In order to test the latex2wp software, I’ve dug out an old file and will post it here. There might be something interesting here. But maybe not.
1. Kashin’s Inequality
Kashin’s Inequality: Let be a Lipschitz function, i.e., there is a constant
such that
Then,
Note that the left hand-side is also known as the Besov semi-norm . The constant
is probably not sharp. However, the function
shows that it’s at least
.
Proof: Use the Lipschitz hypothesis on and integrate in
first to get
Likewise, on we integrate in
first.
Thus,
Now we need to consider . Majorize
by
and split into two integrals. Then, considering
first, change variables with
, and integrate in
for
to get
Likewise, for use the lower bound
instead and integrate in
first.
2. Complex variables approach
Here we follow the lead of Theorem 2-5 page 32 of Ahlfors’s book Conformal Invariants (hat tip to Don Marshall for this reference).
Given , Lipschitz with constant
, let its harmonic extension to the upper half-plane
be:
Then there is analytic on
, which is given by the following formula
To see this one checks that
Consider domains
Now , where
and
.
Note first that
and that
for fixed, as
.
On the other hand, by the residue theorem,
So, for ,
Moreover, Cauchy’s theorem (on similar contours) shows that, for fixed ,
Hence,
Expanding the square we therefore see that
By Lebesgue dominated convergence we can let tend to zero and get that
The middle integral needs justification, e.g., it is ok if .
In particular, we get
3. Remarks
- The Besov semi-norm above can be interpreted as a Dirichlet energy by changing variables:
and, at least for smooth functions,
where the kernel
is quite similar to the Poisson kernel.
- By Plancherel for the Hilbert transform, we have that the Dirichlet integral of
- Green’s theorem applied to the function
and
, and the fact that
for harmonic functions, gives:
where we have skipped many technical details and
is a harmonic conjugate of
. So we also have
However, the Hilbert transform sends
to
so it doesn’t seem possible to replace
by
this way.
- Finally, remark that a simple computation shows
I’m not sure why all the Latex formulas are boxed.
Ok I just changed the theme and the boxes disappeared.