Sublinear Higson Corona and Lipschitz Extensions

The purpose of the paper is to characterize the dimension
of sublinear Higson corona \nu_L(X) of X
in terms of Lipschitz extensions of functions:

Theorem: Suppose (X,d) is a proper metric space.
The dimension of the sublinear Higson corona \nu_L(X) of X
is the smallest integer m>=0 with the following property:
Any norm-preserving asymptotically Lipschitz function f': A -> R^{m+1}, 
A subset of X, extends to a norm-preserving asymptotically Lipschitz
function g': X -> R^{m+1}.

One should compare it to the result of Dranishnikov who characterized 
the dimension of the Higson corona \nu(X) of X is the smallest integer 
n>=0 such that R^{n+1} is an absolute extensor of X in the asymptotic 
category A (that means any proper asymptotically Lipschitz function 
f: A -> R^{n+1}, A closed in X, extends to a proper asymptotically 
Lipschitz function f': X -> R^{n+1}).

Dranishnikov introduced the category \tilde{A} whose objects
are pointed proper metric spaces X and morphisms are asymptotically Lipschitz
functions f: X -> Y such that there are constants b,c > 0 satisfying
|f(x)| >= c |x|-b for all x in X. We show dim(\nu_L(X)) <= n
if and only if R^{n+1} is an absolute extensor of X in the category \tilde{A}.

As an application we reprove the following result of Dranishnikov and Smith:

Theorem: Suppose (X,d) is a proper metric space of finite asymptotic Assouad-Nagata
dimension asdim_{AN}(X). If X is cocompact and connected, then asdim_{AN}(X)
equals the dimension of the sublinear Higson corona \nu_L(X) of X.