Re: extension theoems of lipschitz function.

On 6 May 2007 03:36:23 -0700, Nik <nikita2.evseev@xxxxxxxxx> wrote:

I've read Mcshine-Whitne & Kirszbraun extension theorems (and them
proof).

For example,

http://en.wikipedia.org/wiki/Kirszbraun_theorem

But now I want to understand their importance and application.
Can you tell me?

I'm not familiar with a specific application, but it seems
clear that theorems like this are going to be useful, because
there are a lot of results about and characterizations of
globally defined Lipschitz functions - the theorem allows
you to apply these results to functions defined in a subset
of R^n or your Hilbert space or whatever. Seems pretty clear
this is going to be useful in a lot of places.

Ok. Never having been very scholarly, the only specific
example of an application of something analogous that
springs to mind comes from my own work some years ago:

Say D is a subset of R^n and f : D -> R (or f : D -> C
or whatever). We say f is a Zygmund function if there
exists c such that

|f(x-h) - 2 f(x) - f(x+h)| <= c |h|

whenever x, x+h and x-h all lie in D. (So it's
clear that Lipschitz functions are Zygmund functions,
but the Zygmund class is much larger - it also
often works much better, one reason for which is
that it's a Besov space while Lip_1 is not.)

Now suppose that K is a compact subset of R^n
and let D = R^n \ K. Inventing some temporary
terminology, say that K is strongly removable
if every harmonic Zygmund function in D extends
to a harmonic (Zygmund) function in R^n, and
say that K is weakly removable if every Zygmund
function in R^n which happens to be harmonic
in D is actually harmonic in R^n. The problem
is to characterize removable sets K.

Say m_alpha is Hausdorff measure.

Thm 1. K is strongly removable if and only
if m_{n-1}(K) = 0.

The proof used Thm 2 plus an extension
result analogous to the theorems you mention,
where Thm 2 says that K is weakly
removable if and only if m_{n-1}(K) = 0.

(Carleson proved analogous results for
Lip_alpha, 0 < alpha < 1, but didn't get the
case alpha = 1. Neither did I - the corresponding
question for Lip_1 seems very difficult (it
turns out that removability for Lip_1 is
_not_ equivalent to m_{n-1}(K) = 0). But
I noticed that if 0 < alpha < 1 then Lip_alpha
is equal to a Besov space Lambda_alpha;
on the other hand Lip_1 is not the same as
Lambda_1, in fact Lambda_1 is the Zygmund
class. So it's natural to ask the question
for Lambda_1 = Zyg instead, and _that_
question is easier to answer. When something
works for Lip_alpha, 0 < alpha < 1, but
not for Lip_1, it often turns out that
it does work for the Zygmund class; Krantz
goes almost so far as to claim that the Zygmund
class is what Lip_1 should have been defined
to be...)


************************

David C. Ullrich
.