On the heterogeneous distortion inequality
aa r X i v : . [ m a t h . C V ] F e b ON THE HETEROGENEOUS DISTORTION INEQUALITY
ILMARI KANGASNIEMI AND JANI ONNINEN
Abstract.
We study Sobolev mappings f ∈ W ,n loc ( R n , R n ) , n ≥ , thatsatisfy the heterogeneous distortion inequality | Df ( x ) | n ≤ KJ f ( x ) + σ n ( x ) | f ( x ) | n for almost every x ∈ R n . Here K ∈ [1 , ∞ ) is a constant and σ ≥ is afunction in L n loc ( R n ) . Although we recover the class of K -quasiregularmappings when σ ≡ , the theory of arbitrary solutions is significantlymore complicated, partly due to the unavailability of a robust degree the-ory for non-quasiregular solutions. Nonetheless, we obtain a Liouville-type theorem and the sharp Hölder continuity estimate for all solutions,provided that σ ∈ L n − ε ( R n ) ∩ L n + ε ( R n ) for some ε > . This gives anaffirmative answer to a question of Astala, Iwaniec and Martin. Introduction
Let Ω be a connected, open subset of R n with n ≥ . We study mappings f = ( f , . . . , f n ) : Ω → R n in the Sobolev space W ,n loc (Ω , R n ) that satisfy the heterogeneous distortion inequality (1.1) | Df ( x ) | n ≤ KJ f ( x ) + σ n ( x ) | f ( x ) | n for a.e. (almost every) x ∈ Ω , where K ∈ [1 , ∞ ) and σ ∈ L n loc (Ω) . Here, | Df ( x ) | is the operator norm of the weak derivative Df ( x ) : R n → R n of f at a point x ∈ Ω ; that is, | Df ( x ) | = sup {| Df ( x ) h | : h ∈ S n − } . Moreover, J f ( x ) = det Df ( x ) is the Jacobian determinant of f . It is worth noting thatthe natural Sobolev space in which to seek the solutions is W ,n loc (Ω , R n ) .The reason for this is that this space provides enough regularity to applyintegration by parts to the form J f d x .If σ ≡ in (1.1), we recover the mappings of bounded distortion , alsoknown as K - quasiregular maps. Homeomorphic K -quasiregular maps arealso commonly called K -quasiconformal . The theory of mappings of boundeddistortion arose from the need to generalize the geometry of holomorphicfunctions to higher dimensions, and is by now a central topic in modernanalysis with important connections to partial differential equations, com-plex dynamics, differential geometry and the calculus of variations; see the Mathematics Subject Classification.
Primary 30C65; Secondary 35B53, 35R45,53C21.
Key words and phrases.
Heterogeneous distortion inequality, Quasiregular mappings,Liouville theorem, Hölder continuity, Astala-Iwaniec-Martin question.J. Onninen was supported by the NSF grant DMS-1700274. monographs by Reshetnyak [32], Rickman [33], Iwaniec and Martin [19], andAstala, Iwaniec and Martin [1]. The last 20 years have also seen widespreadstudy of a more general class of deformations, mappings of finite distortion ,where the constant K is replaced by a finite function K : Ω → [1 , ∞ ) ; seee.g. the monographs [19] and [14].A core part of the theory of quasiregular mappings is that the distortionestimate implies several strong topological properties. In higher dimensionsthis was pioneered in a series of papers by Reshetnyak (1966–1969). Accord-ingly, spatial quasiregular mappings enjoy the following properties:(i) A K -quasiregular mapping is locally /K -Hölder continuous [28];(ii) A nonconstant quasiregular mapping is both discrete and open [31,30];(iii) A bounded quasiregular mapping in R n is constant [29].Recall that a mapping f is open if f ( U ) is open for every open U , and that f is discrete if the sets f − { y } consist of isolated points. Generalizations ofthese results also hold for mappings of finite distortion, assuming sufficientintegrability conditions on the distortion function; see e.g. [35, 17, 21, 34, 15].Solutions to the heterogeneous distortion inequality (1.1) generally lackthese powerful conditions, and therefore require new tools for their treat-ment. For instance, if g : Ω → R in W ,n (Ω) and f ( x ) = ( e g ( x ) , , . . . , ,then f satisfies (1.1) with σ = |∇ g | ∈ L n (Ω) . This results in a discontinu-ous solution f by simply choosing a discontinuous g ∈ W ,n (Ω) . Similarly,by choosing a smooth, compactly supported g , we have that f satisfies (1.1)with σ ∈ L p ( R n ) for every p ∈ [1 , ∞ ] , yet f is neither constant nor discrete oropen. Hence, the heterogeneous distortion inequality even with a regular σ is too weak for its solutions to satisfy the above conditions (ii) and (iii), andtherefore the use of e.g. degree theory [9] is unavailable for non-quasiregularsolutions.However, in the planar case n = 2 , Astala, Iwaniec and Martin showedthat entire solutions f of (1.1) which vanish at infinity do satisfy a coun-terpart of the Liouville theorem (iii), provided that σ is sufficiently regular;see [1, Theorem 8.5.1]. This uniqueness theorem has found several importantapplications, such as the nonlinear ∂ -problem in the theory of holomorphicmotions [22] or the solution to the Calderón problem [2]. It is for this reasonthat Astala, Iwaniec and Martin asked in [1, p. 253] whether their form ofthe Liouville theorem and a version of the continuity result (i) remain validin higher dimensions. Astala-Iwaniec-Martin Question.
Suppose that f ∈ W ,n loc ( R n , R n ) sat-isfies the heterogeneous distortion inequality (1.1) with σ ∈ L n + ε ( R n ) ∩ L n − ε ( R n ) for some ε > . Under these assumptions, is the mapping f continuous? Moreover, does f satisfy the following Liouville-type theorem:if f ( x ) → as | x | → ∞ , then f ≡ ? N THE HETEROGENEOUS DISTORTION INEQUALITY 3
Along with their proof of the planar case, Astala, Iwaniec and Martinprovided a counterexample which shows that the assumption σ ∈ L ( C ) isnot enough for the Liouville-type theorem; see [1, Theorem 8.5.2.]. Note thatin the planar case, the heterogeneous distortion inequality (1.1) with σ ∈ L n + ε ( R n ) ∩ L n − ε ( R n ) amounts to saying that the solutions f ∈ W , (Ω , C ) satisfy the homogeneous differential inequality(1.2) | ∂ z f | ≤ k | ∂ z f | + ˜ σ | f | , with k < and ˜ σ ∈ L ε ( C ) ∩ L − ε ( C ) . Moreover, (1.2) can in fact beexpressed as a linear heterogeneous Cauchy-Riemann system, and is thusuniformly elliptic. However, for n ≥ , the theory is nonlinear, and themain inherent difficulty lies in the lack of general existence theorems andcounterparts to power series expansions of the solutions.In this paper, we give an affirmative answer to the Astala–Iwaniec–Martinquestion. The continuity is the more straightforward part of the solution,whereas the main challenge lies in proving the Liouville-type uniquenesstheorem.1.1. Continuity.
The solutions to (1.1) are indeed continuous when σ ∈ L n + ε loc (Ω) . We in fact establish sharp local Hölder continuity estimates forsuch mappings.For γ ∈ (0 , we denote the class of locally γ -Hölder continuous mappings f : Ω → R n by C ,γ loc (Ω , R n ) . We let γ K := 1 /K denote the sharp Hölderexponent of K -quasiregular mappings, and γ ε := ε/ ( n + ε ) the sharp Hölderexponent of W ,n + ε -functions. Our result shows that if γ K = γ ε , then thesharp Hölder exponent γ for solutions to (1.1) is the minimum of the expo-nents γ K and γ ε . There is, however, a somewhat surprising special case: if γ K and γ ε coincide, then we in fact end up with an infinitesimally weakerHölder exponent than γ K = γ ε . Theorem 1.1.
Let Ω ⊂ R n be a domain, K ∈ [1 , ∞ ) , ε > , γ = min( γ K , γ ε ) ,and σ ∈ L n + ε loc (Ω) . Suppose that f ∈ W ,n loc (Ω , R n ) satisfies the heterogeneousdistortion inequality (1.1) with K and σ .If γ K = γ ε , then f ∈ C ,γ loc (Ω , R n ) . Moreover, there exist σ and f satisfyingthe assumptions such that f / ∈ C ,γ ′ loc (Ω , R n ) for any γ ′ > γ .If γ K = γ ε , then f ∈ C ,γ ′ loc (Ω , R n ) for every γ ′ < γ , and there exist σ and f satisfying the assumptions such that f / ∈ C ,γ loc (Ω , R n ) . Liouville Theorem.
Recall that the classical Liouville theorem assertsthat bounded entire holomorphic functions are constant [5]. Its quasiregularcounterpart (iii) follows from the Caccioppoli inequality [3], which controlsthe derivatives of a quasiregular mapping locally in terms of the mapping.Attempting the same approach under the assumptions of the Astala-Iwaniec-Martin question yields that the integral of the Jacobian over the entire space
ILMARI KANGASNIEMI AND JANI ONNINEN R n equals zero, or equivalently, that(1.3) Z R n | Df | n ≤ Z R n | f | n σ n . Analogously to the quasiregular theory [10], the local variants of the energyestimate (1.3) imply a higher degree of integrability for the derivatives ofa solution. These observations can be further combined with the nonlin-ear Hodge theory developed by Iwaniec and Martin [18, 16], which provides L p -estimates for | Df | with exponents p either smaller or larger than the di-mension n . However, this approach alone appears insufficient for the desiredLiouville-type theorem, and although it could be used to show the continu-ity of f , the proof would not yield the sharp Hölder exponents we obtain inTheorem 1.1.Our first Liouville-type result shows that if σ ∈ L n ( R n ) ∩ L n + ε loc ( R n ) , thennontrivial solutions to (1.1) that are bounded in R n do not vanish at anypoint. Note that the standard radial example of a K -quasiregular mapping f ( x ) = | x | K − x shows that this result has no local alternative. Theorem 1.2.
Suppose that f ∈ W ,n loc ( R n , R n ) satisfies the heterogeneousdistortion inequality (1.1) with K ∈ [1 , ∞ ) and σ ∈ L n ( R n ) ∩ L n + ε loc ( R n ) forsome ε > . If f is bounded and f is not identically zero, then / ∈ f ( R n ) . The first and key step in the proof of Theorem 1.2 is to show that asolution to the heterogeneous distortion inequality with σ ∈ L n ( R n ) satisfies(1.4) Z R n J f | f | n = 0 , and therefore, Z R n | Df | n | f | n ≤ Z R n σ n . The proof of this is based on a thorough analysis of the integrals of J f overthe sublevel sets of | f | . In turn, we obtain that |∇ log | f || ∈ L n ( R n ) . Thesecond step is to establish a Morrey-type decay estimate [24] on the integralsof |∇ log | f || n over the balls in R n . Our proof of the decay estimate relies onthe formal identity J f ( x ) | f ( x ) | n vol n = d ω, where ω is a certain differential ( n − -form. The resulting polynomialdecay estimate implies that the function log | f | is Hölder continuous. Thus,the mapping f itself must omit the point from its range, completing theproof of Theorem 1.2.Our final main result is then the Liouville part of the Astala-Iwaniec-Martin question. Theorem 1.3.
Suppose that f ∈ W ,n loc ( R n , R n ) satisfies the heterogeneousdistortion inequality (1.1) for K ∈ [1 , ∞ ) and σ ∈ L n − ε ( R n ) ∩ L n + ε ( R n ) forsome ε > . If lim x →∞ | f ( x ) | = 0 , then f ≡ . We prove Theorem 1.3 by showing that, if there exists a non-identicallyzero solution f : R n → R n to (1.1) with σ ∈ L n + ε ( R n ) ∩ L n − ε ( R n ) , then N THE HETEROGENEOUS DISTORTION INEQUALITY 5 the oscillation of the function log | f | over the entire space R n is uniformlybounded. This clearly leads to a contradiction with the assumption f ( x ) → when | x | → ∞ . The crucial step in obtaining the uniform oscillation boundis to strengthen the estimate (1.4); that is, to prove an integrability estimatebelow the natural exponent n for the expression | Df | / | f | .Our solution is influenced by the case n = 2 . Indeed, in this case, themapping f has no zeros by Theorem 1.2, and hence f has a well-definedcomplex logarithm log f . The mapping log f satisfies the distortion estimate(1.5) | D log f | ≤ KJ log f + σ almost everywhere. This, in turn, gives a nonhomogeneous linear ellipticequation for log f , which implies the desired integrability estimate below thenatural exponent for | D log f | . In higher dimensions, the issue is to con-struct a similar map log f : R n → R n . The Zorich map h Z : R n → R n \ { } provides a well-known n -dimensional generalization of the planar exponen-tial mapping; see [37]. Unfortunately, h Z has a branch set consisting of ( n − -dimensional hyperplanes, which prevents lifting an arbitrary contin-uous f : R n → R n \ { } through h Z .We circumvent the lifting difficulties by moving to the Riemannian mani-fold setting. Indeed, a well defined counterpart for the logarithm exists from R n \ { } to R × S n − , providing us with a mapping “ log f ” : R n → R × S n − .This mapping satisfies a higher dimensional counterpart of the estimate (1.5),and the single Euclidean component R in the target space is sufficient for aCaccioppoli-type inequality to hold, which leads to the desired integrabilityestimate below the natural exponent n . Acknowledgments
We thank Tadeusz Iwaniec and Xiao Zhong for discussions and sharedinsights. 2.
Preliminaries
In this section, we go over some of the tools we require which might beless familiar to readers.2.1.
Sobolev mappings with manifold target.
For the most part of thistext we use the standard Sobolev spaces W ,p (Ω , R k ) and W ,p loc (Ω , R k ) , where Ω ⊂ R n is an open connected set, see e.g. [9, 36]. However, towards the endof the text, we consider a locally Sobolev mapping f : R n → M , where M isan n -dimensional Riemannian manifold.There are various approaches to defining first order Sobolev mappingswith a manifold target; see e.g. [11] or [6]. However, in our case, we onlyhave to consider continuous Sobolev mappings with a manifold target, whichsimplifies the definition significantly. ILMARI KANGASNIEMI AND JANI ONNINEN
Definition 2.1.
Let Ω ⊂ R n be a domain, and let M be a Riemannian k -manifold. We say that a continuous f : Ω → M is in the Sobolev space W ,p loc (Ω , M ) for p ∈ [1 , ∞ ) if, for every x ∈ Ω , there exists a neighborhood U ⊂ Ω of x and a smooth bilipschitz chart ϕ : V → M such that f U ⊂ ϕV and ϕ − ◦ f ∈ W ,n ( U, R k ) .If a continuous function f : Ω → M is in W ,p loc (Ω , M ) , then there exists aweak derivative Df : Ω × R n → T M which satisfies D ( ϕ − ◦ f ) = D ( ϕ − ) ◦ Df for bilipschitz charts ϕ : V → M . This weak derivative is unique up to a setof measure zero, in the sense that if ˜ Df is another such mapping, then ˜ Df = Df outside a set of the form E × R n where the set E has zero n -dimensional Lebegue measure, m n ( E ) = 0 .At a given point x ∈ Ω , we denote by | Df ( x ) | the operator norm of Df ( x ) : R n → T f ( x ) M , where T f ( x ) M is equipped with the norm inducedby the Riemannian metric. It follows that | Df | : Ω → [0 , ∞ ] is in L p loc (Ω) for any continuous f ∈ W ,p loc (Ω , M ) . If dim Ω = dim M and M is oriented,then we also have a measurable Jacobian J f : Ω → R , characterized almosteverywhere by f ∗ vol M = J f vol n .We remark that if M = R k , then the above definition coincides with theusual definition of f ∈ W ,p loc (Ω , R k ) for continuous f . We also remark that ifthe target is a product manifold M = M × M , then given two continuousmappings f : Ω → M and f : Ω → M , we have that ( f , f ) ∈ W ,p loc (Ω , M ) if and only if f ∈ W ,p loc (Ω , M ) and f ∈ W ,p loc (Ω , M ) .Next we recall Sobolev differential forms. Namely, suppose that M is aRiemannian manifold. A measurable k + 1 -form dω ∈ L ( ∧ k +1 M ) is the weak differential of a measurable k -form ω ∈ L ( ∧ k M ) if Z M ω ∧ dη = ( − k +1 Z M dω ∧ η for every η ∈ C ∞ ( ∧ n − k − M ) . We denote by W d,p,q loc ( ∧ k M ) the space of k -forms ω ∈ L p loc ( ∧ k M ) with a weak differential dω ∈ L q loc ( ∧ k +1 M ) . Theversion where ω ∈ L p ( ∧ k M ) and dω ∈ L q ( ∧ k +1 M ) is denoted W d,p,q ( ∧ k M ) .We also use the shorthands W d,p ( ∧ k M ) = W d,p,p ( ∧ k M ) and W d,p loc ( ∧ k M ) = W d,p,p loc ( ∧ k M ) .In particular, we require the following standard result about pull-backs ofcompactly supported smooth forms with Sobolev mappings. We sketch theproof for the convenience of the reader. Lemma 2.2.
Let Ω ⊂ R n be a domain, and let M be a Riemannian m -manifold. Suppose that f ∈ W ,p loc (Ω , M ) , where we assume that f is con-tinuous if M = R m . If ω ∈ C ∞ ( ∧ k M ) and p ≥ k + 1 , then f ∗ ω ∈ W d,p/k,p/ ( k +1)loc ( ∧ k Ω) and df ∗ ω = f ∗ dω . N THE HETEROGENEOUS DISTORTION INEQUALITY 7
Sketch of proof.
The fact that f ∗ ω and f ∗ dω satisfy the correct integrabilitiesfollows from the estimates | ( f ∗ ω ) x | ≤ k ω k ∞ | Df ( x ) | k , | ( f ∗ dω ) x | ≤ k dω k ∞ | Df ( x ) | k +1 for a.e. x ∈ Ω .For df ∗ ω = f ∗ dω , it suffices to consider ω for which the support spt ω of ω is contained in the domain of a bilipschitz chart φ : U → R m , as thegeneral ω is a finite sum of such forms. Moreover, it suffices to consider ω of the form ω dφ ∧ · · · ∧ dφ k , as a general ω with spt ω ⊂ U is againa finite sum of such forms with the coordinates of φ rearranged. We mayalso select φ ′ ∈ C ∞ ( M, R m ) such that φ ′ = φ on spt ω , which lets us write ω = ω dφ ′ ∧ · · · ∧ dφ ′ k in a form where the components are defined on all of N .We then use the chain rule of locally Sobolev and C maps to concludethat f ∗ ω = ω ◦ f ∈ L p loc ( ∧ Ω) , f ∗ dω = d ( ω ◦ f ) ∈ L p loc ( ∧ Ω) , and f ∗ dφ ′ i = d ( φ ′ i ◦ f ) ∈ L p loc ( ∧ Ω) for every i ∈ { , . . . , k } . By the wedge product rules forSobolev forms and the formula d ◦ d = 0 for the weak differential, we concludethat f ∗ dω = f ∗ dω ∧ f ∗ dφ ′ ∧· · ·∧ f ∗ dφ ′ k = d ( f ∗ ω ) ∧ d ( φ ′ ◦ f ) ∧· · ·∧ d ( φ ′ k ◦ f ) = df ∗ ω . (cid:3) Caccioppoli inequality.
The Caccioppoli inequalities are a standardtool in the study of quasiregular mappings. The most basic form of theCaccioppoli estimate for a K -quasiregular mapping f : Ω → R n reads as Z Ω η n | Df | n ≤ n n K n Z Ω | f | n |∇ η | n , where is a real-valued smooth test function with compact support in Ω . Thisfollows from the general inequality Z Ω η n J f ≤ n Z Ω | Df | n − | η | n − |∇ η | | f | which can be proved for arbitrary f ∈ W ,n loc (Ω , R n ) via an integration-by-parts argument. Our arguments, however, require a version with a targetspace other than R n . In general, it is not possible to obtain a Caccioppoli-type estimate for mappings f : R n → M where M is a Riemannian n -manifold. This happens when M is a rational homology sphere , see [11].However, the standard proof generalizes to the case of M = R × N , where N is a Riemannian ( n − -manifold. Lemma 2.3.
Let Ω ⊂ R n be a domain, and let N be a compact orientedRiemannian ( n − -manifold without boundary. Let f ∈ W ,n loc (Ω , R × N ) ,where we assume that f is continuous if N = R n − . Denote by f R : Ω → R and f N : Ω → N the coordinate functions of f . Then for every η ∈ C ∞ (Ω) and every c ∈ R , we have (cid:12)(cid:12)(cid:12)(cid:12)Z Ω η n J f (cid:12)(cid:12)(cid:12)(cid:12) ≤ n Z R n | Df | n − | η | n − |∇ η | | f R − c | . ILMARI KANGASNIEMI AND JANI ONNINEN
Proof.
We define the function f η = ( η n ( f R − c ) , f N ) . Then f η ∈ W ,n loc (Ω , R × N ) . Moreover, since vol R × N = ( π ∗ R vol ) ∧ ( π ∗ N vol N ) , we have f ∗ η vol R × N = d ( η n ( f R − c )) ∧ f ∗ N vol N . We may select a subdomain U with smooth boundary such that U iscompactly contained in Ω and spt η ⊂ U . Since η n ( f R − c ) ∈ W ,n ( U ) withcompact support, we may approximate it in W ,n ( U ) with g i ∈ C ∞ ( U ) .Moreover, we have by Lemma 2.2 that f ∗ N vol N ∈ W d,n/ ( n − , ( ∧ n − Ω) and df ∗ N vol N = f ∗ N d vol N = 0 . Hence, f ∗ N vol N ∈ W d,n/ ( n − ( ∧ n − U ) , and wemay approximate f ∗ N vol N in W d,n/ ( n − ( ∧ n − U ) with ω i ∈ C ∞ ( ∧ n − U ) (seee.g. [20, Corollary 3.6]).By a standard Hölder-type estimate and the Leibniz rule, it thereforefollows that d ( g i ω i ) → d ( η n ( f R − c )) ∧ f ∗ N vol N in L ( ∧ n U ) . However, since g i ω i is smooth and compactly supported for every i , it follows that Z Ω d ( η n ( f R − c )) ∧ f ∗ N vol N = lim i →∞ Z U d ( g i ω i ) = lim i →∞ . Hence, we may estimate that (cid:12)(cid:12)(cid:12)(cid:12)Z Ω η n J f (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω η n d ( f R − c ) ∧ f ∗ N vol N (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ( f R − c ) d ( η n ) ∧ f ∗ N vol N (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R n | f R − c | ( n | η | n − |∇ η | ) | Df | n − . The claim therefore follows. (cid:3)
Jacobians of entire mappings.
To end the preliminaries section, wealso discuss a result regarding the Jacobian of an entire Sobolev mapping.It is the main reason why we stated the Caccioppoli inequality also in thecase where the target is the product of R and an ( n − -manifold. Lemma 2.4.
Let N be a compact oriented Riemannian ( n − -manifoldwithout boundary. Suppose that f ∈ W ,n loc ( R n , R × N ) , where we assume that f is continuous if N = R n − . If | Df | ∈ L n ( R n ) , then Z R n J f = 0 . Proof.
We let η r ∈ C ∞ ( R n , [0 , be such that we have η | B n (0 ,r ) ≡ , η | R n \ B n (0 , r ) ≡ , and |∇ η | ≤ /r . We again denote f = ( f R , f N ) . Wethen use the Caccioppoli inequality of Lemma 2.3 and Hölder’s inequality to N THE HETEROGENEOUS DISTORTION INEQUALITY 9 obtain (cid:12)(cid:12)(cid:12)(cid:12)Z R n η nr J f (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R n η n − r | Df | n − (cid:12)(cid:12) f R − ( f R ) B n (0 , r ) (cid:12)(cid:12) |∇ η r |≤ Z spt |∇ η r | η nr | Df | n ! n − n Z B n (0 , r ) (cid:12)(cid:12) f R − ( f R ) B n (0 , r ) (cid:12)(cid:12) n |∇ η r | n ! n Since f R ∈ W ,n loc ( R n ) and |∇ η | ≤ /r , the Sobolev-Poincaré inequality thenyields that Z spt |∇ η r | η nr | Df | n ! n − n Z B n (0 , r ) (cid:12)(cid:12) f R − ( f R ) B n (0 , r ) (cid:12)(cid:12) n |∇ η r | n ! n ≤ Z spt |∇ η r | η nr | Df | n ! n − n r ) n Z B n (0 , r ) (cid:12)(cid:12) f R − ( f R ) B n (0 , r ) (cid:12)(cid:12) n ! n ≤ C n Z R n \ B n (0 ,r ) | Df | n ! n − n Z B n (0 , r ) |∇ f R | n ! n . Since |∇ f R | ≤ | Df | ∈ L n ( R n ) , the first integral term on the right handside tends to 0 as r → ∞ , while the second term stays bounded. Since | η nr J f | ≤ | Df | n , the claim therefore follows by dominated convergence. (cid:3) Hölder continuity
In this section, we prove the continuity part of Theorem 1.1. Our proofis based on Morrey’s rather elegant ideas in geometric function theory [24,28, 19]. A crucial tool in establishing the sharp Hölder exponent is theisoperimetric inequality in the Sobolev space W ,n loc (Ω , R n ) . For x ∈ Ω andalmost every r > such that B r = B n ( x, r ) compactly contained in Ω , wehave(3.1) (cid:12)(cid:12)(cid:12)(cid:12)Z B r J f (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n n − √ ω n − ) − (cid:18)Z ∂B r (cid:12)(cid:12)(cid:12) D ♯ f (cid:12)(cid:12)(cid:12)(cid:19) nn − , where ω n − is the ( n − -dimensional area of the unit sphere ∂B in R n . Here D ♯ f ( x ) stands for the cofactor matrix of the differential matrix Df ( x ) . For adiffeomorphism f : B r → U this integral form of the isoperimetric inequalityfollows immediately from the familiar geometric form of the isoperimetricinequality n n − ω n − [ m n ( U )] n − ≤ [ m n − ( ∂U )] n , where m n ( U ) stands for the volume of a domain U ⊂ R n and m n − ( ∂U ) is its ( n − -dimensional surface area. For the proof of (3.1) for Sobolevmappings see Reshetnyak [32, Lemma II.1.2.] for a more detailed account. We begin with the primary estimate our proof relies on.
Lemma 3.1.
Let Ω , Ω ′ ⊂ R n be bounded domains with Ω ⊂ Ω ′ . Supposethat f ∈ W ,n (Ω ′ , R n ) satisfies the heterogeneous distortion inequality (1.1) for K ∈ [1 , ∞ ) and σ ∈ L n (Ω ′ ) ∩ L n + ε (Ω ′ ) , where ε > . Let x ∈ Ω , let B r = B ( x, r ) for all r ∈ (0 , ∞ ) , let R > be such that B R ⊂ Ω . Then forevery δ < ε and a.e. r < R we have the estimate Z B r | Df | n ≤ Krn Z ∂B r | Df | n + Cr nδn + δ , where C depends only on n , Ω , f , σ , ε and δ . In particular, C doesn’t dependon x , r and R . Moreover, if f ∈ L ∞ (Ω , R n ) , then the estimate also holds for δ = ε .Proof. By using the heterogeneous distortion inequality, the isoperimetric in-equality (3.1) for W ,n -mappings, Hadamard’s inequality (cid:12)(cid:12) D ♯ f (cid:12)(cid:12) ≤ | Df | n − ,and Hölder’s inequality, we obtain for a.e. r < R the estimate Z B r | Df | n ≤ K Z B r J f + Z B r | f | n σ n ≤ Kn n − √ ω n − (cid:18)Z ∂B r | Df | n − (cid:19) nn − + Z B r | f | n σ n ≤ Krn Z ∂B r | Df | n + Z B r | f | n σ n For the final term, we note that for every p < ∞ , we have f ∈ L p loc (Ω ′ , R n ) by the Sobolev embedding theorem, and consequently also f ∈ L p (Ω , R n ) .Hence, we may use Hölder’s inequality to obtain the desired estimate Z B r | f | n σ n ≤ (cid:18)Z B r σ n + ε (cid:19) nn + ε (cid:18)Z B r (cid:19) δn − δ (cid:18)Z B r | f | ( n + δ )( n + ε ) ε − δ (cid:19) n ( ε − δ )( n + δ )( n + ε ) ≤ (cid:18)Z Ω σ n + ε (cid:19) nn + ε (cid:18)Z Ω | f | ( n + δ )( n + ε ) ε − δ (cid:19) n ( ε − δ )( n + δ )( n + ε ) r nδn + δ . Moreover, if we additionally know that f ∈ L ∞ (Ω , R n ) , we obtain the claimfor δ = ε by estimating Z B r | f | n σ n ≤ k f k ∞ (cid:18)Z Ω σ n + ε (cid:19) nn + ε r nεn + ε . (cid:3) Note that the estimate of Lemma 3.1 is of the form Φ( r ) ≤ Ar Φ ′ ( r )+ Br α .This differential inequality allows us to obtain an estimate for the decay of Φ at 0, which in our case is a decay estimate on the integrals of | Df | n overballs. N THE HETEROGENEOUS DISTORTION INEQUALITY 11
Lemma 3.2.
Suppose that
Φ : [0 , R ] → [0 , S ] is an absolutely continuousincreasing function such that Φ(0) = 0 and (3.2) Φ( r ) ≤ Ar Φ ′ ( r ) + Br α for a.e. r ∈ [0 , R ] , where A, α > and B ≥ . Then there exists a constant C = C ( A, B, α, R, S ) such that the following holds: • if α < A − , then for all r ∈ [0 , R ] we have Φ( r ) ≤ Cr α ; • if α = A − , then for all r ∈ [0 , R ] we have Φ( r ) ≤ Cr α log (cid:16) Re α − r (cid:17) ; • if α > A − , then for all r ∈ [0 , R ] we have Φ( r ) ≤ Cr A − . Proof.
We observe that dd r (cid:16) − Ar − A − Φ( r ) (cid:17) = Φ( r ) − Ar Φ ′ ( r ) r A − for a.e. r ∈ [0 , R ] . Consequently, the estimate (3.2) can be rewritten in theform dd r (cid:16) − Ar − A − Φ( r ) (cid:17) ≤ Br − − ( A − − α ) . We integrate this estimate, obtaining(3.3) Z Rr dd s (cid:16) − As − A − Φ( s ) (cid:17) d s ≤ B Z Rr s − − ( A − − α ) d s. Consider first the case α < A − . Computing the integrals in (3.3) yields A ( r − A − Φ( r ) − R − A − Φ( R )) ≤ BA − − α (cid:16) r − ( A − − α ) − R − ( A − − α ) (cid:17) , and further rearrangement and estimation yields Φ( r ) ≤ r A − (cid:18) R − A − Φ( R ) + B − Aα r α − A − (cid:19) ≤ (cid:18) R − α S + B − Aα (cid:19) r α Suppose then that α = A − . Then (3.3) results in A ( r − A − Φ( r ) − R − A − Φ( R )) ≤ B log( R/r ) , and we may again further estimate Φ( r ) ≤ r A − (cid:18) R − A − Φ( R ) + BA log Rr (cid:19) ≤ (cid:18) R − A − S + BA (cid:19) r A − log (cid:16) Re α − r (cid:17) . Finally, consider the case α > A − . In this case, it follows from (3.3) that A ( r − A − Φ( r ) − R − A − Φ( R )) ≤ Bα − A − (cid:16) R α − A − − r α − A − (cid:17) , and further rearrangement and estimation yields Φ( r ) ≤ r A − R − A − S + BR α − A − Aα − ! . (cid:3) For the remaining component to the proof of Theorem 1.1, we recall awell known fact that the decay estimate on | Df | implies that f belongs to aMorrey–Campanato space [25, 4], and is thus Hölder continuous. The preciseformulation of this fact that we use is as follows. Lemma 3.3.
Let
Ω = B n ( x, R/ for some R > and k ∈ N . Suppose that f ∈ W ,n ( B n ( x, R ) , R k ) satisfies (3.4) Z B r | Df | n ≤ Cr α log β Lr for all B r = B n ( y, r ) ⊂ B n ( x, R ) , where α > , β ≥ , and L > is largeenough that R < Le − β/α . Then | f ( y ) − f ( z ) | ≤ C ′ | y − z | αn log βn L | y − z | for all y, z ∈ Ω , where C ′ depends on n , k , C , A , α and β . Note that the assumption
R < Le − β/α is to ensure that r α log β ( A/r ) isincreasing on [0 , R ] . Lemma 3.3 is merely a small variant of a classical resultof Morrey [24] with an extra logarithmic term, where the logarithmic termbecomes relevant when investigating the exact modulus of continuity. See[26, Theorem 3.5.2] for a proof in the classical case β = 0 . For general β ,we note that Lemma 3.3 also follows from the fractional maximal functionestimate of Sobolev functions: if u ∈ W , ( R n ) and γ ∈ (0 , , then for all y, z ∈ R n outside a set of measure zero, we have(3.5) | u ( y ) − u ( z ) |≤ C n,γ | y − z | − γ (cid:0) M γ, | y − z | | Du | ( y ) + M γ, | y − z | | Du | ( z ) (cid:1) , where M γ,R stands for the restricted fractional maximal function M γ,R | Du | ( y ) = sup Lemma 3.4. Let Ω = B n ( x, R ) for some R > . Suppose that f ∈ W ,n ( B n ( x, R ) , R n ) satisfies the heterogeneous distortion inequality (1.1) with K ∈ [1 , ∞ ) and σ ∈ L n ( B n ( x, R )) ∩ L n + ε ( B n ( x, R )) , where ε > .If K − = ε/ ( n + ε ) , then | f ( y ) − f ( z ) | ≤ C | y − z | min( K − , εn + ε ) for all y, z ∈ Ω , where C = C ( n, K, ε, σ, f ) .If K − = ε/ ( n + ε ) , then | f ( y ) − f ( z ) | ≤ C | y − z | K − log n (cid:16) Re K | y − z | (cid:17) for all y, z ∈ Ω , where C = C ( n, K, ε, σ, f ) .Proof. We first prove a slightly weaker Hölder continuity estimate for f thanis claimed. This in turn implies the local boundedness of f , which lets usapply Lemma 3.1 in full force and to obtain the stated estimates.We set α = min( nε/ ( n + ε ) , n/K ) and choose α ′ ∈ (0 , α ) . ApplyingLemmas 3.1 and 3.2 we conclude that(3.6) Z B r | Df | n ≤ Cr α ′ for all B r = B n ( y, r ) ⊂ B n ( x, R ) . Therefore, it follows from Lemma 3.3that f is ( α ′ /n ) -Hölder continuous in B n ( x, R ) . Since continuity is a localproperty, we conclude that f is continuous, and in particular bounded in B n ( x, R ) .Now, knowing that f is locally bounded we may and do take δ = ε inLemma 3.1. Combining this with Lemma 3.2, we obtain the following decayestimate for the differential:(3.7) Z B r | Df | n ≤ Cr α log β (cid:16) (4 R ) e K r (cid:17) , where ( β = 0 if nεn + ε = nK β = 1 if nεn + ε = nK . for all B r . Thus, the desired Hölder continuity estimates for f follow fromLemma 3.3. (cid:3) Sharpness of the Hölder exponents Having Lemma 3.4, the remaining part of proving Theorem 1.1 is to con-struct solutions which show that the obtained Hölder exponents cannot beimproved. Recalling the notation γ K = K − and γ ε = ε/ ( n + ε ) , we considerthree different cases: γ K < γ ε , γ K > γ ε and γ K = γ ε .For the first case γ K < γ ε , we can simply use the standard radial example f ( x ) = | x | K x | x | . Indeed, the mapping f is K -quasiregular and hence satisfies (1.1) with σ ≡ ,and we also have f / ∈ C ,γ ( B n (0 , for every γ > K − . Next, we discuss the case γ K > γ ε in-depth. Example 4.1. Let K ≥ and ε > such that K − > ε/ ( n + ε ) . We definea mapping f : B n (0 , → R n with only a single non-vanishing coordinatefunction, namely f ( x ) = (cid:18) | x | εn + ε log − n (cid:18) e | x | (cid:19) , , , . . . , (cid:19) . This mapping lies in W ,n ( B n (0 , , R n ) , with ∇ f ( x ) = | x | − nn + ε (cid:18) εn + ε log − n (cid:18) e | x | (cid:19) + 1 n log − n +1 n (cid:18) e | x | (cid:19)(cid:19) x | x | . Furthermore, J f ≡ . Hence, the heterogeneous distortion inequality (1.1)for f reduces to |∇ f | n ≤ | f | n σ n . Since | f ( x ) | = f ( x ) ≥ for every x ∈ B n (0 , , the mapping f solves theheterogeneous distortion inequality for any σ ≥ |∇ f | . We choose σ = |∇ f | = | x | − nn + ε (cid:18) εn + ε log − n (cid:18) e | x | (cid:19) + 1 n log − n +1 n (cid:18) e | x | (cid:19)(cid:19) and then observe that σ n + ε ≤ n + ε | x | − n (cid:18) εn + ε log − n + εn (cid:18) e | x | (cid:19) + 1 n log − ( n +1)( n + ε ) n (cid:18) e | x | (cid:19)(cid:19) . We recall that for any p > , the function | x | − n log − p ( e/ | x | ) is integrableover B n (0 , . Indeed, Z B n (0 , | x | − n log − p e | x | = C n Z r n − d rr n log p er = C n Z dd r (cid:16) log − ( p − er (cid:17) d r < ∞ . Hence, the mapping f solves (1.1) with σ ∈ L n + ε ( B n (0 , . However, forany exponent γ > ε/ ( n + ε ) , the map f fails to be γ -Hölder continuous atthe origin.The remaining part of the proof of Theorem 1.1 is therefore to provide anexample in the special case γ K = γ ε . Example 4.2. Let K ≥ and ε > , and suppose that K − = ε/ ( n + ε ) .We define a mapping f : B n (0 , → R n by f ( x ) = | x | K log nK (cid:18) e | x | (cid:19) x | x | + (2 , , , . . . , . The mapping f is hence obtained by shifting a radially symmetric map ofthe form (Φ( | x | ) / | x | ) x , where Φ( t ) = t /K log / (2 nK ) ( e/t ) . For x ∈ B n (0 , N THE HETEROGENEOUS DISTORTION INEQUALITY 15 we have (cid:12)(cid:12)(cid:12)(cid:12) Φ( | x | ) | x | (cid:12)(cid:12)(cid:12)(cid:12) = | x | − K − K log nK (cid:18) e | x | (cid:19) and (cid:12)(cid:12) Φ ′ ( | x | ) (cid:12)(cid:12) = | x | − K − K (cid:18) K log nK (cid:18) e | x | (cid:19) − nK log − nK − nK (cid:18) e | x | (cid:19)(cid:19) . Using these and the fact that f is orientation preserving, we conclude, seee.g. [19, 6.5.1], that | Df ( x ) | n = max (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) Φ( | x | ) | x | (cid:12)(cid:12)(cid:12)(cid:12) n , (cid:12)(cid:12) Φ ′ ( | x | ) (cid:12)(cid:12) n (cid:19) = | x | − n ( K − K log K (cid:18) e | x | (cid:19) and KJ f ( x ) = K (cid:12)(cid:12)(cid:12)(cid:12) Φ( | x | ) | x | (cid:12)(cid:12)(cid:12)(cid:12) n − (cid:12)(cid:12) Φ ′ ( | x | ) (cid:12)(cid:12) = | x | − n ( K − K (cid:18) log n nK (cid:18) e | x | (cid:19) − n log n − nK − nK − nK (cid:18) e | x | (cid:19)(cid:19) = | x | − n ( K − K (cid:18) log K (cid:18) e | x | (cid:19) − n log − nK − n nK (cid:18) e | x | (cid:19)(cid:19) . Since Φ is increasing on [0 , , we have | f ( x ) | ≥ − Φ(1) = 1 for all x ∈ B n (0 , . Therefore, the heterogeneous distortion inequality (1.1) issatisfied if we choose σ n ( x ) = 12 n | x | − n ( K − K log − nK − n nK (cid:18) e | x | (cid:19) . We then observe that σ n + ε ( x ) = ( σ n ( x )) KK − = 1(2 n ) K − K | x | − n log − (2 nK − n ) K nK ( K − (cid:18) e | x | (cid:19) , and since (2 nK − n ) K nK ( K − 1) = 2 nK − nK nK − nK > , we conclude that σ ∈ L n + ε ( B n (0 , . However, we have | f ( x ) − f (0) || x − | K = log nK (cid:18) e | x | (cid:19) −−−→ x →∞ ∞ , and therefore f / ∈ C ,K − loc ( B n (0 , .The proof of Theorem 1.1 is thus complete. Sublevel sets and the logarithm In this section, we begin studying bounded entire functions f satisfying(1.1), with the goal of eventually reaching the Liouville type theorem statedin the Astala-Iwaniec-Martin question. Our main goal in this section is toshow that if f is not identically zero, then log | f | ∈ W ,n loc ( R n ) . This is alreadynotable, since this condition is not satisfied by all unbounded entire quasireg-ular maps. Our approach does not rely on the theory of partial differentialequations. Instead, the proof is based on two main tools: integration byparts and truncating f with respect to its level sets.5.1. Global integrability. We begin with a simple global integrability re-sult for Df when f is an entire mapping that solves the heterogeneous dis-tortion inequality (1.1). Lemma 5.1. Suppose that f ∈ W ,n loc ( R n , R n ) satisfies the heterogeneousdistortion inequality (1.1) with K ∈ [1 , ∞ ) and σ ∈ L n ( R n ) . If f is bounded,then | Df | ∈ L n ( R n ) .Proof. Let η r : R n → [0 , be a smooth mapping chosen such that η | B n (0 ,r ) ≡ , η | R n \ B n (0 , r ) ≡ , and |∇ η | ≤ /r . Now, by using the heterogeneousdistortion inequality, the Caccioppoli estimate of Lemma 2.3, and Hölder’sinequality, we obtain Z R n η nr | Df | n ≤ K Z R n η nr J f + Z R n η nr | f | n σ n ≤ K Z R n ( η r | Df | ) n − | f | |∇ η r | + k f k n ∞ Z R n η nr σ n ≤ Z B n (0 , r ) | f | n |∇ η r | n ! n (cid:18)Z R n η nr | Df | n (cid:19) n − n + k f k n ∞ k σ k nn ≤ ω n k f k ∞ (cid:18)Z R n η nr | Df | n (cid:19) n − n + k f k n ∞ k σ k nn Hence, we obtain an upper bound on the integral of η nr | Df | n independenton r . Letting r → ∞ yields the claim. (cid:3) Level set methods. We just proved that for a bounded f : R n → R n satisfying (1.1) with σ ∈ L n , the differential | Df | lies in L n ( R n ) . Therefore,by Lemma 2.4, the integral of J f over the entire space R n is zero. We nowproceed to improve this by showing that the integral of the Jacobian alsovanishes over every strict sublevel set of | f | . Lemma 5.2. Let f ∈ W ,n loc ( R n , R n ) . Suppose that | Df | ∈ L n ( R n ) . Thenfor every t > , we have Z { x ∈ R n : | f | Proof. Let t > ε > , and let ψ = ψ t,ε : [0 , ∞ ) → [0 , ∞ ) be a non-decreasingsmooth function such that ψ | [0 ,t − ε ] = id , ψ | [ t, ∞ ) ≡ t , and | ψ ′ | ≤ . Let h t,ε : R n → R n be the radial function defined by h t,ε ( x ) = ψ t,ε ( | x | ) x | x | . Then h t,ε is a smooth and 2-Lipschitz regular mapping. Consequently, thechain rule applies, Df t,ε ( x ) = Dh t,ε ( f ( x )) Df ( x ) for a.e. x , and f t,ε = h t,ε ◦ f lies in W ,n loc ( R n , R n ) , see e.g. [7, p.130].In particular, since | Dh t,ε | ≤ , we have Df t,ε ∈ L n ( R n ) . Therefore,Lemma 2.4 yields that Z R n ( J h t,ε ◦ f ) J f = Z R n J f t,ε = 0 . As ε → , we have J h t,ε → χ [0 ,t ) pointwise where χ E denotes the character-istic function χ E of a set E . Hence, the claim follows by letting ε → andapplying the dominated convergence theorem for the Lebesgue integral. (cid:3) Lemma 5.2 is our main tool in showing that, for an entire non-identicallyzero solution f , the function |∇ log | f || belongs to L n ( R n ) . Towards this, wefirst prove that the function | Df | n / | f | n is globally integrable. Lemma 5.3. Suppose that f ∈ W ,n loc ( R n , R n ) solves the heterogeneous dis-tortion inequality (1.1) with K ∈ [1 , ∞ ) and σ ∈ L n ( R n ) . If f is bounded,then J f / | f | n is integrable, Z R n J f | f | n = 0 , and Z R n | Df | n | f | n ≤ Z R n σ n < ∞ . Here and in what follows, we interpret J f / | f | n = 0 when J f = 0 , andsimilarly, | Df | n / | f | n = 0 when | Df | = 0 . Proof. We split J f into its positive and negative parts J f = J + f − J − f . Theseparts satisfy the inequality(5.1) | Df | n + KJ − f ≤ KJ + f + σ n | f | n . In particular, we have(5.2) J − f ≤ K − | f | n σ n . Indeed, this is trivial when J f ≥ , and if J f < , then (5.2) followsfrom (5.1). Hence, J − f = 0 a.e. where | f | = 0 , and we have(5.3) Z R n J − f | f | n ≤ K Z R n σ n < ∞ . For every t > , we denote the strict sublevel set of | f | at t by L t = { x ∈ R n : | f | < t } . By Lemmas 5.1 and 5.2, we have Z L t J f = 0 for every t > . In particular, we have for every t > that Z L t J + f = Z L t J − f . Multiplying this estimate by t − n − /n , we obtain(5.4) Z R n t − n − n J + f χ { x ∈ R n : | f | With the global integrability of | Df | n / | f | n shown,we now proceed to study the Sobolev regularity of log | f | .Let B R = B n ( x, R ) be a ball in R n with R > . Suppose that f ∈ L ∞ ( B R , R n ) ∩ W ,n loc ( B R , R n ) , and that | Df | / | f | ∈ L n ( B R ) . For every λ > ,we denote by | f | λ the function x max( | f | , λ ) . We then proceed to studythe functions log | f | λ . Since f is bounded and the function h λ : R n → R givenby h λ ( x ) = log max( | x | , λ ) is locally Lipschitz, we may use the chain rule ofLipschitz and Sobolev maps to obtain that log | f | λ = h λ ◦ f ∈ W ,n loc ( B R ) ;see e.g. [36, Theorem 2.1.11]. Moreover, we have the uniform estimate(5.5) |∇ log | f | λ | n = |∇ | f || n | f | n χ {| f | >λ } ≤ | Df | n | f | nλ ≤ | Df | n | f | n < ∞ which is independent of λ .By using these truncated logarithms as a tool, we achieve the followingresult. N THE HETEROGENEOUS DISTORTION INEQUALITY 19 Lemma 5.4. Let f : B R → R n be a bounded and not identically zero map-ping. Suppose that f ∈ W ,n loc ( B R , R n ) and that | Df | / | f | ∈ L n ( B R ) . Then f − { } has zero Lebesgue measure, the measurable function log | f | lies in W ,n loc ( B R ) , and |∇ log | f || ≤ | Df || f | ∈ L n ( B R ) . Proof. By our assumptions, the set | f | − (0 , ∞ ) has positive measure. Hence,there exists t ∈ (0 , such that F t = (cid:8) x ∈ B R : t − > | f ( x ) | > t (cid:9) has pos-itive measure. For every λ > , we denote by f λ : B R → R the function f λ = log | f | λ .Our first goal is to show that log | f | ∈ L ( B R ) . For the proof, we as-sume towards a contradiction that the integral of | log | f || over B R is in-stead infinite. In this case, since the functions f λ are uniformly boundedfrom above and decrease to log | f | monotonically as λ → , we have lim λ → ( f λ ) B R = −∞ ; recall that ( f λ ) B R stands for the integral averagevalue of the function f λ over B R .By the Sobolev-Poincaré inequality and (5.5), we have the upper bound m n ( B R ) Z B R | f λ − ( f λ ) B R | ≤ C n (cid:18)Z B R |∇ f λ | n (cid:19) n ≤ C n (cid:18)Z B R | Df | n | f | n (cid:19) n . This upper bound, independent of λ , is finite by our assumptions. We alsohave the lower bound m n ( B R ) Z B R | f λ − ( f λ ) B R | ≥ m n ( F t ) m n ( B R ) (cid:0) | ( f λ ) B R | − log t − (cid:1) . Since lim λ → ( f λ ) B = −∞ , we arrive at a contradiction. Hence, log | f | ∈ L ( B R ) . In particular, it follows that log | f | is finite almost everywhere, andtherefore f − { } has zero Lebesgue measure.Now, for λ < , we have Z B R | log | f | − f λ | ≤ Z f − [0 ,λ ) | log | f || → when λ → , and Z B R (cid:12)(cid:12)(cid:12)(cid:12) ∇ f λ − ∇ | f || f | (cid:12)(cid:12)(cid:12)(cid:12) n ≤ Z f − [0 ,λ ] | Df | n | f | n → when λ → . Therefore, f λ → log | f | in L ( B R ) and ∇ f λ → ( ∇ | f | ) / | f | in L n ( B R ) .Thus, the weak gradient of log | f | equals ( ∇ | f | ) / | f | . Since ( ∇ | f | ) / | f | ∈ L n ( B R , R n ) , the Sobolev embedding theorem shows that log | f | ∈ L n loc ( B R ) ,and hence log | f | ∈ W ,n loc ( B R ) . (cid:3) Non-existence of zeroes In this section we will show that log | f | is locally Hölder continuous if f is a bounded entire solution to the heterogeneous distortion inequality with σ ∈ L n ( R n ) ∩ L n + ε loc ( R n ) . This will prove Theorem 1.2. Our approachagain mimics the lines of reasoning by Morrey and is based on obtaininga quantitative integral estimate for | Df | n / | f | n over balls. This is done byemploying a suitable isoperimetric inequality.6.1. Logarithmic isoperimetric inequality. We then proceed to showthe following isoperimetric-type estimate for | Df | n / | f | n . As before, we use B r = B n ( x, r ) to denote a ball in R n around a fixed point x . Lemma 6.1. Let f ∈ L ∞ loc ( B R , R n ) ∩ W ,n loc ( B R , R n ) , where R > . If | Df | / | f | ∈ L n loc ( B R ) , then there is a constant C n such that for a.e. r ∈ (0 , R ) , we have (6.1) Z B r J f | f | n ≤ C n r Z ∂B r | Df | n | f | n . The main idea behind the proof is to write J f ( x ) | f ( x ) | n vol n = d ω, where ω is a certain differential ( n − -form, and then to use Stokes’ theorem.This method actually gives similar estimates for integrals of the more generalform ψ ( | f | ) J f over balls. The precise estimate obtained is given by thefollowing lemma, which is a variant of [27, Lemma 2.1] by Onninen andZhong. We provide a proof here due to our assumptions being slightly weakerthan in [27]. Lemma 6.2. Let Ω ⊂ R n be a domain. Suppose that f : Ω → R n is in L ∞ loc (Ω , R n ) ∩ W ,n loc (Ω , R n ) . If Ψ : [0 , ∞ ) → R is a piecewise C -smooth func-tion with Ψ ′ locally bounded, then for every test function η ∈ C ∞ (Ω) , wehave (cid:12)(cid:12)(cid:12)(cid:12)Z Ω η (cid:2) n Ψ( | f | ) + 2 | f | Ψ ′ ( | f | ) (cid:3) J f (cid:12)(cid:12)(cid:12)(cid:12) ≤ √ n Z Ω |∇ η | | f | Ψ( | f | ) | Df | n − . Before jumping into the proof, we comment on the well-definedness of theleft integrand. The function Ψ ′ is defined outside finitely many jump points y ∈ [0 , ∞ ) . Consider the set A y = { x ∈ Ω : | f ( x ) | = y } . Then a.e. on A y ,we have ∇ | f | = 0 ; see e.g. [13, Corollary 1.21]. Since ∇ | f | = P ni =1 f i ∇ f i ,this implies that for a.e. x ∈ A y , we have | f ( x ) | = 0 or some non-zero linearcombination of { ( df i ) x : i = 1 , . . . , n } vanishes. In the latter case, J f ( x ) = 0 .Hence, almost everywhere where Ψ ′ ( | f | ) is not defined, we have | f | = 0 or J f = 0 , making the integrand in the statement well defined. Proof of Lemma 6.2. By switching to a smaller domain Ω which still containsthe support of η , we may assume that Ω is bounded and f ∈ W ,n (Ω , R n ) ∩ L ∞ (Ω , R n ) . By boundedness of | f | and the local boundedness of Ψ and Ψ ′ ,we may also assume that Ψ and Ψ ′ are bounded. N THE HETEROGENEOUS DISTORTION INEQUALITY 21 We consider the function F i = ( f , . . . , f i − , η Ψ( | f | ) f i , f i +1 , . . . , f n ) . Since Ψ ′ is bounded and | f | is Sobolev, the chain rule of Lipschitz andSobolev maps yields that ∇ (Ψ( | f | )) = Ψ ′ ( | f | ) ∇ | f | = 2Ψ ′ ( | f | ) P nj =1 f j ∇ f j a.e. on Ω , see e.g. [36, Theorem 2.1.11]. By further using the product rulesof Sobolev mappings, we see that η Ψ( | f | ) f i has a locally integrable weakgradient given by ∇ ( η Ψ( | f | ) f i )= Ψ( | f | ) f i ∇ η + 2 η Ψ ′ ( | f | ) f i n X j =1 f j ∇ f j + η Ψ( | f | ) ∇ f i . Using the fact that η , Ψ( | f | ) , Ψ ′ ( | f | ) and f i are bounded, we then concludethat this weak gradient is in L n (Ω , R n ) .We therefore have that η n Ψ( | f | ) f i ∈ W ,n (Ω) . Consequently, F i ∈ W ,n (Ω) , and therefore J F i is integrable. Since F i also has a compactlysupported coordinate function, we therefore have Z Ω J F i = 0 . By writing J F i vol n as a wedge product, we obtain Z Ω η (cid:2) Ψ( | f | ) + 2 f i Ψ ′ ( | f | ) (cid:3) J f vol n = − Z Ω Ψ( | f | ) f i df ∧ · · · ∧ df i − ∧ dη ∧ df i +1 ∧ · · · ∧ df n . By summing over i , and by using the fact that | α ∧ · · · ∧ α n | ≤ | α | · · · | α n | for 1-forms α , . . . , α n , the claim follows. (cid:3) With the proof of Lemma 6.2 complete, we then proceed to prove Lemma6.1. Proof of Lemma 6.1. We first prove an isoperimetric estimate of the follow-ing form: for a.e. r ∈ (0 , R ) and every constant c , we have(6.2) Z B r J f | f | n ≤ C n Z ∂B r | Df | n − | f | n − (cid:12)(cid:12) log | f | − c (cid:12)(cid:12) . Hence, fix a c ∈ R and let r ∈ (0 , R ) . For all sufficiently large j ∈ N , weselect cutoff functions η j ∈ C ∞ ( B r ) such that η j ≤ η j +1 ≤ , η j ( x ) = 1 forall x ∈ B r − /j , and sup {|∇ η j ( x ) | : x ∈ B r } ≤ /j . We also fix a > and ε ∈ (0 , , and define a function Ψ a,ε : [0 , ∞ ) → R by Ψ a,ε ( t ) = ( t − n (cid:0) log( t + ε ) − c (cid:1) , t ≥ a a − n (log √ a + ε − c ) , t ≤ a . The function Ψ a,ε is piecewise C and its derivative is locally bounded. More-over, we have n Ψ a,ε ( t ) + 2 t Ψ ′ a,ε ( t ) = ( t − ( n − ( t + ε ) − t > ana n (log √ a + ε − c ) , t < a. Hence, by using Lemma 6.2 with Ψ = Ψ a,ε and η = η j , we obtain that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| f | >a } η j J f | f | n − ( | f | + ε ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C n j Z rr − j Z ∂B s | Df | n − | f || f | na (cid:12)(cid:12)(cid:12)(cid:12) log q | f | a + ε − c (cid:12)(cid:12)(cid:12)(cid:12) ds + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| f | a }∩ B r J f | f | n − ( | f | + ε ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C n Z ∂B r | Df | n − | f || f | na (cid:12)(cid:12)(cid:12)(cid:12) log q | f | a + ε − c (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| f | a }∩ B r J f | f | n − ( | f | + ε ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C n Z ∂B r | Df | n − | f || f | na (cid:12)(cid:12)(cid:12)(cid:12) log q | f | a + ε − c (cid:12)(cid:12)(cid:12)(cid:12) + n (cid:12)(cid:12)(cid:12) log p a + ε − c (cid:12)(cid:12)(cid:12) Z {| f | . This dominant is in L ( B R ) for any C > , since | Df | n − / | f | n − ∈ L n/ ( n − ( B R ) , and since log | f | ∈ L n loc ( B R ) by Lemma 5.4. Consequently, thedominant is also in L ( ∂B r ) for a.e. r ∈ (0 , R ) by the Fubini-Tonelli theorem.Hence, we may apply the dominated convergence theorem as a → in (6.3),and therefore obtain (cid:12)(cid:12)(cid:12)(cid:12)Z B r J f | f | n − ( | f | + ε ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C n Z ∂B r | Df | n − | f | n − (cid:12)(cid:12)(cid:12)(cid:12) log q | f | + ε − c (cid:12)(cid:12)(cid:12)(cid:12) for a.e. r ∈ (0 , R ) .We then let ε → + and again use the dominated convergence theorem,obtaining the claimed inequality (6.2) for a.e. r ∈ (0 , R ) and our fixed value N THE HETEROGENEOUS DISTORTION INEQUALITY 23 of c . Consequently, if S ⊂ R is a countable dense subset, then (6.2) holdsfor a.e. r ∈ (0 , R ) and all c ∈ S . We then obtain (6.2) for all c ∈ R and a.e. r ∈ (0 , R ) by taking limits, since the constant in (6.2) is independent of c and since | Df | n − / | f | n − is integrable over ∂B r for a.e. r ∈ (0 , R ) .It remains to derive the statement of the lemma from (6.2). We denote osc(log | f | , ∂B r ) = sup ∂B r log | f | − inf ∂B r log | f | . Since |∇ log | f || ∈ L n ( B R ) , the Sobolev embedding theorem on spheres [14,Lemma 2.19] implies that, after changing f in a set of measure zero, we have(6.4) osc(log | f | , ∂B r ) ≤ C n r n k∇ log | f |k n ≤ C n r n (cid:18)Z ∂B r | Df | n | f | n (cid:19) n for a.e. r ∈ (0 , R ) . Moreover, if r ∈ (0 , R ) is such that (6.2) is valid, we mayselect b ∈ ∂B r and take c = log | f ( b ) | , in which case (6.2) yields(6.5) Z B r J f | f | n ≤ C n osc(log | f | , ∂B r ) Z ∂B r | Df | n − | f | n − . for a.e. r ∈ (0 , R ) .Now, combining (6.4), (6.5) and Hölder’s inequality, we obtain Z B r J f | f | n ≤ C n r n (cid:18)Z ∂B r | Df | n | f | n (cid:19) n Z ∂B r | Df | n − | f | n − ≤ C n r n (cid:18)Z ∂B r | Df | n | f | n (cid:19) n r n − n (cid:18)Z ∂B r | Df | n | f | n (cid:19) n − n = C n r Z ∂B r | Df | n | f | n . This concludes the proof of Lemma 6.1. (cid:3) Hölder continuity of the logarithmic function. Here we completethe proof of Theorem 1.2. We recall the statement first. Theorem 1.2. Suppose that f ∈ W ,n loc ( R n , R n ) satisfies the heterogeneousdistortion inequality (1.1) with K ∈ [1 , ∞ ) and σ ∈ L n ( R n ) ∩ L n + ε loc ( R n ) where ε > . If f is bounded and f , then / ∈ f ( R n ) . The proof is based on the following logarithmic counterpart of Lemma 3.1,where the use of the isoperimetric inequality is replaced with Lemma 6.1. Lemma 6.3. Suppose that f : R n → R n is in W ,n loc ( R n , R n ) and solves theheterogeneous distortion inequality (1.1) with K ∈ [1 , ∞ ) , and σ ∈ L n ( R n ) .If f is bounded and not the constant function f ≡ , then for every x ∈ R n and almost every ball B r = B n ( x, r ) ⊂ R n , we have Z B r | Df | n | f | n ≤ C n ( K ) r Z ∂B r | Df | n | f | n + Z B r σ n . Proof. By Lemmas (5.1) and Lemma 5.3, we have | Df | / | f | ∈ L n ( R n ) .Hence, the heterogeneous distortion inequality is in this case equivalent withthe inequality(6.6) | Df ( x ) | n | f ( x ) | n ≤ K J f ( x ) | f ( x ) | n + σ n ( x ) for a.e. x ∈ R n , since m n ( f − { } ) = 0 by Lemma 5.4. We then combine (6.6) with Lemma6.1, and the claim follows. (cid:3) Proof of Theorem 1.2. Let B R = B n (0 , R ) for some R > . By Lemmas 5.3and 5.4, we have that log | f | ∈ W ,n ( B R ) , and we moreover have |∇ log | f || ≤| Df | / | f | almost everywhere.Let then B r = B n ( x, r ) ⊂ B R . By Lemma 6.3 and Hölder’s inequality, wehave Z B r | Df | n | f | n ≤ C n ( K ) r Z ∂B r | Df | n | f | n + Z Q r σ n ≤ C n ( K ) r Z ∂B r | Df | n | f | n + (2 r ) nεn + ε (cid:18)Z B R σ n + ε (cid:19) nn + ε . Since this holds for every x ∈ B R and a.e. r ∈ (0 , dist( x, ∂ B )) , Lemma 3.2yields that Z B r |∇ log | f || n ≤ Z B r | Df | n | f | n ≤ Cr α , where C and α are independent of our choice of B r ⊂ B R . Hence, byLemma 3.3, we have that log | f | is Hölder continuous in B R/ . Therefore,the function log | f | locally Hölder continuous in R n . In particular, log | f | islocally bounded. However, if f ( x ) = 0 for some x ∈ R n , then log | f ( x ) | = −∞ . We conclude that f cannot have any zeroes. (cid:3) The Liouville theorem The remaining part of this paper is devoted to proving the Liouville the-orem formulated in Theorem 1.3.We recall from the introduction that our approach is to consider a function“ log f ” from R n to R × S n − . This mapping is well defined and satisfies asimilar distortion inequality as the classical complex logarithm map. Thedifferential inequality makes it possible to show that the weak derivativeof our “ log f ” lies in L n − ε ( R n ) for some ε > . The argument for thisgoes back to two remarkable papers by Iwaniec and Martin [18] (for evendimensions) and Iwaniec [16] (for all dimensions), where they proved localintegral estimates of quasiregular mappings below the natural exponent n .Later, a short proof was given by Faraco and Zhong [8]. We in turn performa global version of the Lipschitz truncation argument of Faraco and Zhongin our setting. N THE HETEROGENEOUS DISTORTION INEQUALITY 25 The logarithm with a manifold target. Let n ≥ . Then thereexists a smooth mapping s : R × S n − → R n \ { } , defined by s ( t, θ ) = e t θ for t ∈ R and θ ∈ S n − ⊂ R n . The map s is conformal, with | Ds ( t, θ ) | n = J s ( t, θ ) = e nt . The inverse of s is given by s − ( x ) = (cid:18) log | x | , x | x | (cid:19) for x ∈ R n \ { } . A simple calculation yields that J s ◦ s − ) J s − = e n log | x | J s − , and therefore (cid:12)(cid:12) Ds − ( x ) (cid:12)(cid:12) n = J s − ( x ) = 1 | x | n . We use the inverse s − to take a “logarithm” of our mapping f . Lemma 7.1. Suppose that f ∈ W ,n loc ( R n , R n ) satisfies the heterogeneousdistortion inequality (1.1) with K ∈ [1 , ∞ ) and σ ∈ L n ( R n ) ∩ L n + ε loc ( R n ) , forsome ε > . Suppose also that f is bounded, and that f is not the constantmapping f ≡ . Denote h = s − ◦ f ( x ) = (cid:18) log | f | , f | f | (cid:19) . Then h has the following properties:(1) h is continuous and h ∈ W ,n loc ( R n , R × S n − ) ;(2) we have | Dh | ∈ L n ( R n ) ;(3) we have | Dh ( x ) | n ≤ KJ h ( x ) + σ n ( x ) for a.e. x ∈ R n .Proof. By Theorem 1.1, we have that f is continuous, and by Theorem 1.2,we have that the image of f does not meet zero. Hence, h is well definedand continuous. We also easily see that h ∈ W ,n loc ( R n , R × S n − ) , since if B is a ball compactly contained in R n \ { } , then s | B is a smooth bilipschitzchart. Hence, we have (1).We then note that since s − is conformal, we have | Dh | = (cid:0)(cid:12)(cid:12) Ds − (cid:12)(cid:12) ◦ f (cid:1) | Df | = | Df || f | . Hence, we have by Lemma 5.3 that | Dh | ∈ L n ( R n ) , proving (2). Finally, weprove (3) by computing that | Dh | n = (cid:0)(cid:12)(cid:12) Ds − (cid:12)(cid:12) ◦ f (cid:1) | Df | ≤ ( J s − ◦ f )( KJ f + σ n | f | n )= K ( J s − ◦ f ) J f + σ n | f | n | f | n = KJ h + σ n . (cid:3) Integrability below the natural exponent. According to Lemma 7.1,the logarithmic mapping h : R n → R × S n − lies in W ,n loc ( R n , R × S n − ) andit solves the distortion inequality,(7.1) | Dh ( x ) | n ≤ KJ h ( x ) + σ n ( x ) for K ∈ [1 , ∞ ) and a.e. x ∈ R n . Since | Dh | ∈ L n ( R n ) , the integral of the Jacobian J h over R n vanishes.Therefore, the natural integral estimate for the logarithmic map over theentire space reads as follows Z R n | Dh | n ≤ Z R n σ n . The next lemma gives the key global integrability estimate for the differentialbelow the natural exponent n . Lemma 7.2. Suppose that a mapping h : R n → R × S n − is continuous, andthat h ∈ W ,n loc ( R n , R × S n − ) with | Dh | ∈ L n ( R n ) . If h satisfies the distortioninequality (7.1) with σ ∈ L n − ε ( R n ) ∩ L n ( R n ) for some ε > , then there exists ε ′ = ε ′ ( n, K, ε ) ∈ (0 , ε ) such that | Dh | ∈ L n − ε ′ ( R n ) . In particular, we havethe estimate Z R n | Dh | n − ε ′ ≤ C ( ε ′ ) Z R n σ n − ε ′ , where C ( ε ′ ) → as our choice of ε ′ tends to .Proof. We may assume ε < . We denote h = ( h R , h S n − ) , where h R : R n → R and h S n − : R n → S n − . Let g = | Dh | + σ, and for every λ > , let F λ = { x ∈ R n : M ( g )( x ) ≤ λ } . Suppose that x, y ∈ F λ . Then by a pointwise Sobolev estimate, we havefor every i ∈ { , . . . , n } that | h R ( x ) − h R ( y ) | ≤ C n | x − y | ( M ( |∇ h R | )( x ) + M ( |∇ h R | )( y )) ≤ C n | x − y | ( M ( g )( x ) + M ( g )( y )) ≤ C n λ | x − y | . Hence, h R is C n λ -Lipschitz in F λ . Consequently, by using the McShaneextension theorem [23], we find a C n λ -Lipschitz map h R ,λ : R n → R suchthat h R ,λ | F λ = h R | F λ . We denote h λ = ( h R ,λ , h S n − ) .We point out that we have | Dh λ | ≤ (1 + C n ) M ( g ) a.e. in R n . Indeed, we have | Dh λ | = | Dh | ≤ g ≤ M ( g ) a.e. in F λ , andsince |∇ h R ,λ | ≤ C n λ , we also have | Dh λ | ≤ | Dh | + C n λ ≤ (1 + C n ) M ( g ) a.e. in R n \ F λ . Since g ∈ L n ( R n ) , we also have M ( g ) ∈ L n ( R n ) , and N THE HETEROGENEOUS DISTORTION INEQUALITY 27 therefore | Dh λ | ∈ L n ( R n ) . Hence, we may apply the case of Lemma 2.4 witha manifold target, obtaining that Z R n J h λ = 0 . For r > , we denote B r = B n (0 , r ) . Since J h = J h λ in F λ , we maytherefore estimate that (cid:12)(cid:12)(cid:12)(cid:12)Z B r ∩ F λ J h (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R n \ ( B r ∩ F λ ) J h λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B r \ F λ J h λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R n \ B r J h λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 + C n ) n Z B r \ F λ λM ( g ) n − + Z R n \ B r M ( g ) n ! . Moreover, since | Dh | n ≤ KJ h + σ n , we have Z B r ∩ F λ | Dh | n ≤ K (cid:12)(cid:12)(cid:12)(cid:12)Z B r ∩ F λ J h (cid:12)(cid:12)(cid:12)(cid:12) + Z B r ∩ F λ σ n . We now chain these estimates together, and multiply by λ − − ε ′ , where ε ′ ∈ (0 , ε ) . We obtain(7.2) Z B r | Dh | n λ − − ε ′ χ F λ ≤ Z B r σ n λ − − ε ′ χ F λ + (1 + C n ) n K Z B r λ − ε ′ M ( g ) n − χ R n \ F λ + λ − − ε ′ Z R n \ B r M ( g ) n ! . Let t > . We now integrate (7.2) from t to ∞ with respect to λ , and usethe Fubini–Tonelli theorem to switch the order of integration. Observe that χ F λ ( x ) = 0 if λ < M ( g )( x ) , and χ F λ ( x ) = 1 otherwise. Hence, Z ∞ t λ − − ε ′ χ F λ ( x ) d λ = Z ∞ max( t,Mg ( x )) λ − − ε ′ d λ = [max( t, M g ( x ))] − ε ′ ε ′ ≤ [ M g ( x )] − ε ′ ε ′ for a.e. x ∈ R n . Moreover, we also have Z ∞ t λ − ε ′ χ R n \ F λ ( x ) d λ = Z M ( g )( x ) t λ − ε ′ if M g ( x ) > t if M g ( x ) ≤ t , and hence Z ∞ t λ − ε ′ χ R n \ F λ ( x ) d λ = χ R n \ F t ( x ) Z Mg ( x ) t λ − ε ′ d λ = χ R n \ F t ( x ) [ M g ( x )] − ε ′ − t − ε ′ − ε ′ ≤ χ R n \ F t ( x ) [ M g ( x )] − ε ′ − ε ′ for a.e. x ∈ R n . In conclusion, we obtain the estimate(7.3) Z B r | Dh | n [max( M ( g ) , t )] − ε ′ ≤ Z B r σ n M ( g ) − ε ′ + (1 + C n ) n K ε ′ − ε ′ Z B r \ F t M ( g ) n − ε ′ + t − ε ′ Z R n \ B r M ( g ) n ! . We then further estimate some of the terms in (7.3). On the left hand side,we observe that if x / ∈ F t , then M ( g )( x ) > t , and therefore max( M ( g )( x ) , t ) = M ( g )( x ) . Hence, we have Z B r | Dh | n [max( M ( g ) , t )] − ε ′ ≥ Z B r \ F t | Dh | n M ( g ) − ε ′ . For the first term on the right hand side, we see from the definition of g that σ ≤ g ≤ M ( g ) , and therefore obtain Z B r σ n M ( g ) − ε ′ ≤ Z B r σ n − ε ′ . For the remaining terms, we use the strong Hardy–Littlewood maximal in-equality, where the same constant M n can be used for all exponents in theinterval [ n − , n ] . Moreover, we also estimate the third term by g n − ε ′ ≤ n ( | Dh | n − ε ′ + σ n − ε ′ ) . After all these estimates of individual terms, we ob-tain a total estimate of the form(7.4) Z B r \ F t | Dh | n M ( g ) − ε ′ ≤ (cid:18) C n ) n Kε ′ − ε ′ (cid:19) Z B r σ n − ε ′ + (2 + 2 C n ) n Kε ′ − ε ′ Z B r \ F t | Dh | n − ε ′ + (2 + 2 C n ) n Kt − ε ′ Z R n \ B r g n . We then use Young’s inequality to obtain the estimate Z B r \ F t | Dh | n − ε ′ = Z B r \ F t (cid:16) | Dh | n − ε ′ M ( g ) − ε ′ ( n − ε ′ ) n (cid:17)(cid:16) M ( g ) ε ′ ( n − ε ′ ) n (cid:17) ≤ ( n − ε ′ ) n Z B r \ F t | Dh | n M ( g ) − ε ′ + ε ′ n Z B r \ F t M ( g ) n − ε ′ ≤ Z B r \ F t | Dh | n M ( g ) − ε ′ + ε ′ M n n Z B r \ F t (cid:16) | Dh | n − ε ′ + σ n − ε ′ (cid:17) . N THE HETEROGENEOUS DISTORTION INEQUALITY 29 Hence, combining this with (7.4), we now have(7.5) Z B r \ F t | Dh | n − ε ′ ≤ (1 + δ ) Z B r σ n − ε ′ + δ Z B r \ F t | Dh | n − ε ′ + (2 + 2 C n ) n Kt − ε ′ Z R n \ B r g n , where δ = 2 n ε ′ ((1 + C n ) n K/ (1 − ε ′ ) + M n ) . We then select ε ′ small enoughthat δ < . Since B r \ F t is of finite measure, | Dh | n − ε ′ is integrable overit, and we may absorb its term from the right hand side of (7.5) to the lefthand side. We obtain the estimate Z B r \ F t | Dh | n − ε ′ ≤ δ − δ Z B r σ n − ε ′ + (2 nC n ) n Kt − ε ′ − δ Z R n \ B r g n . We let r → ∞ . Since g ∈ L n ( R n ) , this makes the final term vanish, yielding Z R n \ F t | Dh | n − ε ′ ≤ δ − δ Z R n σ n − ε ′ < ∞ . Note that we may assume that S t> R n \ F t = R n . Indeed, otherwise M ( g ) has a zero; this is possible only if g ≡ , in which case h is constant and theclaim is trivial. Hence, by letting t → + , the claim follows. (cid:3) Proof of the Liouville theorem. It remains to complete the proofof Theorem 1.3. We recap the statement before the proof. Theorem 1.3. Suppose that f ∈ W ,n loc ( R n , R n ) satisfies the heterogeneousdistortion inequality (1.1) with K ∈ [1 , ∞ ) and σ ∈ L n − ε ( R n ) ∩ L n + ε ( R n ) ,for some ε > . If f is bounded and lim x →∞ | f ( x ) | = 0 , then f ≡ .Proof. Suppose towards to a contradiction that f is bounded and lim x →∞ | f ( x ) | =0 , but f is not identically zero. By Theorems 1.1 and 1.2, we have that f is continuous and has no zeros. Hence, we may define the “logarithmic”mapping h : R n → R × S n − by h ( x ) = (cid:18) log | f | , | f || f | (cid:19) . By Lemma 7.1, we have that h ∈ W ,n loc ( R n , R × S n − ) , | Dh | ∈ L n ( R n ) ,and | Dh | n ≤ KJ h + σ n . Combining this with Lemma 7.2 we conclude that | Dh | ∈ L n − ε ′ ( R n ) for some ε ′ > . In particular, since |∇ log | f || ≤ | Dh | , wehave(7.6) Z R n |∇ log | f || n − ε ′ ≤ Z R n | Dh | n − ε ′ ≤ C ( ε ′ ) Z R n σ n − ε ′ . Consider now balls of the form B i = B n (0 , i ) . Our goal is to show thatthe integral average of | log | f || over B i , denoted by (log | f | ) B i , is bounded independently of i ∈ N ∪ { } . By the Sobolev-Poincaré inequality [7, 4.5.2]and (7.6) we have (cid:12)(cid:12) (log | f | ) B i − − (log | f | ) B i (cid:12)(cid:12) ≤ n m n ( B i ) Z B i | log | f | − (log | f | ) B i |≤ C n n i (cid:18) m n ( B i ) Z B i |∇ log | f || n − ε ′ (cid:19) n − ε ′ ≤ C n n i − nin − ε ′ ω − n − ε ′ n (cid:18)Z R n |∇ log | f || n − ε ′ (cid:19) n − ε ′ ≤ C n ′ C ( ε ′ )2 n max(1 , ω − nn )2 − ε ′ n − ε ′ i (cid:18)Z R n σ n − ε ′ (cid:19) n − ε ′ . Consequently, we have that | (log | f | ) B i − (log | f | ) B |≤ C n C ( ε ′ )2 n max(1 , ω − nn ) (cid:18)Z R n σ n − ε ′ (cid:19) n − ε ′ ∞ X i =0 − ε ′ n − ε ′ i < ∞ . Since log | f | ∈ L by Lemma 5.4, we have | (log | f | ) B | < ∞ . However,since lim x →∞ f ( x ) = 0 , we have lim i →∞ (log | f | ) B i = −∞ . 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