# some questions about properties of harmonic measure

The original post

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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