quasiregularity in metric spaces

QUASIREGULARITY IN METRIC SPACES *> M IH AI CR ISTEA

We show that if / : X —►Y is a continuous, open and discrete map of finite mul­ tiplicity N ( f ) between two p-regular metric spaces, then / satisfies the modular inequality M p( r ) < K N ( f ) M p( f ( V ) ) for every path family F from X if and only if H ( x , } ) < I I for every x 6 X . A M S 2000 Subject Classification: 30C65. Key words: quasiregular map, metric space.

1. I N T R O D U C T I O N I f X , Y axe metric spaces and / : X —■» Y is continuous, open, discrete, x 6 X and r > 0, we let L{x,f,r) —

sup

d{f(y),f(x)),l(x,f,r)=

yeS(x,r)

inf

d(f(y),f(x))

y es(x,r)

and define the linear dilatation of / at x as H (x, f ) — limsup r —+0

For a > 1 wc also put ha(x, /) — lim inf sup r< t< a r

and if A C X we let N ( f , A) = sup Card

h(x, f ) = hi ( x, /), ’ *

D A and N ( f ) = N ( f , X ) .

yeY

If / : X —> Y is a map, we say that / is open if / carries open sets into open sets, and we say that / is discrete if f ~ l (y) is an isolated set for every y e YI f D C R " is open, n > 2, a map f : D —* R n is quasiregular if / € W ^ n(£>,Rn) n C{ D, R n) and ||/'(x)Hn < K ■ Jf {x) a.e. for some K > 1, and this is known as the analytic definition of quasiregularity. If / is continuous, open, discrete, with N ( f ) < oo, then / is quasiregular if and only if there exists H > 1 such that H ( x , f ) < H for every x € D (see [12], Theorems 4.5 and 4.13). Then, at least for mappings of finite multiplicity, we can say that a continuous, open and discrete map / : X —> Y between two metric spaces is quasiregular (the metric definition) if there exists H > 1 such that H ( x , f ) < H for every x € X . Homeomorphisms between metric Paper presented at the 17th R. Nevanlinna Colloquium, Jyvaskyla, 2003. REV. R O U M A IN E M ATH . PURES A P P L ., 51 (2006), 3, 291-310

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Mihai Cristea

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spaces are called quasiconformal if there exists H > 1 such that H( x, f ) < H for every x G X (the metric definition of quasiconformality). Such maps have been recently considered in [1], [7], [9], [10], [16]. If is a metric measure space and F is a family of nonconstant paths in X , we let F ( F ) = {p : X —* [0, oo] a Borel map |/ pds > 1 for every locally rectifiable 7 G T }, and if p > 0 we define the p-modulus of F by

M p(r ) =

inf

p eF ( r ) JX

^(zld/x.

I f D, D ' are domains in R ra, K > 1 and / : D —* D' is a homeomorphism, we say that / is K -quasiconformal if ■M n( T) < M n(/ ( r )) < K M n( T) for every path family T in D (the geometric definition of quasiconformality). We also say that / is If-quasiconformal, under the analytic definition of quasi­ conformality, if / € R n) and ||//(^)||n' < K J f ( x ) a.e. It is known that for domains D, D ' in R n and a homeomorphism / : D —►D\ these three definitions of quasiconformality are equivalent (see [17], Theorems 34.1 and 34.6). It is shown in [11] that these definitions of quasiconformality are also equivalent on arbitrary metric spaces satisfying a few conditions of regularity. I f D G R " is open, ti ^ 2, f \D * is continuous, open, discrete with N ( f ) < 00, then / is quasiregular with K o ( f ) < K if and only if / satisfies the so called K q{ } ) inequality, i.e., M n( T) < K N ( f , D ) • M n( f ( T ) ) for every path family T in D (see [14], Theorem 6.7, page 44). Here K q( J) is the smallest K > 1 such that \\f'(x)\\n < K J f ( x ) a.e. We shall say that a continuous, open and discrete map f : X —* Y between two metric measure spaces with N { f ) < 00 is A'-quasiregular (under the geometric definition) if there exists p > 0 such that M P( F) < K N ( f ) ■M p( f ( T ) ) for every path family T in D. In [8] are considered quasiregular maps / : U —> G under the analytic definition of quasiregularity, where G is a Carnot group and U C G is open. Their methods seem to be enough difficult to be transported on arbitrary me­ tric spaces, where the metric and the geometric definitions of quasiregularity are very natural, at least for mappings of finite multiplicity. However, a theory of quasiregular maps on arbitrary metric spaces is necessary, since there are plenty of such mappings defined on sets which are not Riemannian manifolds. Indeed, if D, D' are domains in R n, / : D —» D' is quasiregular and surjective with N ( f , D ) < 00, H( x, f ) < H for every x G D, A c D, B C D ' are such that A = then f\A : A —►B is open, discrete, with N( f \A, A) < 00, H( x , f\A) < H( x , f ) < H for every x G A and the sets A and B can be taken complicated enough. Probably, this is the easiest way to produce quasiregular maps defined on some sets which axe not manifolds. In [1] are constructed examples of quasiconformal mappings in metric spaces by taking sets A C B C R n, spaces X = B LU B and mappings / : X —» X, f = Id x • We can use this technique of gluing spaces and mappings

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to obtain some other examples of quasiregular mappings. If X, Y are metric spaces, A is a closed subset of X and Y, we let X U ^ Y the disjoint union of X and Y, with points in the two copies of A identified. Then X LU Y is a metric space, where d(x, y) is the distance from X if x, y € X, d(x, y) is the distance from Y if x, y € Y and d(x, y) = inf d(x, a) + d{a, y) if x, y are from aeA

two different parts of the union X LU Y. Let D , D ' be domains in R n, A closed in D and in D ' , f : D —> R ” , g : D ’ —> R n quasiregular and nonconstant such that f\A = g\A, f ( A ) = g( A) is a closed subset of f ( D ) and g( D' ) , A = / _ 1(/ (A )) = g~1( g( A) ) and let X = D LU D Y = f ( D ) Uf(A) 9{D' ) . Then D and D ' are the two parts of X, f ( D ) and f ( D ' ) are the two parts of Y and we define F : X —> Y by F\D — /, F\D' — g. (We can take for instance D — D ' — R 2,A = {(a;,?/) € R 2|y = 0 } , f ( z ) = g(z) = z2 for 2 e R 2.) We can easily see that F is discrete and we show that F is open. Let x € D\A. Then, for small r, B x ( x , r ) = B p ( x , r ) , where B x { x , r ) is the ball of center x and radius r from X and By( x , r ) is the ball of center x and radius r from D. Let S > 0 be small enough such that B y ( f ( x ) , S ) = B f ^ ( f ( x ) , 6 ) and B f ^ { f { x ) , 8 ) C f ( B D ( x , r ) ) . T h en F ( B x ( x , r ) ) = f ( B D ( x , r ) ) D B f {D)( f { x ) , 5 ) = B Y ( f ( x) 6) , hence F is open at x. and in the same way we show that F is open at x if x 6 D' \A. If x € A, then B x ( x , r) = B D ( x , r ) U A B £ >>(x,r) for r > 0 and if 8 > 0 is taken small enough such that B f ^ ( f { x ) , 5 ) C f ( B o ( x , r ) ) , B g^ ^ { g { x ) , 8 ) C g ( B Dr( x, r) ) , then F ( B x ( x , r ) ) = F ( B D ( x , r ) ) U AB D>{x,r) = f ( B D( x , r ))U /(A) g { B D’ ( x , r ) ) D B f ( D) ( f ( x ) , 8 ) Uf(A) Bg(D,}( g( x) , 8) = B v ( F ( x ) , 8 ) , hence F is also open at x. We proved that F is an open map on X , and we see that F is continuous on X , that N ( F ) < max { N (/), N ( g ) } and that H ( x , F ) < max { H( x , /), H ( x , g ) } for every x G X . It follows that if N ( f ) < 00, N ( g ) < 00, then there exist H\, H 2 > 1 such that H ( x , f ) < H i for x G D and H ( x , g ) < M2 for x € D 1, hence H ( x , F ) < max { H i , } for every x e X and N ( F ) < 00, hence F is quasiregular under the metric definition. We see that F is not a local homeomorphism if / and g are not local homeomorphisms, that X is not manifold and it is a Loewner space (see [10], 6.14, page 42), and this construction is also valid for our main results, Theorems 1 and 2. This procedure allows us to construct a lot of quasiregular maps on rather general metric spaces, using in a canonical way two arbitrary quasiregular maps defined on some open subsets from R n. In this manner we can also produce a lot of open, discrete mappings with uniformly bounded linear dilatation without finite multiplicity, so our paper may be a starting point for some further researches on this kind of mappings. A metric space X endowed with a Borel measure fi is called an Ahlfors Q regular space if there exists a constant C > 1 such that C _ 1r® < n ( B r ) < Cr® for every ball B r of radius r from X .

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Mihai Cristea

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If (X , n ) is a metric measure space and E , F , G are subsets of X , we let A ( E , F , G ) — {7 : [a, 6] —» G |7 is a path and 7 (a) € E,-y(b) e F } and if G = X , we put A ( E , F ) = A ( E , F , X ) . We say that (X,/x) is a Loewner space if it is a connected p-regular space with p > 1 and there exists a function $ : (0 ,00) —►(0 ,00) such that $ (t) < M P( A ( E , F ) ) for every nondegenerate continua E and F in X with min^ (E )4 (F)} - L A metric space X is called linearly locally connected of constant c > 1 (c-LLC) if there exists c > 1 such that any two points in B( x , r) can be joined by a path in B( x, cr), and any two points in X \ B( x, r ) can be joined by a path in X \ B( x , £), for every ball B ( x , r ) in X. We shall prove in our main results (Theorems 1 and 2) that the metric and geometric definitions of quasiregularity are equivalent for continuous, open and discrete maps of finite multiplicity between regular spaces. T h e o r e m 1. Let X , Y be locally compact metric spaces, c > 1, Y a

c-LLC space, // a Borel measure on X , v a Borel measure on Y such that there exist constants Co, C\ and p > 1 such that Cq 1 rp < fx(Br ) < Corp and C f Jrp < v(B'r ) < C\rp f or every ball B r of radius r in X and every ball B'r of radius r in Y . Let f : X —* Y be continuous, open, discrete such that N ( f ) < 00, and assume there exists H > 1 such that H ( x , f ) < H f or every x £ X . Then there exists a constant K depending on Co, C\, c,p, H such that M p( T) < K N ( f ) M p( f ( T ) ) f or every path family F in X . THEOREM 2. Let X be a locally compact p-Loewner space, Y a c-LLC p-regular space such that there exists C\ fo r which fx(B(y, r )) < C\rp f or every ball B ( y , r ) from Y , and let f : X —» Y be continuous, open and discrete. Suppose that D CL X is open, N ( f , D ) < 00 and there exists K > 1 such that M p( r ) < K N ( f , D ) M p ( f ( T ) ) f or every path family T in D. Then there exists a constant H = H ( K , p , N ( f , D ), C i, c) such that H( x , f ) < H f or every x E D. A known theorem of Renggli [13] and Caraman [2] says that if n > 2, D , D ' are domains in R n and / : D —►D ' is a homeomorphism, then / is quasiconformal if and only if / carries path families T in D of infinite modulus into path families from D ' of infinite modulus. A generalization of this result for open discrete maps is given in [4], [5]. We give the following version of the theorem of Renggli and Caraman for open, discrete maps on metric spaces. T h e o r e m 3. Let X be a locally compact p-Loewner space, Y a c-LLC p-regular space, and let f : X —> Y be continuous, open and discrete and suppose that there exists 8 > 0 such that M p( f ( F ) ) > 5 f or every path family r in D with M P( T) = 00. Then there exists H > 1 such that H( x, f ) < H for every x € X .

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Quasiregularity in metric spaces

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As a corollary we obtain that an open, discrete map / : X —> Y between metric spaces which satisfies the modular inequality M p( r ) < K M p( f ( T ) ) for every path family T in X is such that the linear dilatation H( x, /) is uniformly bounded on X. We cannot estimate the constant H > 1 such that H( x , f ) < H for every x € X in terms of K and the spatial constants of X and Y. T h e o r e m 4. Let X be a locally compact p-Loewner space, Y a c-LLC p-regular space and let. f : X —* Y be continuous, open and discrete such that there exists K > 1 fo r which M P( T) < K M p( f { T ) ) ) f or every path family T in X . Then there exists H > 1 such that H( x , /) < H f or every x 6 X .

For open, discrete mappings / : X —* Y between metric spaces satisfying the modular inequality M P(T ) < K M p( f ( T )) for every path family T in X , we have the following estimate of the multiplicity function N ( f ) . T heorem 5. Let X be a locally compact p-Loewner space, Y a c-LLC p-regular space such that there exists C\ > 0 such that p( B( y, r ) ) < C\rp f or every ball B ( y , r ) from Y , and let f : X —►Y be continuous, open and discrete and suppose that there exists K > 1 such that M P( T) < K M p( f ( T ) ) for every path family F in X . Then, fo r every A > 1, there exists at most

n = [ ^ iT (S g $ - i 1 Pomts X\,...,xn from X such that f ( x k) = f ( x i) and h(xk, f ) >