LIPSCHITZ AND LOWER SEMI-CONTINUOUS FUNCTIONS

KEVIN ROGOVIN

Abstract. We show that for a complete geodisic metric space X, given a bounded lower semi-continuous function ρ there exists a Lipschitz function f for which lip(f) ≥ ρ pointwise.

1. Preliminaries and Notation

We write the distance between two points x, y ∈ X where X is a metric space as |x − y|. For a function f : X → ℝ we define:

lip(f)(x) := lim inf_{r→0} sup_{|x−y|≤r} \frac{|f(x) − f(y)|}{r}

and

Lip(f)(x) := lim sup_{r→0} sup_{|x−y|≤r} \frac{|f(x) − f(y)|}{r}.

For a pair of points x, y ∈ X we set Γ_{x,y} as the set of all rectifiable curves joining x to y.

2. Main Theorem

Theorem 2.1. Let X be a proper geodesic matric space and ρ : X → [0, M] be lower semi-continuos, then the for each x_{0}, the function

f(x) := inf_{γ∈Γ_{x_{0},x}} ∫_{γ} ρ

is M-Lipschitz and lip(f) ≥ ρ pointwise.

Proof. That X is geodesic immediately implies that f is finite valued at every point in X and is M-Lipschitz since ρ is bounded from above by M. For each ϵ > 0, let

f_{ϵ}(x) := inf_{γ∈Γ_{x_{0},x}} ∫_{γ}(ϵ + ρ).

Then each f_{ϵ} is an M + ϵ Lipschitz-function. We first give two lemmas which will be used in the proof of the theorem.

Lemma 2.2. For each ϵ > 0 and x ≠ x_{0}, there exists γ ∈ Γ_{x_{0},x} so that

f_{ϵ}(x) = ∫_{γ}(ϵ + ρ).

Proof. Let γ_{n} ∈ Γ_{x_{0},x} so that

f_{ϵ}(x) = lim_{n→∞} ∫_{γ_{n}}(ϵ + ρ).

We then see that for n large,

l(γ_{n}) ≤ \frac{1}{ϵ}f_{ϵ}(x) + 1.

Thus there exist a curve γ ∈ Γ_{x_{0},x} and a subsequece {γ_{n_{j}}} so that γ_{n_{j}} → γ uniformly. Hence, since ρ is lower semi-continuous,

f_{ϵ}(x) ≤ ∫_{γ} ρ ≤ lim inf_{j→∞} ∫_{γ_{n_{j}}} (ϵ + ρ) = f_{ϵ}(x)

as needed. □

Lemma 2.3. f_{ϵ} ↓ f as ϵ ↓ 0 uniformly on compact sets.

Proof. Clearly from the definitions of f and f_{ϵ}, we see that f ≤ f_{ϵ} and f_{ϵ_{1}} ≤ f_{ϵ_{2}} whenever ϵ_{1} ≤ ϵ_{2}. Fix x ∈ X and for each δ > 0 let γ_{δ} ∈ Γ_{x_{0},x} so that

f(x) ≥ −δ + ∫_{γ_{δ}} ρ.

Then whenver ϵ < \frac{δ}{1+l(γ_{δ})},

f_{ϵ}(x) ≤ ∫_{γ_{δ}}(ϵ + ρ) ≤ δ + ∫_{γ_{δ}} ρ ≤ 2δ + f(x).

Hence,

lim_{ϵ↓0} f_{ϵ}(x) ≤ 2δ + f(x)

for all δ > 0. Letting δ → 0 shows us that f_{ϵ} ↓ f pointwise. However, since f is continuous (it is in fact Lipschitz), and so are each of the f_{ϵ} we can apply Dini’s lemma to conclude that f_{ϵ} ↓ f as ϵ ↓ 0 uniformly on compact sets. □

With those two lemmas proven we are now ready to prove the theorem. Fix x ∈ X. Since ρ is lower semi-continuous, for each δ > 0 we an find an open set W_{δ} containing x so that ρ ≥ ρ(x) − δ on W_{δ}. By Lemma 2.2, for each ϵ > 0 there γ_{ϵ} ∈ Γ_{x,x_{0}} so that

f_{ϵ}(x) = ∫_{γ_{ϵ}}(ϵ + ρ).

We will parameterize γ_{ϵ} by arclength so that γ_{ϵ}(0) = x and γ_{ϵ}(l(γ_{ϵ})) = x_{0}. From the definition of f_{ϵ} we see that

f_{ϵ}(x) − f_{ϵ}(γ_{ϵ}(t)) = ∫_{t=0}^{l(γ_{ϵ})}(ϵ + ρ ∘ γ_{ϵ}(t))dt.

Thus whenever B(x, r_{0}) ⊆ W_{δ} we have for each ϵ > 0 and 0 < r < r_{0},

f_{ϵ}(x) − f_{ϵ}(γ_{ϵ}(r)) ≥ (ρ(x) + ϵ − δ) · r.

Since f_{ϵ} ↓ f uniformly on compact sets and X is locally compact we can choose r_{0} > 0 so that B(x, r_{0}) ⊆ W_{δ} and f_{ϵ} ↓ f uniformly on B(x, r_{0}). Now,

lip(f)(x) := lim inf_{r→0} sup_{|x−y|≤r} \frac{|f(x) − f(y)|}{r}
≥ lim inf_{r→0} lim sup_{ϵ↓0} \frac{|f(x) − f(γ_{ϵ}(r))|}{r}
≥ lim inf_{r→0} \frac{1}{r} lim sup_{ϵ↓0} (|f_{ϵ}(x) − f_{ϵ}(γ_{ϵ}(r))| − sup_{y∈B(x,r_{0})} |f(y) − f_{ϵ}(y)|)
≥ lim inf_{r→0} \frac{1}{r} lim sup_{ϵ↓0}(ρ(x) + ϵ − δ) · r
= ρ(x) − δ.

Letting δ ↓ 0 completes the proof of the theorem. □

If X is as in Theorem 2.1 and ρ : X → [0, M] is upper semi-continuous, it is a straightforward exercise to see that f is M-Lipschitz with Lip(f)(x) ≤ ρ(x) pointwise. Hence we can conclude the following:

Theorem 2.4. Let X be a proper geodesic matric space and ρ : X → [0, M] be continuos, then the for each x_{0}, the function

f(x) := inf_{γ∈Γ_{x_{0},x}} ∫_{γ} ρ

is M-Lipschitz and Lip(f) =lip(f) = ρ pointwise.

University of Jyväskylä

Department of Mathematical Information Technology

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