On the radial limits of mappings on Riemannian manifolds

We study the geometric properties of the mappings for which a generalized inverse Poletsky modular inequality holds. Our approach is on Riemannian manifolds and we generalize some known theorems from the theory of analytic mappings concerning radial limits, such as the theorems of Fatou and M. and F. Riesz and their extensions given for quasiregular mappings by Martio and Rickman.


Introduction
Throughout this paper X , Y will be connected Riemannian n-manifolds and D ⊆ X an open subset of X . We set by d the geodesic distance on X and Y and if A ⊂ X , we set by d(A) the diameter of A. If x ∈ X , we say that B(x, 1) is a -neighbourhood of x if : B n → B(x, 1) is a C 1 diffeomorphism such that (0) = x and (S(0, t)) = S(x, t) for every 0 < t ≤ 1. Here B n is the unit ball from R n endowed with the Euclidean metric and B(x, 1) is the ball from X endowed with the geodesic distance. We know from [38] that every point x ∈ X has locally a -neighbourhood. A Riemannian metric on a smooth manifold X of dimension n is a positive definite smooth symmetric tensor field of type (0, 2) g i j . The n-dimensional volume of a domain A in X is given by μ(A) = A det g i j dx 1 . . . dx n and X endowed with the measure μ is a metric measure space. We denote by ν the corresponding measure on Y . By an area element of a smooth hypersurface S in X we mean the expression of the

ω(x)ρ p (x)dμ i f ∈ A(D).
If F( ) = φ we set M p ω ( ) = 0. If ω = 1 we put One of the tools in the study of quasiregular mappings is the modular inequality of Poletsky M n ( f ( )) ≤ K M n ( ) f or ever y ∈ A(D) valid for K -quasiregular mappings (see the monographs [53,54,69,70] for more information about such mappings). Let n ≥ 2, D ⊂ R n be open and p ≥ 1. We denote by W 1, p loc (D, R n ) the Sobolev space of all mappings f : D → R n which are locally in L p together with their first order distributional derivatives. Such mappings have a.e. first order partial derivatives and f (x) is the matrix of the first order partial derivatives of f at x and J f (x) = det f (x) a.e. If f : E → F is a mapping, we set I m f = f (E).
A mapping f ∈ W 1,n loc (D, R n )∩C(D, R n ) is quasiregular if there exists K ≥ 1 such that | f (x)| n ≤ K J f (x) a.e. and the smallest such K ≥ 1 is the outer dilatation K 0 ( f ) of f . If f is quasiregular, the smallest K ≥ 1 such that J f (x) ≤ K l( f (x)) n a.e. is the inner dilatation K I ( f ) of f and K ( f ) = max{K I ( f ), K 0 ( f )} is the maximal dilatation of f . Here, if A ∈ L(R n , R n ), we set l(A) = inf |x|=1 |A(x)| and |A| = sup |x|=1 |A(x)| and if x = (x 1 , . . . , x n ) ∈ R n , we set |x| = ( n i=1 x 2 i ) 1 2 . A mapping f : D → R n is called K -quasiregular if K ( f ) ≤ K . Quasiregular mappings form a class with geometric properties close to those of analytic mappings.
Several generalizations of quasiregular mappings were developed in the last 35 years and the most important is the class of mappings of finite distortion. Let n ≥ 2 and D ⊂ R n be open. A mapping f ∈ W 1,1 loc (D) and there exists K : D → [0, ∞] measurable and finite a.e. such that a.e. and we set the outer dilatation We recommend the reader the books [28,32] for more information about such mappings and for some particular classes of mappings of finite distortion a Poletsky modular inequality holds (see [9,37]).
In the last 20 years the classes of mappings satisfying some weight modular inequalities were intensively studied. For some weights ω and some p, q > 1, such mappings are defined by the modular inequality This approach and the systematic use of the weight modulus in mapping theory was proposed by Martio and the resulting monograph of Martio et all [47] summarized the work for mappings between open sets in R n (see also [4, 9, 10, 12, 13, 15-21, 33-35, 45-47, 55, 56, 60-62]). The homeomorphisms and open, discrete mappings satisfying a modular inequality of type (1.1) and defined on general metric measure spaces, other than R n with the Euclidean metric are studied in [2,3,5,14,15,29,30,57,58,63,64]. It must be mentioned that in such classes of mappings we can give analoguous of Liouville, Montel, Picard type theorems, boundary extension and equicontinuity results and estimates of the modulus of continuity.
As in the classical Euclidean case, even on very general metric measure spaces X and Y , a continuous, open and discrete mapping f : D → Y is quasiregular if and only if it is geometrically quasiregular, i.e. M n ( ) ≤ K Y N (y, f , G)ρ n (y)dν for every open set G D, every ∈ A(G) and every ρ ∈ F( f ( )) (see [8] or [23]). Inspired by this property of quasiregular mappings defined on general metric measure spaces, in the last 10 years several authors studied a class of mappings satisfying a generalized inverse Poletsky modular inequality of type for every G D. Some modulus of continuity, equicontinuity results and boundary extension theorems are established for such mappings in [11] and [65][66][67][68].
If X , Y are Riemannian n-manifolds and f : D → Y is continuous, open and discrete, we say that f is analytically K -quasiregular if f ∈ N 1,n loc (D, Y ) and Here |∇ f | is the minimal n-weak upper gradient of a Sobolev mapping f ∈ N 1,n loc (X , Y ). We see from Theorem 6.25, page 56 in [23] that f is analytically K -quasiregular if and only if is geometrically quasiregular.
It must be mentioned that the class of mappings satisfying relation (1.2) for some arbitrary weight ω is strictly larger than the class of quasiregular mappings, where relation 1.2 holds for a particular weight ω G (y) = N (y, f , G) for every G D and every y ∈ Y .
We also note that if a Poletsky modular inequality is known for some general classes of mappings of finite distortion (see [9] or [37]), an inverse Poletsky modular inequality with a very general weight is not yet known in the class of mappings of finite distortion. Let  and open, we see that g is strictly increasing, that g (x) = 0 a.e. in E, g (x) = 1 a.e. in E, μ 1 (g( E)) = μ 1 ( E) and since f (g(x)) = x for every x ∈ [0, 1], we see that f (y) = 1 a.e. in [0, μ 1 ( E)]. We see that g is absolutely continuous and since , G is the inverse of F and satisfies condition (N ) and F (y) = 1 a.e. and then . We can easy modify this example to produce an open, discrete mapping F with B F = φ such that F is not a mapping of finite distortion and satisfies an inverse Poletsky modular inequality. We continuous and is absolutely continuous on almost every line segment parallel to the coordinate axes.

We say that f is open if f carries open sets into open sets and we say that
In this paper we study the existence H n−1 a.e. of the radial limits of some continuous, open discrete mappings f : We have in mind two important theorems concerning bounded analytical mappings f : B 2 → C, the theorem of Fatou, respectively the theorem of F. and M. Riesz. The theorem of Fatou states that such mappings have a.e. radial limits and the theorem of F. and M. Riesz states that if w ∈ C and Let us see the generalizations of these theorems in the class of quasiregular mappings. First, we remark that it is not yet known whether a bounded quasiregular mapping f : B 3 → R 3 has at least a radial limit. Rajala showed in [52] that if in addition f is a local homoemorphism, then f has at least a radial limit. Miklykov proved in [48] that if f : B n → R n is a bounded quasiregular mappings and B n | f ' (x)| n dx < ∞, then f has non-tangential limits with the possible exception of a set of n-capacity zero.
We remind that a mapping f : B n → R n has a non-tangential limit at a point x ∈ ∂ B n if for every finite cone with vertex at x such that there exists a finite cone Using the modulus method, Manfredi and Villamor showed in [40] has non-tangential limits with the possible exception of a set of p-capacity zero (see [41] and [49] for related results). We remind that if D ⊂ R n is a domain, a continuous then it results that f has radial limits. Mizuta [49] extended this result in dimension n ≥ 3 for mappings defined on the upper half space R n + and satisfying the condition In the next theorem we partially extend these results on Riemannian n-manifolds and for open, discrete mappings which may not belong to the Sobolev space .
Our result may be applied to the class of bi-conformal energy. If X , Y are bounded domains in R n , a homeomorphism f : X → Y is of bi-conformal energy if f ∈ W 1,n loc (X , Y ) and f −1 ∈ W 1,n loc (Y , X ) and the energy E X , Such mappings form an important class with connections to mathematical models of Nonlinear Elasticity (see [31]). It results that if X = B n and f is of bi-conformal energy, then f has a.e. radial limits. In fact, if f : B n → Y is a homeomorphism such that f is a.e. differentiable and there exist q > 1 and 0 < α < q − 1 such that , then f −1 satisfies condition (N ) and hence f has a.e. radial limits.
Let us consider the following class of mappings of finite distortion: Such a mapping f is continuous, open and discrete, is a.e. differentiable, μ n (B f ) = 0, μ n ( f (B f )) = 0 and f satisfies condition (N −1 ) (see [28]). Let G D. Then In fact, using Hölder's inequality, we can replace n by some q ∈ (n − 1, n). We denote by μ n the Lebesgue measure in R n .
We established a first general inverse Polestky modular inequality for a large class of mappings of finite distortion as is the class satisfying condition (1.3). It results that if f : B n → R n is a mapping of finite distortion satisfying relation (1.3) and such that B n | f (x)| n dμ n < ∞, then f has a.e. radial limits and this is a direct generalization of Miklyukov's result from [48] established for bounded quasiregular mappings. Some other properties of the class of mappings of finite distortion satisfying relation (1.3) simply result due to the inverse Poletsky modular inequality (1.4) and we will study them in some future paper.
In Theorem 5.1 in [44] it is proved that if 0 < b < n − 1, f : B n → B n is a locally injective quasiregular mapping such that there exists a constant C > 0 such that where E( f ) = {y ∈ S(0, 1)| f has no radial limits at y}. We see from [39] that if f : B n → R n is a bounded quasiregular mapping such that there exist C > 0 and 0 < b < n − 1 such that N ( f , B(0, r ) where T ( f ) = {y ∈ S(0, 1)| the non-tangential limits does not exists}. Using the modulus method, Martio and Rickman [43] generalized the theorems of Fatou and M. and F. Riesz. They showed that if f : B n → R n is quasiregular and there exist C > 0 and 0 < b < n − 1 such that R n N (y, f , B(0, r ))dy ≤ C(1 − r ) −b for every 0 < r < 1, then f has a.e. radial limits and if w ∈ R n and E w = {y ∈ S(0, 1)| lim t→1 f (t y) = w}, then H n−1 (E w ) = 0. Akkinen showed in [1] that if f : B n → R n is a mapping of finite distortion with exponentially integrable outer distortion and there exist C > 0 and 0 < b < n − 1 such that B n J f (x)dx ≤ C(1 − r ) −b for every 0 < r < 1, then it results that f has a.e. radial limits.
Our Theorem 2 extends the results from [43]. Indeed,if f is quasiregular, condition b) from Theorem 2 holds for a = n and then condition c) from Theorem 2 is "bq < n(n − 1)" which obviously holds.
The main tool in the proof of Theorem 1 and Theorem 2 is the use of the weighted modulus and the mapping f : B(x, 1) → Y from these theorems satisfy a generalized inverse Poletsky modular inequality of type 1.2. We improve the classical modulus estimates and the arguments from Theorem 1 in [40] and Theorem 5.15 and Theorem 5.17 in [43]. We obtain in this way extensions of these theorems on Riemannian n-manifolds in a larger class of mappings. We also point out that our results are valid for quasiregular mappings between Riemannian n-manifolds, since in this case μ(B f ) = 0, ν( f (B f )) = 0 and f satisfies condition (N −1 ) (see [22] and Lemma 6.39 in [23]).
For a large period of time, the modulus method, initiated by Beurling and Ahlfors, was an indispensable tool in the study of quasiconformal and quasiregular mappings. The present paper shows that the use of the weighted modulus and of some generalized inverse Poletsky modular inequalities is a good method for studying the geometric behaviour of more general classes of mappings than the class of quasiregular mappings.  {a 1 , . . . , a m }, there exists V = B(y, ρ) and open sets U i such that f |U i : U i → V is a homeomorphism for every i = 1, . . . , m and

Lemma 1 Let n ≥ 2, q > 1, D ⊂ X be open, suppose that D with the induced metric is separable and let f : D → Y be continuous, open and discrete such that
j=1 U i j . We denote by g i j : V i → U i j the inverse of f |U i j : U i j → V i for i ∈ N, j = 1, . . . , j(i) and we can also suppose that J g i j (y) < ∞ for every y ∈ V i \G k , every i ∈ N and every j = 1, . . . , j(i).
, ω(y) = 0 otherwise and the definition is correct and ω k ω ν a.e. Since f satisfies condition (N −1 ), we have Letting k → ∞ and using the monotone convergence theorem and the fact that We easy see that also f (Q) ω(y)dν ≤ Q L q (z, f )dμ for every open set Q ⊂ D. Let ∈ A(D) and = {γ ∈ |γ is rectifiable and f • γ 0 is absolutely continuous} and using Fuglede's theorem from [14], Theorem 2.1, we see that M q ( ) = M q ( ). Let η ∈ F( f ( )) and let ρ : Letting k → ∞ and using the monotone convergence theorem and the fact that μ(B f ) = 0, we find that D ρ q (z)dμ ≤ Y η q (y)ω(y)dν and since η ∈ F( f ( )) was arbitrarily chosen, we find that Proof We take ∈ A(D) and = {γ ∈ | f • γ 0 is absolutely continuous}. We see from Theorem 2.1 in [14] that M q ( ) = M q ( ) and if η ∈ F( f ( )) is fixed and Since η ∈ F( f ( )) was arbitrarily chosen, the proof is finished.
Integrating on A we have  ( ( (t, w)))|J (t, w)|J ( (t, w))dtdw ≤ (using polar coor dinates in R n ) Letting C = 2 n−1 B q M m and since ρ ∈ F( ) was arbitrarily chosen, we proved that

Proof of the results
Proof of Theorem 1 We show first that H n−1 (F) = 0. We can suppose that b > 0. Let 0 < δ < 1, let λ > 1 be such that b = q−1 λ and let q q−1 < m < λq q−1 . Then mb < q and let t k = 1 − δ km and Q k = B(x, t k+1 ) for every k ∈ N. We see from the proof of Lemma 1 that there exists ω : for every k ∈ N. We set γ y,k = γ y |[t k , t k+1 ] for every y ∈ S(x, 1) and every k ∈ N.
We see from Lemma 2 that there exists ω k : f (Q k ) → [0, ∞], ω k (y) = N (y, f , Q k ) for every y ∈ f (Q k ) such that M q ( k ) ≤ for every k ∈ N.
< H n−1 (E w ). It results that we can find at least a point y ∈ E w \ ∞ k=0 F k and since t k 1 and f • γ y,k (t k+1 ) / ∈ B(w, β 2 ) for every k ∈ N, this contradicts the fact that lim t→1 f (γ y (t)) = w.
We therefore proved that H n−1 (E w ) = 0.