Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 389–404

DEFINITIONS OF QUASIREGULARITY

M. Cristea

University of Bucharest, Faculty of Mathematics, Str. Academiei 14, R-70109 Bucharest, Romania.

Abstract. We show that if D ⊂ R^{n} is open, f: D → R^{n} is continuous, open and discrete such that for some h > 0 the linear dilatation satisfies h(x, f) < h for every x ∈ D, then f is K(h, n)-quasiregular. Here “lim inf” is used in the definition for h(x, f). A removability result on quasiregular mappings is obtained, and we enlarge the notion of the discrete modulus introduced by J. Heinonen and P. Koskela in [7]. We also give some bounds for the discrete modulus.

1. Introduction

In a recent paper [7], J. Heinonen and P. Koskela showed that if f: R^{n} → R^{n} is a homeomorphism such that for some h > 0, h(x, f) < h holds for every x ∈ R^{n}, then f is K(h, n)-quasiconformal. Here

L(x, f, r) = sup_{|y−x|=r} |f(y) − f(x)|,
l(x, f, r) = inf_{|y−x|=r} |f(y) − f(x)|,
h(x, f) = lim inf_{r→0} \frac{L(x, f, r)}{l(x, f, r)},
H(x, f) = lim sup_{r→0} \frac{L(x, f, r)}{l(x, f, r)}.

This generalizes the classical result which says that f is quasiconformal if H(x, f) is uniformly bounded. In [3] we introduced the inferior linear dilatation h(x, f) for mappings f: D → R^{n}, D ⊂ R^{n} open, f continuous, open and discrete, and we studied some properties of such mappings. We show now that if D ⊂ R^{n} is open, n ≥ 2, f: D → R^{n} is continuous, open and discrete such that there exists h > 0 such that h(x, f) < h for every x ∈ D, then f is K(h, n)-quasiregular and H(x, f) = h(x, f) a.e. in D. In this way, the problem of finding the best constant of quasiregularity K(h, n) is reduced to the classical case when H(x, f) < H for every x ∈ D.

A removability result of J. Heinonen and P. Koskela [7] shows that if n ≥ 2, f: R^{n} → R^{n} is a homeomorphism such that there exists a closed set H ⊂ R^{n} such that f is K -quasiconformal on R^{n} \H and a > 1 such that for every x ∈ H there exists a sequence r_{j} → 0 depending on x such that (B(x, ar_{j})\\overline{B} (x, r_{j})) ∩ H = ∅ for every j ∈ N, then it follows that f is K -quasiconformal. We shall show that if D ⊂ R^{n} is open, n ≥ 2, f: D → R^{n} is continuous such that there exists H ⊂ D closed in D such that f is K -quasiregular on D \ H and there exists 0 < a < 1 and ℬ = (B_{i})_{i∈N} an a-porous base of H, then, if f is open and discrete or if int f(H) = ∅, it follows that f is K -quasiregular on D. Here, if D ⊂ R^{n} is open, n ≥ 2, A ⊂ D, 0 < a < 1, and ℬ = (B_{i})_{i∈N} is a covering of A, we say that ℬ is an a-porous covering of A if (B_{i} \ a\overline{B}_{i}) ∩ A = ∅ for every i ∈ N and we say that ℬ is an a-porous base of A if ℬ is an a-porous covering of A and for every x ∈ A and U ∈ V (x), there exists i ∈ N such that x ∈ B_{i} ⊂ U.

1991 Mathematics Subject Classification: Primary 30C; Secondary 30C65.

We showed in [5] that if D ⊂ R^{n} is open, n ≥ 2, f: D → R^{n} is continuous, open and discrete such that there exists 0 < a ≤ 1 and H > 0 with lim sup_{r→0} d(f(B(x, ar)))^{n}/μ_{n}(f(B(x, r))) ≤ H for every x ∈ D, then f is K(H, n, a)-quasiregular. We shall partially generalize this result, giving at the same time a generalization of the removability result of J. Heinonen and P. Koskela from [7]. We show that if D ⊂ R^{n} is open, n ≥ 2, f: D → R^{n} is continuous, open and discrete, there exists K ⊂ D and h, t > 0 such that for every x ∈ D\K, lim inf_{r→0} d(f(B(x, r)))^{n}/μ_{n}(f(B(x, r))) < h, and there exists 0 < a < 1 and ℬ = (B_{i})_{i∈N} an a-porous base of K such that d(f(aB_{i}))^{n}/μ_{n}(f(B_{i})) < t for every i ∈ N, then it follows that f is K(h, t, n)-quasiregular and K_{0}(f) < V_{n} ·H, where H = max{h, t}.

As in [7], the principal instrument used in proving such theorems is the new discrete modulus introduced by J. Heinonen and P. Koskela. We give a slightly modified version of this notion, and obtain bounds for this class of discrete modulus. For these bounds we modify some covering theorems from [10] and, for the sake of completeness, we give a detailed proof of one of them (Theorem 3).

We use the notation from [15], [9] and [12]. If f: D → R^{n} is a map, D ⊂ R^{n} is open, x ∈ D, U(x, f, r) is the component of f^{−1}(B(f(x), r)) containing x. For D ⊂ R^{n} open and f: D → R^{n} a map, we say that f is discrete if f^{−1}(y) is isolated for every y ∈ R^{n}, and we say that f is light if f^{−1}(y) is a set of topological dimension zero for every y ∈ R^{n}.

For x ∈ R^{n} we denote by |x| = (∑_{i=1}^{n} x_{i}^{2})^{1/2} , and if A, B ⊂ R^{n}, we denote by d(A) = sup_{x,y∈A} |x − y| and by d(A, B) = inf_{x∈A, y∈B} |x − y|. If A ∈ L(R^{n}, R^{n}), let |A| = sup_{|h|=1} |A(h)|, and l(A) = inf_{|h|=1} |A(h)|.

If D ⊂ R^{n} is a domain, n ≥ 2, and f: D → R^{n} is a map, we say that f is quasiregular if f is ACL^{n} and there exists K ≥ 1 such that |f′(x)|^{n} ≤ K · J_{f}(x) a.e. in D. For a quasiregular map we also have J_{f}(x) ≤ K′ · l(f′(x))^{n} a.e. in D for some K′ ≥ 1, and we denote by K_{0}(f) the smallest K ≥ 1 such that |f′(x)|^{n} ≤ K · J_{f}(x) a.e. in D, and by K_{I}(f) the smallest K ≥ 1 such that J_{f}(x) ≤ K · l(f′(x))^{n} a.e. in D. We say that a quasiregular map f: D → R^{n}, n ≥ 2, is K -quasiregular if K_{0}(f) ≤ K, K_{I}(f) ≤ K.

If Γ is a path family, we denote by

F (Γ) = \{ρ: \overline{R}^{n} → [0, ∞] a Borel map : ∫_{γ} ρ ds ≥ 1 for every γ ∈ Γ\}

and by M(Γ) = inf_{ρ∈F (Γ)} ∫_{R^{n}} ρ^{n}(x) dx.

If E, F ⊂ R^{n}, D ⊂ R^{n} is open such that E ∪ F ⊂ \overline{D} , let Δ(E, F, D) be those paths, open or not, which join E with F in D. If f: D → R^{n} is a map, we let N(f, D) = sup_{y∈R^{n}} Card f^{−1}(y). A domain A ⊂ \overline{R}^{n} is a ring if CA has exactly two components, C_{0} and C_{1} , and we denote A = R(C_{0}, C_{1}), C_{0} being bounded, and Γ_{A} = Δ(C_{0}, C_{1}, A). For r > 0, let Φ_{n}(r) be the set of all rings A = R(C_{0}, C_{1}) in \overline{R}^{n} such that 0 ∈ C_{0} and there exists a ∈ C_{0} with |a| = 1 and such that ∞ ∈ C_{1} , and there exists b ∈ C_{1} with |b| = r, and we let ℋ (r) = inf M(Γ_{A}), the infimum being taken over all rings A ∈ Φ_{n}(r). We denote by V_{n} the volume of the unit ball in R^{n}, by μ_{n} the Lebesgue measure in R^{n} and by ω_{n−1} the area of the unit sphere from R^{n}. For 0 < a < λ < 1, we write N(a, λ) = 2K · (4/λ · (λ − a))^{n}, where K = [log \frac{1}{2}(1 − a)/ log λ] + 1. For ε > 0 we let N(ε) = (3+ε)^{n}·(\frac{1}{2}(2 + ε))^{n}. In [10, p. 29] the following lemma is proved: There exists a positive integer B(n) such that if a_{1}, …, a_{k} ∈ R^{n}, r_{1}, …, r_{k} ∈ (0, ∞), a_{i} ∉ B(a_{j}, r_{j}) for i ≠ j, i, j = 1, …, k and ⋂_{i=1}^{k} B(a_{i}, r_{i}) ≠ ∅, then it follows that k ≤ B(n). Throughout this paper we shall denote by B(n) the number from this lemma.

2. Preliminaries

Theorem 1. Let D ⊂ R^{n} be open, A ⊂ D closed in D and such that A is bounded or A = D, ℬ a collection of balls from D such that every ball from ℬ is of the form B(x, r) with x ∈ A and r ∈ I_{x} ⊂ R with

(a) 0 ∈ I′_{x} for every x ∈ A,

(b) if x_{k} ∈ A, r_{x_{k}} ∈ I_{x_{k}} for k ∈ N, x_{k} → x, r_{x_{k}} → r and \overline{B} (x, r) ⊂ D, then r ∈ I_{x}. Then there exists an at most countable collection of balls (B_{i})_{i∈N} such that B_{i} = B(x_{i}, r_{i}) ∈ ℬ for i ∈ N, A ⊂ ⋃_{i=1}^{∞} B_{i}, r_{k} ≥ r_{k+1} and x_{k} ∉ ⋃_{i=1,i≠k}^{∞} B_{i} for k ≥ 1. Hence every point from A belongs to at most B(n) balls and \frac{1}{2}B_{i} ∩ \frac{1}{2}B_{j} = ∅ for i, j ∈ N, i ≠ j.

Theorem 2. Let D ⊂ R^{n} be open, A ⊂ D such that either A is bounded or A = D, ℬ a collection of balls from D such that every ball B ∈ ℬ is of the form B = B(x, r) with x ∈ A and r ∈ I_{x} ⊂ R_{+} such that 0 ∈ I′_{x} for every x ∈ A. Then, for every ε > 0 there exists an at most countable collection of balls (B_{i})_{i∈N} such that B_{i} = B(x_{i}, r_{i}) ∈ ℬ for i ∈ N, A ⊂ ⋃_{i=1}^{∞} B_{i}, B_{i}/(2 + ε) ∩ B_{j}/(2 + ε) = ∅ for i, j ∈ N, i ≠ j and every point from A belongs to at most N(ε) · B(n) balls B_{i}.

Theorem 3. Let D ⊂ R^{n} be open, A ⊂ D, 0 < a < 1 and ℬ = (B_{i})_{i∈N} an a-porous covering of A. Then, if A is bounded or if A = D and ℬ is an a-porous base of A, for every 0 < a < λ < 1 we can find 𝒞 = (B_{i_{k}})_{k∈N} a subcovering of A such that every point from A belongs to at most N(a, λ) balls B_{i_{k}} and \frac{1}{2}(λ − a)B_{i_{k}} ∩ \frac{1}{2}(λ − a)B_{i_{l}} = ∅ for k, l ∈ N, k ≠ l, where N(a, λ) = 2k · (4/λ(λ − a))^{n}, with k = [log \frac{1}{2}(1 − a)/ log λ] + 1.

Proof. We suppose first that A is bounded. Hence we can suppose that M = sup_{i∈N} r_{i} < ∞, where B_{i} = B(x_{i}, r_{i}) for i ∈ N. Let 0 < a < λ < 1 and

A_{1} = {x ∈ A |there exists i ∈ N and λM < r_{i} ≤ M
such that B(x_{i}, r_{i}) ∈ ℬ and x ∈ B(x_{i}, r_{i})}.

If A_{1} ≠ ∅, we take x_{11} ∈ D and λM < ρ_{11} ≤ M such that B(x_{11}, ρ_{11}) ∈ ℬ and A ∩ B(x_{11}, ρ_{11}) ≠ ∅. If A_{1} ⊄ B(x_{11}, ρ_{11}), we take x_{12} ∈ D and λM < ρ_{12} ≤ M such that B(x_{12}, ρ_{12}) ∈ ℬ and (A_{1} \ B(x_{11}, ρ_{11})) ∩ B(x_{12}, ρ_{12}) ≠ ∅. If A_{1} ⊄ B(x_{11}, ρ_{11}) ∪ B(x_{12}, ρ_{12}), we continue the process and we show that this process must end in a finite number of steps. Indeed, let B(x_{i}, r_{i}), B(x_{j}, r_{j}) be two balls which cover A_{1} obtained as before. Then there exists a point x ∈ A such that x ∈ B(x_{j}, r_{j}) \ B(x_{i}, r_{i}), hence |x − x_{j}| ≤ ar_{j} ≤ aM (because (B(x_{j}, r_{j}) \ \overline{B} (x_{j}, ar_{j})) ∩ A = ∅) and |x − x_{i}| > r_{i} ≥ λM, which implies that

|x_{i} − x_{j}| ≥ |x − x_{i}| − |x − x_{j}| > (λ − a)M.

Let ρ = \frac{1}{2}(λ − a) and suppose that there exists a point z ∈ B(x_{j}, ρr_{j})∩B(x_{i}, ρr_{i}). Then 2ρM ≥ ρr_{j} + ρr_{i} ≥ |x_{j} − z| + |x_{i} − z| ≥ |x_{i} − x_{j}| > (λ − a)M, which represents a contradiction. It follows that B(x_{i}, \frac{1}{2}(λ − a)r_{i}) ∩ B(x_{j}, \frac{1}{2}(λ − a)r_{j}) = ∅. Using the boundedness of A we can find m(1) ∈ N, x_{11}, …, x_{1m(1)} ∈ D, real numbers ρ_{11}, …, ρ_{1m(1)} ∈ (λM, M] such that B(x_{1l}, ρ_{1l}) ∈ ℬ for l = 1, …, m(1), (A_{1} \ ⋃_{l=1}^{k−1} B(x_{1l}, ρ_{1l})) ∩ B(x_{1k}, ρ_{1k}) ≠ ∅ for k = 2, …, m(1) and A_{1} ⊂ ⋃_{l=1}^{m(1)} B(x_{1l}, ρ_{1l}). Also, a point from A_{1} may belong to at most m = (4/λ(λ − a))^{n} balls B(x_{1l}, ρ_{1l}). Indeed, let B(x_{1i}, ρ_{1i}) be a fixed ball of this type and suppose that it is intersected by m balls B(x_{1j}, r_{1j}), j ∈ C ⊂ {1, …, m(1)}. Then every such ball is contained in B(x_{1i}, 2M); hence

\begin{aligned}V_{n} · (2M)^{n} & = μ_{n}(B(x_{1i}, 2M)) ≥ ∑_{j∈C} μ_{n}((B(x_{1j}, \frac{1}{2}(λ − a)ρ_{1j})) \\ & ≥ V_{n} · m · \frac{1}{2^{n}} ((λ − a) · λM))^{n} \end{aligned}

and this implies that m ≤ (4/λ(λ − a))^{n}. We have completed the first step of our inductive process.

At step j, we take

A_{j} = {x ∈ A \ ⋃_{k=1}^{j−1} ⋃_{l=1}^{m(k)} B(x_{1l}, ρ_{1l}) | there exists i ∈ N

and λ^{j}M < r_{i} ≤ λ^{j−1} · M such that B(x_{i}, r_{i}) ∈ ℬ and x ∈ B(x_{i}, r_{i})}.

If A_{j} ≠ ∅, we take x_{j1} ∈ A_{j} and λ^{j}·M < ρ_{j1} ≤ λ^{j−1}·M such that B(x_{j1}, ρ_{j1}) ∈ ℬ and A_{j} ∩ B(x_{j1}, ρ_{j1}) ≠ ∅. If A_{j} ⊄ B(x_{j1}, ρ_{j1}), we continue the process from step j, and if A_{j} ⊄ ⋃_{l=1}^{k−1} B(x_{jl}, ρ_{jl}), we take x_{jk} ∈ D and λ^{j} · M < ρ_{jk} ≤ λ^{j−1} · M such that B(x_{jk}, ρ_{jk}) ∈ ℬ and (A_{j} \ ⋃_{l=1}^{k−1} B(x_{jl}, ρ_{jl})) ∩ B(x_{jk}, ρ_{jk}) ≠ ∅.

Using the boundedness of A_{j}, we see as in step 1 that this process must end in a finite number of steps. Hence we find m(j) ∈ N, x_{j1}, …, x_{jm(j)} ∈ D, ρ_{j1}, …, ρ_{jm(j)} ∈ (λ^{j} · M, λ^{j−1} · M] such that B(x_{jl}, ρ_{jl}) ∈ ℬ, l = 1, …, m(j),

(A_{j} \ ⋃_{l=1}^{k−1} B(x_{jl}, ρ_{jl})) ∩ B(x_{jk}, ρ_{jk}) ≠ ∅

for k = 1, …, m(j) and A_{j} ⊂ ⋃_{l=1}^{m(j)} B(x_{jl}, ρ_{jl}).

The process will have an infinite number of steps. We show that A ⊂ ⋃_{j=1}^{∞} ⋃_{l=1}^{m(j)} B(x_{jl}, ρ_{jl}). Indeed, if this is not true, we can find a point x ∈

A \ ⋃_{j=1}^{∞} ⋃_{l=1}^{m(j)} B(x_{jl}, ρ_{jl}). Since (B_{i})_{i∈N} is a covering of A, we can find i ∈ N such that x ∈ B_{i} and j ∈ N with λ^{j} · M < r_{i} ≤ λ^{j−1} · M. Using the definition of A_{j}, we obtain that x ∈ A_{j}, which represents a contradiction, since x ∉ ⋃_{l=1}^{m(j)} B(x_{jl}, ρ_{jl}) and we proved that A_{j} ⊂ ⋃_{l=1}^{m(j)} B(x_{jl}, ρ_{jl}). It follows that A ⊂ ⋃_{j=1}^{∞} ⋃_{l=1}^{m(j)} B(x_{jl}, ρ_{jl}).

Let now k ∈ N be such that a ≤ −2λ^{k}+1. We show that two balls B(x_{il}, ρ_{il}), B(x_{jq}, ρ_{jq}), l ∈ {1, …, m(i)}, q ∈ {1, …, m(j)} with j − i − 1 ≥ k cannot have a common point from A. Indeed, suppose that this is not true and pick two balls as before, B(x_{il}, ρ_{il}), B(x_{jq}, ρ_{jq}), with j − i − 1 ≥ k such that there exists a point x ∈ A ∩ B(x_{il}, ρ_{il}) ∩ B(x_{jq}, ρ_{jq}). From the construction of the balls B(x_{kl}, ρ_{kl}) we can find a point z ∈ (A \ B(x_{il}, ρ_{il})) ∩ B(x_{jq}, ρ_{jq}), and since A ∩ (B(x_{il}, ρ_{il}) \ \overline{B} (x_{il}, a · ρ_{il})) = ∅, we see that the point x ∈ B(x_{il}, a · ρ_{il}). Then 2M · λ^{j−1} ≥ 2ρ_{jq} ≥ |x − z| ≥ (1 − a)ρ_{il} > (1 − a)M · λ^{i}, and hence \frac{1}{2}(1 − a) < λ^{j−i−1} < λ^{k}, i.e., a > −2λ^{k} + 1, which contradicts the way we chose k. It follows that if j − i − 1 ≥ k, then any two balls B(x_{il}, ρ_{il}), B(x_{jq}, ρ_{jq}) with j − i − 1 ≥ k cannot have a common point with A. Since we showed that any point from A_{j} can belong to at most (4/λ(λ − a))^{n} balls B(x_{jl}, ρ_{jl}), j ∈ N, l = 1, …, m(j), it follows that any point from A can belong to at most 2k · (4/λ(λ − a))^{n} balls B(x_{il}, ρ_{il}), i ∈ N, l ∈ \{1, …, m(i)\}, where k = [log \frac{1}{2}(1 − a)/ log λ] + 1. As in step 1, we show that \frac{1}{2}(λ − a) · B(x_{il}, ρ_{il}) ∩ \frac{1}{2}(λ − a) · B(x_{jq}, ρ_{jq}) = ∅, for i ≠ j, l ∈ {1, …, m(i)}, q ∈ {1, …, m(j)} and the theorem is proved if A is bounded.

If A = D and ℬ is an a-porous base of A, we leave the proof to the reader.

3. Discrete modulus. Bounds for discrete modulus

We shall use, now in a slightly enlarged form, the concept of the discrete modulus introduced by J. Heinonen and P. Koskela in [7]. If D ⊂ R^{n} is open and ℬ = (B_{i})_{i∈N} is a covering with balls of D, we say that ℬ is a (λ, μ)-covering of D if every point of D belongs to at most λ balls B_{i} and μB_{i} ∩ μB_{j} = ∅ for i, j ∈ N, i ≠ j and 0 < μ < 1. From Theorem 1, we always have (B(n), \frac{1}{2}) coverings of an open set D ⊂ R^{n}; hence we always have (λ, μ)-coverings of an open set D ⊂ R^{n} if λ ≥ B(n), 0 < μ ≤ \frac{1}{2} , and we denote by E(λ, μ, D) all the (λ, μ)-coverings of the open set D from R^{n}. If ℬ = ℬ^{1} ∪ ⋯ ∪ ℬ^{m} is a covering with balls of D, 0 < μ < 1, D open in R^{n}, ℬ^{k} = (B_{i}^{k})_{i∈N} , k = 1, …, m, such that every point from D belongs to at most λ balls B_{i}^{k} for k = 1, …, m, and for every k ∈ {1, …, m}, we have μB_{i}^{k} ∩ μB_{j}^{k} = ∅, i, j ∈ N, i ≠ j, we say that ℬ is a (m, λ, μ)-covering of D and we denote by E(m, λ, μ, D) all the (m, λ, μ)- coverings of D. In this definition it is possible that some families of balls ℬ^{i}, ℬ^{j} are the same. If Γ is a path family such that Im γ ⊂ D for every γ ∈ Γ and ℬ = (B_{i})_{i∈N} is a collection of balls from D, a subcollection 𝒜 = (B_{i})_{i∈I} with I ⊂ N will be called a chain of balls from the collection ℬ along the path γ if Im γ ∩ B_{i} ≠ ∅ for every i ∈ I and Im γ ⊂ ⋃_{i∈I} B_{i}.

Let now m, λ ∈ N, 0 < μ < 1, ℬ = ℬ^{1} ∪ ⋯ ℬ^{m}, ℬ ∈ E(m, λ, μ, D) and v_{k}: ℬ^{k} → [0, ∞) be set functions which assign to each ball B_{i}^{k} ∈ ℬ^{k} a positive number for k = 1, …, m. We say that v = (v_{1}, …, v_{m}) is (m, λ, μ, D)-admissible for the (m, λ, μ) covering of D,

ℬ = (B_{i})_{i∈N}, ℬ = ℬ^{1} ∪ ⋯ ∪ ℬ^{m}, ℬ^{k} = (B_{i})_{i∈I_{k}}, k = 1, …, m

and for the path family Γ such that Im γ ⊂ D for every γ ∈ Γ, if

∑_{k=1}^{m} ∑_{i∈I_{k}∩I} v_{k}(B_{i}) ≥ 1,

for every 𝒜 = (B_{i})_{i∈I} chain of balls along some path γ ∈ Γ. We denote by F_{(m,λ,μ,D)}(ℬ, Γ) the class of all (m, λ, μ, D)-admissible m-tuples of set functions v = (v_{1}, …, v_{m}) for the collection ℬ ∈ E(m, λ, μ, D) and the path family Γ. For δ > 0 we denote by

δ − Mod_{(m,λ,μ)}(Γ) = inf ∑_{ℬ∈E(m,λ,μ,D)} ∑_{v=(v_{1},...,v_{m})∈F_{(m,λ,μ,D)}(ℬ,Γ)} ∑_{k=1}^{m} ∑_{i∈I_{k}} v_{k}(B_{i})^{n},

where the infimum is taken along all the collections

ℬ ∈ E(m, λ, μ, D), ℬ = ℬ^{1} ∪ ⋯ ∪ ℬ^{m}, ℬ = (B_{i})_{i∈N}, ℬ^{k} = (B_{i})_{i∈I_{k}},

k = 1, …, m, and all the (m, λ, μ, D)-admissible m-tuples of set functions v = (v_{1}, …, v_{m}) ∈ F_{(m,λ,μ,D)}(ℬ, Γ), D being any open set in R^{n} such that Im γ ⊂ D for every γ ∈ Γ and d(B_{i}) < δ for i ∈ N. The function δ → δ −Mod_{(m,λ,μ)}(ℬ, Γ) is clearly decreasing; hence it has a limit in 0 and we denote by Mod_{(m,λ,μ)}(Γ) = lim_{δ→0} δ−Mod_{(m,λ,μ)}(Γ) and call the number Mod_{(m,λ,μ)}(Γ) the (m, λ, μ) discrete modulus of the path family Γ. Of course, this notion has sense if E(m, λ, μ, D) ≠ ∅. For λ ≥ B(n) and 0 < μ ≤ \frac{1}{2} , m ∈ N, the (m, λ, μ) discrete modulus is defined. For m = 1 we note Mod_{(λ,μ)}(Γ) instead of Mod_{(1,λ,μ)}(Γ) and by F_{(λ,μ,D)}(ℬ, Γ) instead of F_{(1,λ,μ,D)}(ℬ, Γ) for ℬ ∈ E(λ, μ, D).

Using the method from [7], we can prove the following relation between the classical modulus and the (m, λ, μ) discrete modulus:

Theorem 4. Let Γ be a path family in R^{n}. Then there exists a constant R(n, m, μ) such that M(Γ) ≤ R(n, m, μ) · Mod_{(m,λ,μ)}(Γ), where R(n, m, μ) = V_{n} · 2^{n} · n^{n^{2}}(3^{n}/(μ(n − 1))^{n} · m · n)^{n−1} .

Theorem 5. Let n ≥ 2, D ⊂ R^{n} be open, K ⊂ D, f: D → R^{n} continuous such that there exists h > 0 with

lim inf_{r→0} \frac{d(f(B(x, r)))^{n}}{μ_{n}(f(B(x, r)))} < h

for every x ∈ D \ K and N(f, D) < ∞. If K = ∅, for E, F ⊂\overline{D} , we have

Mod_{(B(n),1/2)}(Δ(E, F, D)) ≤ h · N(f, D) · B(n) · \frac{μ_{n}(f(D))}{d(f(E), f(F))^{n}} .

If K ≠ ∅, D is bounded and there exists t > 0 and 0 < α ≤ 1 such that for an α-porous base of K, ℬ̃ = (B(x_{i}, r_{i}))_{i∈N} the inequality

\frac{d(f(α · B(x_{i}, r_{i})))^{n}}{μ_{n}(f(B(x_{i}, r_{i})))} < t

holds for i ∈ N, then, for every ε > 0 and 0 < α < λ < 1 it follows that

Mod_{(2,M,P)}(Δ(E, F, D)) ≤ 2H · N(f, D) · M · \frac{μ_{n}(f(D))}{d(f(E), f(F))^{n}} ,

where

P = min\{\frac{1}{2 + ε}, \frac{λ − α}{2}\}, H = max\{h, t\}, M = max\{B(n) · N(ε), N(α, λ)\}.

Proof. Suppose first that K = ∅. We may assume that r = d(f(E), f(F)) > 0, since otherwise the theorem is clear. If x ∈ D, let

I_{x} = \{r ∈ R_{+} | \frac{d(f(B(x, r)))^{n}}{μ_{n}(f(B(x, r)))} < h\}.

Then 0 ∈ I′_{x}, and let ℬ^{1} = ⋃_{x∈D,r∈I_{x}} B(x, r). We define for B ∈ ℬ^{1} a set function v by v(B) = d(f(B))/d(f(E), f(F)). Now ℬ^{1} satisfies the conditions from Theorem 1, hence, if δ > 0 is fixed, we can find ℬ ∈ E(B(n), \frac{1}{2}, D), ℬ = (B_{i})_{i∈N} , a subcollection of ℬ^{1} such that d(B_{i}) < δ, \overline{B}_{i} ⊂ D, d(f(B_{i}))^{n}/μ_{n}(f(B_{i})) < h for i ∈ N.

Let γ ∈ Δ(E, F, D) and 𝒜 = (B_{i})_{i∈I} be a chain of balls from the collection ℬ along the path γ. Then ⋃_{i∈I} f(B_{i}) is a connected set which covers the path Γ = f ∘ γ, hence ∑_{i∈I} d(f(B_{i})) ≥ l(Γ) ≥ d(f(E), f(F)), and this implies that ∑_{i∈I} v(B_{i}) ≥ 1 for every 𝒜 = (B_{i})_{i∈I} chain of balls from the collection ℬ along every path γ ∈ Δ(E, F, D), i.e., v ∈ F_{(B(n),1/2,D)}(ℬ, Δ(E, F, D)).

We have

(1) δ − Mod_{(B(n),1/2)}(Δ(E, F, D)) ≤ ∑_{i=1}^{∞} v(B_{i})^{n} ≤ ∑_{i=1}^{∞} \frac{d(f(B_{i}))^{n}}{d(f(E), f(F))^{n}} .

Now every point y ∈ f(D) has at most N(f, D) points x_{k} ∈ D such that f(x_{k}) = y, and since such a point x_{k} may belong to at most B(n) balls B_{i}, it follows that every point y ∈ f(D) belongs to at most B(n) · N(f, D) sets f(B_{i}). This implies that

\begin{aligned}∑_{i=1}^{∞} μ_{n}(f(B_{i})) & = ∑_{i=1}^{∞} ∫_{f(D)} 𝒳_{f(B_{i})}(x) dx = ∫_{f(D)}(∑_{i=1}^{∞} 𝒳_{f(B_{i})})(x) dx \\ & ≤ ∫_{f(D)} B(n) · N(f, D) dx = B(n) · N(f, D) · μ_{n}(f(D)), \end{aligned}

and using (1), we have

δ − Mod_{(B(n),1/2)}(Δ(E, F, D)) ≤ ∑_{i=1}^{∞} \frac{d(f(B_{i}))^{n}}{d(f(E), f(F))^{n}}

≤ h · ∑_{i=1}^{∞} \frac{μ_{n}(f(B_{i}))}{d(f(E), f(F))^{n}}

≤ h · B(n) · N(f, D) · \frac{μ_{n}(f(D))}{d(f(E), f(F))^{n}} .

Letting δ tend to zero, we obtain that

Mod_{(B(n),1/2)}(Δ(E, F, D)) ≤ h · B(n) · N(f, D) · \frac{μ_{n}(f(D))}{d(f(E), f(F))^{n}} .

Suppose now that K ≠ ∅ and let 0 < α < λ < 1 and ε > 0 and let δ > 0. Applying Theorem 2 to the base of balls 𝒟 = ⋃_{x∈D\K,r∈I_{x}} B(x, r), we can consider ℬ^{1} = (B_{i}^{1})_{i∈N} a subcovering of 𝒟 which covers D \ K such that d(B_{i}^{1}) < δ, \overline{B} _{i}^{1} ⊂ D, d(f(B_{i}^{1}))^{n}/μ_{n}(f(B_{i}^{1})) < h for i ∈ N and every point from D \ K belongs to at most N(ε) · B(n) balls B_{i}^{1} and B_{i}^{1}/(2 + ε) ∩ B_{j}^{1}/(2 + ε) = ∅ for i ≠ j, i, j ∈ N.

Applying Theorem 3 to the α-porous covering of K, 𝒞 = (B(x_{i}, r_{i}))_{i∈N} , we can consider ℬ^{2} = (B_{i}^{2})_{i∈N} a subcollection of 𝒞 which covers K such that

d(B_{i}^{2}) < δ,  \overline{B} _{i}^{2} ⊂ D,  \frac{d(f(αB_{i}^{2}))^{n}}{μ_{n}(f(B_{i}^{2}))} < t

for i ∈ N, and every point from K belongs to at most N(α, λ) balls B_{i}^{2} and \frac{1}{2}(λ − α) · B_{i}^{2} ∩ \frac{1}{2}(λ − α) · B_{j}^{2} = ∅ for i ≠ j, i, j ∈ N. Since (B_{i}^{2} − α\overline{B} _{i}^{2}) ∩ K = ∅ for i ∈ N, we see that (αB_{i}^{2})_{i∈N} is also a covering of K; hence ℬ = ℬ^{1} ∪ αℬ^{2} ∈ E(2, M, P, D). We define set functions v_{k} on the balls B_{i}^{k}, k = 1, 2, i ∈ N by v_{1}(B_{i}^{1}) = d(f(B_{i}^{1}))/d(f(E), f(F)) for i ∈ N and v_{2}(αB_{i}^{2}) = d(f(αB_{i}^{2}))/d(f(E), f(F)) for i ∈ N. Let us show that

v = (v_{1}, v_{2}) ∈ F_{(2,M,P,D)}(ℬ, Δ(E, F, D)).

Indeed, let γ ∈ Δ(E, F, D) and let 𝒜 = (B_{i})_{i∈N} be a chain of balls from ℬ along the path γ, where I = I_{1} ∪ I_{2} and B_{i} ∈ ℬ^{1} for i ∈ I_{1} , B_{i} ∈ αℬ^{2} for i ∈ I_{2} . Then ⋃_{i∈I_{1}} f(B_{i}) ∪ ⋃_{i∈I_{2}} f(αB_{i}) is a connected set which covers the path Γ = f ∘ γ, hence ∑_{i∈I_{1}} d(f(B_{i}^{1})) + ∑_{i∈I_{2}} d(f(αB_{i}^{2})) ≥ l(Γ) ≥ d(f(E), f(F)) and this implies that v = (v_{1}, v_{2}) ∈ F_{(2,M,P,D)}(ℬ, Δ(E, F, D)). We have

δ − Mod_{(2,M,P)}(Δ(E, F, D)) ≤ ∑_{i∈I_{1}} v_{1}(B_{i}^{1})^{n} + ∑_{i∈I_{2}} v_{2}(αB_{i}^{2})^{n}

= ∑_{i∈I_{1}} \frac{d(f(B_{i}^{1}))^{n}}{d(f(E), f(F))^{n}} + ∑_{i∈I_{2}} \frac{d(f(αB_{i}^{2}))^{n}}{d(f(E), f(F))^{n}}

≤ \frac{H}{d(f(E), f(F))^{n}} (∑_{i∈I_{1}} μ_{n}(f(B_{i}^{1})) + ∑_{i∈I_{2}} μ_{n}(f(B_{i}^{2})))

≤ \frac{H · M · N(f, D)}{d(f(E), f(F))^{n}}(μ_{n}(f( ⋃_{i∈I_{1}} B_{i}^{1})) + μ_{n}(f( ⋃_{i∈I_{2}} B_{i}^{2})))

≤ \frac{2 · H · M · N(f, D)}{d(f(E), f(F))^{n}} · μ_{n}(f(D)).

Letting δ tend to zero, we complete the proof.

4. Conditions of quasiregularity and a removability result

We shall first prove the following theorem.

Theorem 6. Let D ⊂ R^{n} be open, n ≥ 2, a > 1, x ∈ D and r > 0 such that \overline{B} (x, a.r) ⊂ D and f: D → R^{n} continuous, open and discrete. If f is K -quasiregular on B(x, a.r) \ \overline{B} (x, r), then it follows that

\frac{d(f(B(x, r)))^{n}}{μ_{n}(f(B(x, a.r)))} ≤ C(a, K),

where C(a, K) = 1/L(a, K) and

L(a, K) = min\{\frac{V_{n}}{2^{n}} · exp(\frac{−2K · n · ω_{n−1}}{c_{n} · (log a)^{n−1}} ),

\frac{V_{n} · n · K · ω_{n−1}}{2^{n} · c_{n} · (log a)^{n−1}} · exp(\frac{−2K · n · ω_{n−1}}{c_{n} · (log a)^{n−1}} )\}.

Proof. Suppose first that μ_{n}(f(S(x, a.r))) = 0. Let s = L(x, f, r), δ = K · ω_{n−1}/(log a)^{n−1} and let α > 0 be such that 2δ = c_{n} · log(1/α), where c_{n} is the constant from Theorem 10.12, [15, p. 31].

Now α = exp(−2K · ω_{n−1}/c_{n} · (log a)^{n−1}); hence 0 < α < 1.

\begin{aligned}\frac{μ_{n}(f(B(x, a.r)))}{d(f(B(x, r)))^{n}} & ≥ \frac{μ_{n}(f(B(x, a.r)))}{2^{n} · L(x, f, r)^{n}} \\ & ≥ V_{n} · \frac{l(x, f, r)^{n}}{2^{n} · L(x, f, r)^{n}} ≥ V_{n} · \frac{α^{n}}{2^{n}} ≥ L(a, K), \end{aligned}

if L(x, f, r)/l(x, f, r) ≤ 1/α. Suppose that L(x, f, r)/l(x, f, r) > 1/α. If Q = f(S(x, r)), then Q ∩ S(f(x), t) ≠ ∅ for every t ∈ [αs, s] and let Q_{p}, p ∈ P be the components of Cf(\overline{B} (x, a.r)) which intersect B(f(x), s) \ \overline{B} (f(x), αs). Since S(x, r) ⊂ Int \overline{B} (x, ar) and f is open, it follows that Q ∩ ∂Q_{p} = ∅ for p ∈ P, and let B_{p} = {t ∈ (αs, s) | S(f(x), t) ∩ Q_{p} ≠ ∅}, p ∈ P, E = ⋃_{p∈P} Q_{p} and B = {t ∈ (αs, s) | S(f(x), t) ∩ E ≠ ∅}. Since B is open, B = ⋃_{i∈I}(a_{i}, b_{i}), I ⊂ N and (a_{i}, b_{i}) ∩ (a_{j}, b_{j}) = ∅ for i, j ∈ I, i ≠ j. When i ∈ I, we denote D_{i} = B(f(x), b_{i}) \ \overline{B} (f(x), a_{i}) and

J_{i} = {j ∈ P | there exists t ∈ (a_{i}, b_{i}) such that S(f(x), t) ∩ Q_{j} ≠ ∅}

and let U_{i} = ⋃_{j∈J_{i}} ∂Q_{j}. Let now i ∈ I and t ∈ (a_{i}, b_{i}). Then there exists j ∈ J_{i} such that S(f(x), t) ∩ Q_{j} ≠ ∅. Since S(f(x), t) ∩ Q ≠ ∅ and Q ⊂ f(\overline{B} (x, ar)) ⊂ CQ_{j}, we see that S(f(x), t) ∩ CQ_{j} ≠ ∅; hence S(f(x), t) ∩ ∂Q_{j} ≠ ∅. If Δ_{i} is the family of all paths γ: [0, 1] → R^{n} which joins Q with U_{i} in D_{i}, then D_{i} ∩ U_{i} and D_{i} ∩Q are nonempty, disjoint sets which intersect S(f(x), t) for every t ∈ (a_{i}, b_{i}). Using Theorem 10.12, [15, p. 31], we obtain M(Δ_{i}) ≥ c_{n} · log(b_{i}/a_{i}). Let now γ ∈ Δ_{i} be such that γ(0) ∈ Q, γ(1) ∈ U_{i}. We can find b_{γ} a subpath of γ, such that there exists a path a_{γ}: [α_{γ}, β_{γ}] → R^{n} with 0 ≤ α_{γ} ≤ β_{γ} ≤ 1, a_{γ}(α_{γ}) ∈ S(x, r), a_{γ}(β_{γ}) ∈ S(x, ar), a_{γ}(α_{γ}, β_{γ}) ⊂ B(x, ar) \ \overline{B} (x, r) and f ∘ a_{γ} = b_{γ}. If Γ_{i} = {a_{γ} | γ ∈ Δ_{i}}, Γ′_{i} = {b_{γ} | γ ∈ Δ_{i}}, then Γ′_{i} = f(Γ_{i}) and the paths from Γ′_{i} are shorter than the paths from Δ_{i}, and let Γ′ = ⋃_{i∈I} Γ′_{i}, Γ = ⋃_{i∈I} Γ_{i} and A be the spherical ring B(x, ar) \ \overline{B} (x, r).

Then Γ′ = f(Γ) and

δ = \frac{K · ω_{n−1}}{(log a)^{n−1}} = K · M(Γ_{A}) ≥ K · M(Γ) ≥ M(Γ′) ≥ ∑_{i∈I} M(Γ′_{i})

≥ ∑_{i∈I} M(Δ_{i}) ≥ ∑_{i∈I} c_{n} · log \frac{b_{i}}{a_{i}} .

We used here a path inequality of Poleckii [11]. Let now J ⊂ I be finite and suppose that a_{i_{1}} < b_{i_{1}} ≤ a_{i_{2}} < b_{i_{2}} ≤ ⋯ ≤ a_{i_{p}} < b_{i_{p}} is an ordering of the endpoints of the intervals counted by J. We take b_{i_{∘}} = αs, a_{i_{p+1}} = s, and since

c_{n} · ∑_{j=1}^{p} log \frac{b_{i_{j}}}{a_{i_{j}}} + c_{n} · ∑_{j=0}^{p} log \frac{a_{i_{j+1}}}{b_{i_{j}}} = c_{n} · log \frac{1}{α} = 2δ,

it follows that

c_{n} · ∑_{j=0}^{p} log \frac{a_{i_{j+1}}}{b_{i_{j}}} ≥ δ,

and let A_{J} = ⋃_{j=0}^{p}(\overline{B} (f(x), a_{i_{j+1}}) \ B(f(x), b_{i_{j}})). We take θ_{j} ∈ (b_{i_{j}}, a_{i_{j+1}}) such that log(a_{i_{j+1}}/b_{i_{j}}) = (a_{i_{j+1}} − b_{i_{j}}) · 1/θ_{j} for j = 0, 1, …, p. Then

V (A_{J}) = V_{n} · ∑_{j=0}^{p}(a_{i_{j+1}}^{n} − b_{i_{j}}^{n}) = V_{n} · ∑_{j=0}^{p}(a_{i_{j+1}} − b_{i_{j}}) · (∑_{k=0}^{n−1} a_{i_{j+1}}^{n−k−1} · b_{i_{j}}^{k})

≥ n · V_{n} · α^{n−1} · s^{n−1} · ∑_{j=0}^{p}(a_{i_{j+1}} − b_{i_{j}})

= n · V_{n} · α^{n−1} · s^{n−1} · ∑_{j=0}^{p} θ_{j} log \frac{a_{i_{j+1}}}{b_{i_{j}}}

≥ n · V_{n} · α^{n} · s^{n} · ∑_{j=0}^{p} log \frac{a_{i_{j+1}}}{b_{i_{j}}} ≥ \frac{n · V_{n} · δ · α^{n} · s^{n}}{c_{n}} .

Let A = ⋂_{J⊂I} A_{J}, J ⊂ I being finite. Then A is Lebesgue measurable and V (A) ≥ n · V_{n} · δ · α^{n} · s^{n}/c_{n} and A = ⋂_{t∈[αs,s]\B} S(f(x), t), and we have

V (C) = V (A) ≥ \frac{n · V_{n} · δ · α^{n} · s^{n}}{c_{n}} ,

if C = ⋂_{t∈(αs,s)\B} S(f(x), t). Since S(f(x), t)∩E = ∅ for every t ∈ (αs, s)\B, it follows that S(f(x), t) ⊂ f(\overline{B} (x, ar)) for t ∈ (αs, s) \ B; hence C ⊂ f(\overline{B} (x, ar)).

We have now that

\begin{aligned}\frac{μ_{n}(f(B(x, ar)))}{d(f(B(x, ar)))^{n}} & ≥ \frac{μ_{n}(f(\overline{B} (x, ar)))}{2^{n} · L(x, f, r)^{n}} ≥ \frac{V (C)}{2^{n} · s^{n}} \\ & ≥ n · V_{n} · α^{n} · s^{n} · \frac{δ}{c_{n} · 2^{n} · s^{n}} = n · V_{n} · α^{n} · \frac{δ}{c_{n} · 2^{n}} ≥ L(a, K), \end{aligned}

i.e., d(f(B(x, r)))^{n}/μ_{n}(f(B(x, ar))) ≤ C(a, K).

Finally, if μ_{n}(f(S(x, ar))) ≠ 0, we let ε > 0. Since f is K -quasiregular on B(x, ar) \ \overline{B} (x, r), we have μ_{n}(f(S(x, (a − ε)r)) = 0 and this implies that d(f(B(x, r)))^{n}/μ_{n}(f(B(x, (a − ε)r))) ≤ C(a − ε, K). Letting ε tend to zero, we complete the proof.

Theorem 7. Let D ⊂ R^{n} be open, n ≥ 2, f: D → R^{n} continuous, open and discrete such that there exists K ⊂ D, and h, t > 0 such that

lim inf_{r→0} \frac{d(f(B(x, r)))^{n}}{μ_{n}(f(B(x, r)))} < h

for every x ∈ D \ K, and there exists 0 < a < 1 and ℬ = (B_{i})_{i∈N} an a-porous base of K with d(f(aB_{i}))^{n}/μ_{n}(f(B_{i})) < t for i ∈ N. Then f is quasiregular and K_{0}(f) ≤ V_{n} · H where H = max{h, t}.

Proof. Let g: (1, ∞) → (1, ∞) be defined by g(t) = (t^{n} − 1)/(t − 1)^{n} for t ∈ (1, ∞). Then g′(t) < 0 for t > 1, lim_{t→1} g(t) = ∞, lim_{t→∞} g(t) = 1, hence g is a bijection of (1, ∞) onto (1, ∞). Let x ∈ D be fixed. We can find U ∈ V (x), V ∈ V (f(x)) such that f | U: U → V is a proper map, f(∂U) = ∂V , \overline{U} is compact and N(f, U) = |i(f, x)|. Let r > 0 be small enough such that \overline{B} (x, r) ∪ U(x, f, L_{α}) ⊂ U, 0 < α ≤ 1 and L_{α} = L(x, f, αr), l = l(x, f, r) and suppose that L_{α}/l ≥ 1. Let A = B(f(x), L_{α}) \ \overline{B} (f(x), l) and B = U(x, f, L_{α}) \ \overline{U} (x, f, l).

From [15, p. 9], for r > 0 small enough, B is also a ring of components C_{0} = \overline{U} (x, f, l), C_{1} = CU(x, f, L_{α}) and f(B) = A. Also, the component C_{0} contains the point x and a point a such that |x − a| = r and the component C_{1} contains ∞ and a point b such that |x − b| = αr, hence from [15, p. 36] it follows that M(Γ_{B}) ≥ ℋ_{n}(|x − b|/|x − a|) = ℋ_{n}(α). Let ε > 0 and 0 < a < λ < 1 and let M = max{B(n) · N(ε), N(a, λ)}, P = min{1/(2 + ε), (λ − a)/2}.

Using Theorem 4 and Theorem 5, we obtain

ℋ_{n}(α) ≤ M(Γ_{B}) ≤ R(n, 2, P) · Mod_{(2,M,P)}(Γ_{B})

≤ 2 · R(n, 2, P) · H · N(f, B) · M · \frac{μ_{n}(f(B))}{d(\overline{B} (f(x), l), CB(f(x), L_{α})))^{n}}

≤ 2 · R(n, 2, P) · H · M · N(f, U) · \frac{μ_{n}(A)}{(L_{α} − l)^{n}}

= 2 · R(n, 2, P) · H · M · N(f, U) · V_{n} · \frac{(L_{α}^{n} − l^{n})}{(L_{α} − l)^{n}}

= 2 · R(n, 2, P) · H · M · N(f, U) · V_{n} · g(L_{α}/l).

Since lim_{α→0} ℋ_{n}(α) = ∞, we can find 0 < α < 1 small enough such that

ℋ_{n}(α) > 2 · R(n, 2, P) · H · M · N(f, U) · V_{n}.

Then

\frac{L_{α}}{l} ≤ g^{−1}(\frac{ℋ_{n}(α)}{2 · R(n, 2, P) · H · M · N(f, U) · V_{n}} ).

We denote

C(n, h, t, α, λ, ε, U) = max\{1, g^{−1}(\frac{ℋ_{n}(α)}{2 · R(n, 2, P) · H · M · N(f, U) · V_{n}} )\}.

It follows that

\frac{L(x, f, αr)}{l(x, f, r)} ≤ C(n, h, t, α, λ, ε, U)

for every r > 0 such that \overline{B} (x, r) ∪ U(x, f, L_{α}) ⊂ U. This yields that

H_{α}(x, f) = lim sup_{r→0} \frac{L(x, f, αr)}{l(x, f, r)} ≤ C(n, h, t, α, λ, ε, U).

This inequality is also valid for every point z ∈ U; hence from Theorem 1 [4] it follows that f is quasiregular on U. We have therefore proved that f is locally quasiregular on D; hence f is ACL^{n} on D and f is a.e. differentiable on D and J_{f}(x) ≠ 0 a.e. in D.

Let us now fix a point x ∈ D \ K such that f is differentiable in x and J_{f}(x) ≠ 0 and let 0 < ε < |f′(x)| be fixed. Then there exists r_{ε} > 0 such that |f(z) − f(x) − f′(x)(z − x)| ≤ ε · |z − x| for |z − x| ≤ r_{ε}. Then

(2)  (|f′(x)| − ε) · r ≤ L(x, f, r)  for 0 < r < r_{ε}.

Since J_{f}(x) ≠ 0, we have l(f′(x)) > 0, and let s = (l(f′(x)) + ε)/l(f′(x)) and φ: R^{n} → R^{n} be defined by φ(z) = f(x) + f′(x)(z − x) for z ∈ R^{n}.

We show that f(B(x, r)) ⊂ φ(B(x, rs)) for 0 < r ≤ r_{ε}. Indeed, let 0 < r < r_{ε} be fixed and y ∈ f(B(x, r)). Then there exists a ∈ B(x, r) such that y = f(a), and since φ: R^{n} → R^{n} is a bijection, we can find z ∈ R^{n} such that y = φ(z).

We have

|z − a| ≤ |f′(x)(z − a)|/l(f′(x)) = |φ(z) − φ(a)|/l(f′(x))
= |f(a) − (f(x) − f′(x)(x − a))|/l(f′(x))
≤ ε · |x − a|/l(f′(x)) < ε · r/l(f′(x));

hence |z − x| ≤ |x − a| + |z − a| ≤ r + rε/l(f′(x)) = rs, and this shows that z ∈ B(x, rs). We proved that y = f(a) = φ(z) ∈ φ(B(x, rs)), and since y was arbitrary in f(B(x, r)), it follows that f(B(x, r)) ⊂ φ(B(x, rs)); hence μ_{n}(f(B(x, r))) ≤ μ_{n}(φ(B(x, rs))) = μ_{n}(f′(x)(B(x, rs))) = V_{n} · (rs)^{n} · |J_{f}(x)|. From (2) we have

\begin{aligned}\frac{l(f′(x))^{n} · (|f′(x)| − ε)^{n}}{(l(f′(x)) + ε)^{n} · V_{n} · |J_{f}(x)|} & = \frac{(|f′(x)| − ε)^{n} · r^{n}}{V_{n} · (rs)^{n} · |J_{f}(x)|} \\ & ≤ \frac{L(x, f, r)^{n}}{μ_{n}(f(B(x, r)))} ≤ \frac{d(f(B(x, r)))^{n}}{μ_{n}(f(B(x, r)))} \end{aligned}

for 0 < r ≤ r_{ε} and we obtain that

(3)  \frac{(|f′(x)| − ε)^{n}}{|J_{f}(x)|} ≤ V_{n} · (\frac{l(f′(x)) + ε}{l(f′(x))})^{n} · \frac{d(f(B(x, r)))^{n}}{μ_{n}(f(B(x, r)))}

for 0 < r ≤ r_{ε}. Since lim inf_{r→0} d(f(B(x, r)))^{n}/μ_{n}(f(B(x, r))) ≤ h, we can find r_{p} → 0 such that d(f(B(x, r_{p})))^{n}/μ_{n}(f(B(x, r_{p}))) ≤ h for every p ∈ N; hence replacing r by r_{p} in (3) and letting p tend to infinite, we obtain that

\frac{(|f′(x)| − ε)^{n}}{|J_{f}(x)|} ≤ V_{n} · h · (\frac{l(f′(x)) + ε}{l(f′(x))})^{n},

and letting now ε tend to zero, we find that |f′(x)|^{n}/|J_{f}(x)| ≤ V_{n} · h. Since μ_{n}(K) = 0 (no point of K can be a point of density of K), f is a.e. differentiable in D and J_{f}(x) ≠ 0 a.e. in D, it follows that |f′(x)|^{n} ≤ V_{n} · h · |J_{f}(x)| a.e. in D and since f is ACL^{n}, it follows from [9, p. 9] that f is quasiregular on D and K_{0}(f) ≤ V_{n} · h.

Corollary 1. Let D ⊂ R^{n} be open, K ⊂ D, n ≥ 2, f: D → R^{n} continuous, open and discrete such that there exists h > 0 such that h(x, f) < h for every x ∈ D \ K, and there exists t > 0, 0 < a < 1 and ℬ = (B_{i})_{i∈N} an a-porous base of K with d(f(aB_{i}))^{n}/μ_{n}(f(B_{i})) < t for every i ∈ N. Then f is quasiregular and H(x, f) = h(x, f) < h a.e. in D.

Proof. From Theorem 10 it follows that f is quasiregular; hence f is a.e. differentiable and J_{f}(x) > 0 a.e. in D. Let x ∈ D be fixed such that f is differentiable in x and J_{f}(x) > 0 and let 0 < ε < l(f′(x)). Since f is differentiable in x, there exists r_{ε} > 0 such that |f(z) − f(x) − f′(x)(z − x)| ≤ ε · |z − x| for |z − x| ≤ r_{ε}. Then

(4)  \frac{|f′(x)| − ε}{l(f′(x)) + ε} ≤ \frac{L(x, f, r)}{l(x, f, r)} ≤ \frac{|f′(x)| + ε}{l(f′(x)) − ε}

for 0 < r ≤ r_{ε}. We can find r_{p} → 0 such that L(x, f, r_{p})/l(x, f, r_{p}) → h(x, f), and let p_{ε} ∈ N be such that 0 < r_{p} ≤ r_{ε} for p ≥ p_{ε}. From (4) we have for p ≥ p_{ε} that

\frac{|f′(x)| − ε}{l(f′(x)) + ε} ≤ \frac{L(x, f, r_{p})}{l(x, f, r_{p})} ≤ \frac{|f′(x)| + ε}{l(f′(x)) − ε},

and letting first p tend to ∞ and then letting ε tend to zero, we obtain that |f′(x)|/l(f′(x)) = h(x, f). Using (4) again, we see that

\frac{|f′(x)| − ε}{l(f′(x)) + ε} ≤ H(x, f) ≤ \frac{|f′(x)| + ε}{l(f′(x)) − ε},

and letting ε tend to zero we obtain that H(x, f) = |f′(x)|/l(f′(x)); hence H(x, f) = h(x, f). We have therefore proved that H(x, f) = h(x, f) a.e. in D.

Remark. In 1994, at a seminar held in Helsinki, J. Väisälä raised the following problem:

It is known that if f: R^{n} → R^{n} is a homeomorphism such that there exists h > 0 such that h(x, f) < h for every x ∈ R^{n}, then f is K(h, n)-quasiconformal (it is the result of J. Heinonen and P. Koskela from [7]). Find the best estimation of the constant of quasiconformality K(h, n). Corollary 1 shows that this problem is reduced to the classical problem when the inferior linear dilatation h(x, f) is replaced by the classical linear dilatation H(x, f). Indeed, we proved that H(x, f) = h(x, f) a.e. in D. Since f is ACL^{n}, hence a.e. differentiable in D, the problem of finding the best constant of quasiregularity K = K(h, n) is reduced in this way to the known case when we have that H(x, f) < H for every x ∈ D. For interesting estimations of this kind, see the papers of M. Vuorinen [1], [2], [13], [14].

As an open problem, we raise the following question: If D ⊂ R^{n} is open, n ≥ 2, f: D → R^{n} is continuous, open and discrete such that there exists h > 0 with h(x, f) < h for every x ∈ D, does it follow that H(x, f) < h for every x ∈ D? We can now prove the following generalization of a removability result of J. Heinonen and P. Koskela from [7]:

Theorem 8. Let D ⊂ R^{n} be open, n ≥ 2, f: D → R^{n} continuous, such that there exists H ⊂ D closed in D such that f is K -quasiregular on D \H and there exists 0 < a < 1, and ℬ = (B_{i})_{i∈N} , an a-porous base of H. Then, if f is open and discrete, or if int f(H) = ∅, it follows that f is K -quasiregular on D.

404 M. Cristea

Proof. We see that μ_{n}(H) = 0 and dim(H) = 0, hence f is a light map. Suppose that int f(H) = 0. Using the fact that f is quasiregular on D \ H, it follows that f is differentiable on D \ H and J_{f}(x) ≥ 0 in D \ H, and since int f(H) = ∅, we apply Theorem 10 [6] to obtain that f is open and discrete on D. Using Theorem 6, we can find a constant C(a, K) such that d(f(aB_{i}))^{n}/μ_{n}(f(B_{i})) ≤ C(a, K) for every i ∈ N. We apply now Theorem 7 to see that f is quasiregular on D. Since μ_{n}(H) = 0 and f is K -quasiregular on D \ H, it follows that f is K - quasiregular on D.

Another generalization of the removability result from [7] may be found in [8].

Acknowledgement. I wish to thank the referee for his comments.

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Received 16 September 1998