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The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" http://www.math.msu.edu/~fedja/Published/paper.ps where he proves a (weak type) Bernstein Inequality using certain properties of `harmonic measure'.

I am not familiar with harmonic measure (I have checked wiki and got hold of a mammoth book by koosis (logarithmic integral) which has a section about harmonic measure). What I am looking for is the statements of theorems and principles about harmonic measures which are being used in the following argument.

Let $h(\xi)$ be the harmonic measure of the set $\mathbb R$ \ $[-y,y]$ with respect to the upper half-plane and a point $\xi \in \mathbb C_+$.

Let $z_1, z_2, \dots, z_{n_1}$ be such that $ Im (z_j) \leq 0$, and define $\sum_1(z):=\sum_{j=1}^{n} \frac{1}{z-z_j}$.

Define $u(z) := h(-\sum_1(z))$.

The function $u(z)$ is harmonic in $C_+$, $0\leq u(z) \leq 1$, $u(it) \lim_{t\rightarrow + \infty} 0$, and $u(z) \geq 1/2$ if $|\sum_1(z)| \geq y$ (the latter fact follows from the geometric description of the harmonic measure as a ratio to $\pi$ of the angle at which a subset of $\mathbb R$ is seen from the point $\xi$).

Moreover, we have

$$\lim_{t\rightarrow +\infty} \pi t u(it) =\int_{\mathbb R} u(x) dx \ \ (Why?) $$
$$\geq \frac{1}{2} \mu \{ x \in \mathbb R : |\sum_1(x)|>y \}.$$

On the other hand, an easy computation shows that

$\lim_{t\rightarrow \infty} \pi t u(it) =\lim_{t \rightarrow +\infty} \pi t h(\iota n/t + O (1/t^2)) = 2n/y.$

(As you can see by the end of this the author has obtained a weak type Bernstein Inequality).

Thankyou for your time and patience.


Question


The only place I am stuck now is the place which I have highlighted(see above). My guess is it is some version of mean value property of harmonic function. So, one has to extend the harmonic function at infinity then ? It should be possible, as in this case $u(it)->0$ as $t -> \infty$. Is it so ? Can someone suggest a reference ?

Any suggestion?

I would like to know if there are some lecture notes about harmonic measures available which is self contained and fits the category `every analysis student must know'. The wiki article is not very helpful there are many treatise available but I couldnot find an exposition which introduces the concept and its importance.

Afterthought


I now realise the first part of the argument is quite easy, they simply follows from properties of harmonic functions (like maximum modulous principle, composition of harmonic functions etc. ) and the geometric description mentioned within quotes about harmonic measure of the set $\mathbb R \setminus [−y,y].$

For example to check that $u(z)\geq 1/2$ iff $\sum_1(z) \geq y$ just draw a semicircle of radius $y$ centered at $0$ and observe that for any point which is outside it the angle (which is the harmonic measure !!) is more than $\frac{\pi}{2}$.

The geometric description is easy to obtain:- The harmonic measure of an interval $[a,b]$ is simply the harmonic extension of $\chi_[a,b]$ on the upper half plane, so

$\int_a^b P_y(x-t) dt = \frac{1}{\pi} \int_a^b \frac{y}{(t-x)^2+y^2} = \int_a^b \frac{1}{\pi} Im(\frac{1}{t-z})= \frac{1}{\pi} Im (log (\frac{b-z}{a-z}) )$

Using limiting argument one can find harmonic measure of $\chi_{\mathbb R} = 1$ and hence get the geometric description of harmonic measure of $\mathbb R \setminus [-y,y]$.


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