Abstract:
We show a discussion of the following questions: for a given real –valued
function
(i)
(
f ∈ ∞ Rn
L
)
there can not be
bi-Lipschitz homeomorphism
determinant, det
Dφ f
=
φR
:
n
→n
R
such that the Jacobian
,
(ii) a Lipschitz or quasiconformal vector field with div
(iii) for a given separated net
except for
n=
1
y⊂ n
R
a bi-Lipschitz map
u =f
φy
:
,
→n
Z
or if the Lipschitz condition is relaxed to a
Hölder condition.
We show also an extent to certain metric measure spaces, a
generalization of the theorem of Rademacher which asserts that : a
real-valued Lipschitz function on
Rn
is differentiable almost
everywhere with respect to Lebesgue measure and the blow ups of a
real-valued Lipschitz function converge to a unique linear function.
