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.