area distortion result | f λ ( E ) | ≤ πM | E | /M for quasi-conformal maps was used to estimate R ( n ) by considering the solution f = f λ to the Beltramiequation f z = λµf z , with | λ | < µ S ) n µ of the Neumann series f λz = P ∞ n =0 λ n +1 ( µ S ) n µ. bya Cauchy integral ( µ S ) n µ = 12 πi Z | λ | = ρ λ n +2 f λz dλ, multiplying by the characteristic function χ B n and by forcing the Jabobian to appear under theintegral. This yielded (see [3, p. 8]) for any µ with just k µ k ∞ ≤
1, any
M >
1, and any E ⊂ D the general estimates k χ E ψ n k L ≤ π (cid:18) M + 1 M − (cid:19) n ( M + 1) | E | /M and(14) k χ E S ( ψ n ) k L ≤ π (cid:18) M + 1 M − (cid:19) n +2 ( M + 1) | E | /M . (15)Choosing E = B n this yields p R ( n ) ≤ √ π (cid:18) M + 1 M − (cid:19) n ( M + 1)2 | B n | / M ≤ √ π (cid:18) M + 1 M − (cid:19) n ( M + 1)2 (cid:18)Z D e pK α (cid:19) / M e − np/ ( βM ) . In our situation we may actually slightly improve this by invoking the Eremenko and Hamiltonform of the area distortion estimate stating for any measurable E ⊂ D the inequality(16) | g ( E ) | ≤ M /M π − /M | E | /M ≤ πe / ( πe ) | E | /M . This leads to p R ( n ) ≤ √ πe / (2 πe ) (cid:18) M + 1 M − (cid:19) n r ( M + 1) M (cid:18)Z D e pK α (cid:19) / M e − np/ ( βM ) . We want to choose
M > R ( n ) decay exponentially. For that we need tohave log (cid:18) M + 1 M − (cid:19) − M pβ ≤ − δ < . Choose M = 2 p/ ( p − β ) and estimatelog (cid:18) M + 1 M − (cid:19) − M pβ ≤ M − − M pβ = 2( p − β ) p + β − p − β )2 β = − ( p − β ) β ( p + β ) =: − δ. Noting that ( M +1) M ≤ M and √ πe / (2 πe ) ≤ p R ( n ) ≤ e Ce − δn with e C := r pp − β (cid:18)Z D e pK α (cid:19) ( p − β ) / p . Hence, if we denote b := 2 − /α β /α , we obtain n X j =1 exp (cid:18) − /α β /α − /α (cid:16) ( j + 1 + β/ − /α − (1 + β/ − /α (cid:17)(cid:19) p R ( j ) ≤ e Ce e B ∞ X j =1 e − jδ/ ≤ δ − e Ce e B , where e B := sup j ≥ (cid:16) b − /α (cid:0) ( j + 1 + β/ − /α − (1 + β/ − /α (cid:1) − ( δ/ j (cid:17) . An elementary com-putation where one simply differentiates with respect to j shows that e B ≤ B := max (cid:16) b − /α (cid:0) (2 b/δ ) α − − (1 + β/ − /α (cid:1) , (cid:17) . In view of (13) we then obtain k ψ n − g n k L ≤ δ − e Ce B exp (cid:18) − − /α β /α − /α (cid:0) ( n + 1 + β/ − /α − ( β/ − /α (cid:1)(cid:19) . Together with (12) this proves the lemma. (cid:3) Proof of Lemma 7 and Theorem 3
We start with an area distortion result that generalizes [3, Cor. 3.2 and Thm. 5.1] to therange α ≥
1. In case α = 1 we need to keep careful track of the constants, which is somewhatnon-trivial in this situation, and hence for the readers sake we give the details here although thebasic idea proof follows that in [8, 3]. OCALIZED REGULARITY 11
Proposition 10.
Let µ and < β < p and f be as in the previous lemma. (i) In case α > we have the area distortion estimate (17) | f ( E ) | ≤ c exp (cid:0) − c ′ log − /α ( e + 1 / | E | ) (cid:1) with some constants c, c ′ > . (ii) In case α = 1 , under the additional assumption / < β < p < it holds that (18) | f ( E ) | ≤ A | E | δ + A δ − β log − β (1 + 1 / | E | ) (cid:18)Z D e pK (cid:19) / , E ⊂ D , where we denoted δ := p − β and A is a universal constant.Proof of Proposition 10. We start by observing that our maps are Sobolev homeomorphismsthat satisfies Lusin’s condition N . Especially, we obtain (using the notation of the previoussection) | f ( E ) | = Z E | f z | − | f z | ≤ | E | + 2 Z E | f z − | = 2 | E | + 2 k χ E (cid:0) f z − (cid:1) k (19) ≤ | E | + 2 (cid:16) ∞ X n =0 k χ E Sψ n k (cid:17) . We estimate the last written sum in two parts, and fix for that end index m ≥ M = 3 yields m − X n =0 k χ E Sψ n k ≤ m − X n =0 √ π n +2 | E | / ≤ m +3 | E | / . (20)In case α > β = p/ ∞ X n = m k χ E Sψ n k ≤ c exp( − c m − /α ) . Here and later c j :s are constants that may depend on β, p, α , and whose exact value is of nointerest to us. By choosing m = ⌊
112 log 2 log 1 / | E |⌋ we obtain in view of (19) and the previousestimates | f ( E ) | ≤ | E | + c | E | / + 4 c exp (cid:0) − c log − /α ( e + 1 / | E | ) (cid:1) , which proves part (i).In case (ii) we have α = 1. In this case we first assume that 2 < β < p and application ofLemma 9 yields in this case (cid:16) ∞ X n = m k χ E ψ n µ k (cid:17) ≤ C ∞ X n = m (cid:16) n + β/ β/ (cid:17) − β/ ! ≤ C ( β/ ( β − (cid:16) m + β/ β/ (cid:17) − β where the expression for the constant C = C ( β, p ) is given in (10). In view of (20) we thushave | f ( E ) | ≤ | E | + 2 m +7 | E | / + 8 C ( β/ ( β − (cid:16) m + β/ β/ (cid:17) − β . Choosing m = (cid:6)
112 log 2 log(1 + 1 / | E | ) (cid:7) and noting that | E | ≤ | E | / and 12 log 2 ≤ | f ( E ) | ≤ | E | / + 8 C ( β/ ( β − (cid:16) log((1 / | E | + 1) / ) + β/
41 + β/ (cid:17) − β (21)Our next step is to apply G. David’s factorisation trick to improve the above bound andextend it to all values of p . We assume thus that f is as in the statement of the proposition(with the general assumption 1 / < β < p <
4) and recall from [3] that for any M ≥ f as f = g ◦ F , where g and F are principal mappings, g is M -quasiconformal and F satisfies I M := Z D e pMK ( z,F ) ≤ e M Z D e pK ( z,f ) . Denote β := ( p + β ) /
2, and M = 2 / ( β − β ) = 4 / ( p − β ) ≥
1. We will apply (21) withparameters (
M β , M p ) instead of ( β, p ) in order to estimate | F ( E ) | . This is possible since bythe assumption p < < βM = β M < pM . Thus, | F ( E ) | ≤ | E | / + 8 C ( M β , M p, I M )( M β / ( M β − (cid:16) log((1 + 1 / | E | ) / ) + M β /
41 +
M β / (cid:17) − Mβ . Above the notation C ( M β , M p, I M ) recalls the dependences of the constant C . As g is aprincipal quasiconformal mapping, we obtain from the standard area distortion estimate (16) | f ( E ) | = | g ◦ F ( E ) | ≤ | F ( E ) | /M . By noting that 2 − M β = − M β , and
M β / p + βp − β > ( p/β − − , combing the last twoinequalities leads to | f ( E ) | ≤ | E | / M ++4 · /M (cid:16) C ( M β , M p, I m )( M β / ( M β ) (cid:17) /M (cid:16) ((log(1 + 1 / | E | )) / ) + ( p/β − − p/β − − (cid:17) − β (22)Here, since M p/M β − p − β ) / ( p + β ) ≥ ( p − β ) / p and ( p/β − / M ≤ / /M (cid:16) C ( M β , M p, I m )( M β / ( M β ) (cid:17) /M ≤ A (( p/β − − β (cid:18)Z D e pK (cid:19) / , where A is an absolute constant. In the simplification we applied our assumption on the rangeof p and β and observed that ( p − β ) − ( p − β ) has a universal upper bound. We also observein (22) that ( p/β − β has a universal upper bound, and by increasing A we may replacelog(1 + 1 / | E | )) / ) by log(1 + 1 / | E | ). In addition, in our situation p/β − ≍ ( p − β ). Combiningthese estimates completes the proof of part (ii). (cid:3) We then turn to the goals stated in the title of this section. As expected, we will firstestimate integrability of the Jacobian using the estimates for area-distortion we just proved. Forthat purpose we will first state a general lemma that yields (essentially optimal) integrabilityestimates from estimates of area distortion.
OCALIZED REGULARITY 13
Lemma 11.
Assume that f is a principal mapping of finite distortion and g : [0 , π ) → [0 , ∞ ) is concave with g (0) = 0 , satisfying for any measurable subsets E ⊂ B (0 , the area distortionestimate (23) | f ( E ) | ≤ g ( | E | ) . Then for any convex and increasing H on [0 , ∞ ) it holds that Z B (0 , H ( J f ( z )) dA ( z ) ≤ Z π H ( g ′ ( t )) dt. Proof.
Let us denote by h : (0 , π ) → R + the decreasing rearrangement of J f . By assuming firstthat g is differentiable on (0 , π ), our assumption may be rewritten as Z x h ( t ) dt ≤ Z x g ′ ( t ) dt for all x ∈ (0 , π ) . The statement now follows from a continuous version of the Hardy-Littlewood P´olya (or Kara-mata) inequality, see [7, Theorem 2.1] or [11]. (cid:3)
Proof of Theorem 3.
It follows from and Proposition 10(i) that in our situation the higher inte-grability of J f is at least as good as that of the derivative h ′ on the interval (0 , π ) where h ( x ) := exp (cid:0) − c ′ log − /α ( e + 1 /x ) (cid:1) on the interval (0 , π ). Namely, h is clearly decreasing near the origin which is enough for us inorder to apply Lemma 11. We may leave safely to the reader the checking that φ ( h ′ ) is integrablenear origin with φ ( y ) := y exp(log β ( e + y )) for β < − /α. In other words, we have Z D J f exp(log β ( e + J f )) < ∞ for β < − α − . By recalling that | Df | = KJ f , the stated integrability of the derivative follows immediately bythe elementary inequality xy exp (cid:0) log β ′ ( e + xy ) (cid:1) ≤ C (cid:16) exp( px α ) + y exp (cid:0) log β ( e + y ) (cid:1)(cid:17) , x, y ≥ . for any 0 < β ′ < β < p >
0, and where C = C ( p, β, β ′ , α ) . The latter inequality followseasily by examining separately the cases x < exp((1 /
2) log β ( e + y )) and x ≥ exp((1 /
2) log β ( e + y )). (cid:3) Proof of Lemma 7.
Easy estimates that just apply differentiation show that the function x (1 + δ log(1 + 1 /x )) − β is concave for x > δ < (1 + β ) − , which in our situation holds at least if δ < / . We now fix p = 1 + 2 ε , β = 1 + ε , with ε ∈ (0 , /
10) in Proposition 10 (ii) and note that Lemma11 yields the integrability Z D J ( z, f ) log( e + J ( z, f )) ≤ Z π h ′ ( x ) log( e + h ′ ( x )) dx, where h ( x ) := A x ε + A ε − − ε (cid:16) log(1 + 1 /x ) (cid:17) − − ε (cid:0)R D e pK (cid:1) / . Hence, if we denote A := A ′ (cid:0)R D e pK (cid:1) / with another universal constant A ′ we have h ′ ( x ) ≤ A (cid:18) εx − ε + ε − x (cid:16) log(1 + 1 /x ) (cid:17) − − ε (cid:19) , Obviously log h ′ ( x ) ≤ log( A )+3 log(1 /ε )+3 log(10 /x ) , so that noting that R π h ′ ( x ) dx = h ( π ) ≤ A ε − we obtain that Z D J ( z, f ) log( e + J ( z, f )) ≤ Z π h ′ ( x ) log( e + h ′ ( x )) dx ≤ A ε − (cid:0) log( A ) + log(1 /ε ) (cid:1) + 3 A Z π (cid:18) εx − ε + ε − x (cid:16) log(1 + 1 /x ) (cid:17) − − ε (cid:19) log(10 /x ) dx ≤ A log( A )10 ε − . In the estimation of the last written integral we noted that R π εx − ε log(10 /x ) dx ≤ ε − and estimated the second integral from the above by 2 log(20) (cid:16) ε − R / log(1 /x ) − − ε dxx +3 ε − ) ≤ ε − . We next note the well-known inequality stating that for any ε ∈ (0 ,
1) and reals x, y > xy ≤ x log( e + x ) + e (1+ ε ) y (one simply checks that is true for ε = 0). The choice x = J j ( z ), y = K := K f ( z ), andintegration over D finally yields that Z D | Df | ≤ A (cid:0) Z D e (1+ ε ) K (cid:1) / log (cid:0) Z D e (1+ ε ) K (cid:1) ε − + Z D e (1+ ε ) K ≤ A ε − Z D e (1+ ε ) K , (24)where A , A are universal constants.We are now ready to complete the proof of Lemma 7. For that end we need to establish forany w with Re w = 1 the key estimate Z D | g w | ≤ Cε , with constant C does not depend on ε . Note that this estimate implies the bound M ≤ C ε for some constant C . Morever, estimating the integrability of | g w | reduces to that of | ( f w ) z | because | g w | = | ( f w ) z | a.e. since we have Re w = 1 . Let us first estimate the distortion K ( z, f w ). Assume that ε ∈ (0 , /
2) and consider thefunction r ( x ) := 1 − x ε − (1+ ε/ − x ) . We claim that r ( x ) ≥ x ∈ [1 / , r is concavewith r (1) = 0 and r ′ (1) = − ε/ <
0, it is enough to check that r (1 / ≥ , or equivalently that1 + ε/ ≤ − − ε . In turn this follows from the concavity of ε R ( ε ) := 2 − − ε − (1 + ε/ R (0) = R (1 /
2) = 0.
OCALIZED REGULARITY 15
We thus have that 1 − | ν ( z ) | ε ≥ (1 + ε/ − | ν ( z ) | assuming that | ν ( z ) | ≥ /
2, and wemay estimate the distortion as follows: K ( z, f w ) = 1 + | ν ( z ) | ε − | ν ( z ) | ε = − − | ν ( z ) | ε ≤ − ε/ − | ν ( z ) | ) , − / ε ) ≤ − ε/ − | ν ( z ) | ) + 21 − / ε ≤ ε/ − | ν ( z ) | )= 11 + ε/ (cid:18) − − | ν ( z ) | (cid:19) + 3 + 11 + ε/ ≤ K ( z, f )1 + ε/ . It follows that R D e (1+ ε/ K ( z,f w ) ≤ R D e K ( z,f )+4+2 ε ≤ e R D e K ( z,f ) .In conclusion, an application of inequality (24) (with ε/ ε ) yields the desireduniform bound Z D | ( f w ) z | ≤ Z D | Df w | ≤ Cε . (cid:3) Proof of Theorem 4
Throughout this section we assume that f : D → D is a radial homeomorphism of the form f ( z ) = z | z | φ ( z ) , where φ : [0 , → [0 ,
1] is an increasing homeomorphism. We also assume that f is a map withexponentially integrable distortion (satisfying (1) ). By the Lusin condition this implies that φ is absolutely continuous, and the assumed exponential integrability of K f can expressed as Z (cid:16) e p rφ ′ ( r ) φ ( r ) + e p φ ( r ) rφ ( r ) (cid:1) rdr = C < ∞ . (25)Our aim is to first prove an area distortion estimate for these maps. Proposition 12.
Let f : D → D be a radial homeomorphism of exponentially integrable distor-tion ( see (1)) . Then for any measurable subset E ⊂ D . | f ( E ) | ≤ C (cid:0) log(1 + 1 / | E | ) (cid:1) − p . The constant C = C ( p, C ) is uniform for fixed C and p ∈ [1 , . Proof.
We shall denote by C constants whose actual size if of no interest to us, and their valuemay change from line to line. We call the set E ⊂ D ’radial’ if one has that z ∈ E if and onlyif | z | ∈ E . By a standard approximation argument it is enough to prove the claim in the casewhere E is a disjoint union of sets of the form { a < | z | < b, α < arg( z ) < α } , and this caseis easily reduced to the case of radial sets. Thus, we may assume that E = {| z | ∈ F, } where F ⊂ (0 ,
1) is a disjoint union of open intervals.For n = 1 , , . . . we denote the dyadic annuli A n := { − n ≤ | z | ≤ − n } . Our first goal isto estimate φ ( r ) from the above. For that end, fix n ≥ inequality applied on the probability measure r − dr on ( e − n , e − n ) and on the convex function x e p/x yieldslog (cid:0) φ ( e − n ) /φ ( e − n ) (cid:1) = Z e − n e − n rφ ′ ( r ) φ ( r ) drr ≥ p log Z e − n e − n exp (cid:18) p φ ( r ) rφ ′ ( r ) (cid:19) drr !! − ≥ p log e n Z e − n e − n exp (cid:18) p φ ( r ) rφ ′ ( r ) (cid:19) rdr !! − ≥ p log C + 2 n . Applying this for first n annuli yields(26) φ ( e − n ) ≤ exp n X k =1 p log C + 2 n ! ≤ Cn − p/ . We next produce a very crude estimate of area distortion for radial sets E ⊂ A n . Write E = {| z | ∈ F, } where F ⊂ ( e − n , e − n ) and note that | F | ≤ e n | E | . Let us observe first that (25),Jensen’s inequality and the convexity of the map x exp( p √ x + 1) on [0 , ∞ ) yield that Z e − n e − n (cid:16) rφ ′ ( r ) φ ( r ) (cid:17) drr ≤ p log Z e − n e − n exp (cid:18) p s(cid:16) rφ ′ ( r ) φ ( r ) (cid:17) + 1 (cid:19) drr !! − ≤ p log e n Z e − n e − n exp (cid:18) p rφ ′ ( r ) φ ( r ) + p (cid:19) rdr !! − ≤ p − (log C + 2 n + p ) ≤ Cn . We may then compute using the above estimate, the bound (26) and Cauchy-Schwarz to obtainfor radial subsets of E ⊂ A n | f ( E ) | = 2 π Z F φ ( r ) φ ′ ( r ) dr ≤ π ( φ ( e n +1 )) Z F rφ ′ ( r ) φ ( r ) drr ≤ Cn p sZ F drr sZ e − n e − n (cid:16) rφ ′ ( r ) φ ( r ) (cid:17) drr ≤ Cn p p | F | e n/ n ≤ Cn − p e n √ E (27)We finally observe that in the general case we may assume that | E | = e − N for some integer N ≥
1. By using the estimates (26) and (27) it follows that | f ( E ) | ≤ | f ( {| z | ≤ e − N } ) | + N X n =1 | f ( E ∩ A n ) | ≤ π ( φ ( e − N )) + C N X n =1 e n n − p √ e − N ≤ CN p + N e − N ≤ C ′ N p , as was to be shown. (cid:3) OCALIZED REGULARITY 17
Proof of Theorem 4.
One simply applies the area distortion estimate we just proved and obtainsthe analogue of (24) now with term 1 /ε instead of 1 /ε . The claim follows then directly fromLemma 8. (cid:3) References
1. L. V. Ahlfors,
Lectures on Quasiconformal Mappings,
Van Nostrand, Princeton, 1966; reprinted by Amer.Math. Soc., 2006.2. Kari Astala,
Area distortion of quasiconformal mappings,
Acta Math. 173 (1994), no. 1, 37–60.3. K: Astala, J.T. Gill, S. Rohde and E. Saksman,
Optimal regularity for planar mappings of finite distortion,
Ann. Inst. H. Poincar´e Anal. Non Lineaire 27 (2010), 1–19.4. K. Astala, T. Iwaniec, and G. J. Martin,
Elliptic partial differential equations and quasiconformal mappingsin the plane , Princeton University Press, 2009.5. K. Astala, T. Iwaniec, I. Prause and E. Saksman,
Burkholder integrals, Morrey’s problem and quasiconformalmappings,
J. Amer. Math. Soc. 25 (2012), 507–531.6. M.A. Brakalova and J.A. Jenkins,
On solutions of the Beltrami equation,
J. Anal. Math. 76 (1998) 67–92.7. K.M. Chong,
Some extensions of a theorem of Hardy, Littlewood and P´olya and their applications,
Canad. J.Math. 26 (1974) 1321–1340.8. G. David,
Solutions de l’equation de Beltrami avec k µ k ∞ = 1, Ann. Acad. Sci. Fenn. Ser. AI Math. 13 (1988),25–70.9. D. Faraco, P. Koskela and X. Zhong, Mappings of finite distortion: The degree of regularity,
Adv. Math. 190(2005), 300–318.10. C-Y Guo, P. Koskela, Pekka, J. Takkinen:
Generalized quasidisks and conformality,
Publ. Mat. 58 (2014),193–212.11. G.H. Hardy, J.E. Littlewood, and G. P´olya:
Inequalities.
Cambridge University Press 1934.12. T. Iwaniec, P. Koskela and G. Martin,
Mappings of BMO-distortion and Beltrami-type operators,
J. Anal.Math. 88 (2002), 337–381.13. T. Iwaniec, P. Koskela, G. Martin and C. Sbordone,
Mappings of finite distortion: L n log L integrability, J.London Math. Soc. (2) 67 (2003), 123–136.14. P. Koskela:
Planar mappings of finite distortion , Comput. Methods Funct. Theory 10 (2010), 663–678.15. V. Ryazanov, U. Srebro and E. Yakubov,
BMO-quasiconformal mappings,
J. Anal. Math. 83 (2001), 1–20.
University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68 , FIN-00014University of Helsinki, Finland
E-mail address : [email protected] Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101Joensuu, Finland
E-mail address : [email protected] University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68 , FIN-00014University of Helsinki, Finland
E-mail address ::