Harmonic Functions

Suppose $u$ and $v$ are real harmonic functions in a plane region $\Omega$. Under what conditions is $u v$ harmonic? (Note that the answer depends strongly on the fact that the question is one about real functions). Show that $u^{2}$ cannot be harmonic in $\Omega$, unless $u$ is constant. For which $f \in H(\Omega )$ is $\left|f\right|^{2}$ harmonic?

Suppose $f$ is a complex function in a region $\Omega$, and both $f$ and $f^{2}$ are harmonic in $\Omega$. Prove that either $f$ or $\bar{f}$ is holomorphic in $\Omega$.

If $u$ is a harmonic function in a region $\Omega$, what can you say about the set of points at which the gradient of $u$ is 0? (This is the set on which $u_{x} = u_{y} = 0$).

Prove that every partial derivative of every harmonic function is harmonic. Verify, by direct computation, that $P_{r}(\theta -t)$ is, for each fixed $t$, a harmonic function of $r e^{i \theta }$. Deduce (without referring to holomorphic functions) that the Poisson integral $P[d \mu ]$ of every finite Borel measure $\mu$ on $T$ is harmonic in $U$, by showing that every partial derivative of $P[d \mu ]$ is equal to the integral of the corresponding partial derivative of the kernel.

Suppose $f \in H(\Omega )$ and $f$ has no zero in $\Omega$. Prove that $\log \left|f\right|$ is harmonic in $\Omega$, by computing its Laplacian. Is there an easier way?

Suppose $f \in H(U)$, where $U$ is the open unit disc, $f$ is one-to-one in $U, \Omega = f(U)$, and $f(z) = \sum c_{n} z^{n}$. Prove that the area of $\Omega$ is

Hint: The Jacobian of $f$ is $\left|f^{\prime }\right|^{2}$.

Suppose $\Omega$ is a region, $f_{n} \in H(\Omega )$ for $n = 1,2,3, \ldots , u_{n}$ is the real part of $f_{n},\left\{ u_{n}\right\}$ converges uniformly on compact subsets of $\Omega$, and $\left\{ f_{n}(z)\right\}$ converges for at least one $z \in \Omega$. Prove that then $\left\{ f_{n}\right\}$ converges uniformly on compact subsets of $\Omega$.

Suppose $u$ is a Lebesgue measurable function in a region $\Omega$, and $u$ is locally in $L^{1}$. This means that the integral of $\left|u\right|$ over any compact subset of $\Omega$ is finite. Prove that $u$ is harmonic if it satisfies the following form of the mean value property:

whenever $\bar{D}(a ; r) \subset \Omega$.

Suppose $I = [a, b]$ is an interval on the real axis, $\varphi$ is a continuous function on $I$, and

Show that

exists for every real $x$, and find it in terms of $\varphi$. How is the result affected if we assume merely that $\varphi \in L^{1}$? What happens then at points $x$ at which $\varphi$ has right- and left-hand limits?

Suppose that $I = [a, b], \Omega$ is a region, $I \subset \Omega , f$ is continuous in $\Omega$, and $f \in H(\Omega -I)$. Prove that actually $f \in H(\Omega )$.

Replace $I$ by some other sets for which the same conclusion can be drawn.

(Harnack's Inequalities) Suppose $\Omega$ is a region, $K$ is a compact subset of $\Omega , z_{0} \in \Omega$. Prove that there exist positive numbers $\alpha$ and $\beta$ (depending on $z_{0}, K$, and $\Omega$) such that

for every positive harmonic function $u$ in $\Omega$ and for all $z \in K$.

If $\left\{ u_{n}\right\}$ is a sequence of positive harmonic functions in $\Omega$ and if $u_{n}\left(z_{0}\right) \rightarrow 0$, describe the behavior of $\left\{ u_{n}\right\}$ in the rest of $\Omega$. Do the same if $u_{n}\left(z_{0}\right) \rightarrow \infty$. Show that the assumed positivity of $\left\{ u_{n}\right\}$ is essential for these results.

Suppose $u$ is a positive harmonic function in $U$ and $u(0) = 1$. How large can $u\left(\frac{1}{2}\right)$ be? How small? Get the best possible bounds.

For which pairs of lines $L_{1}, L_{2}$ do there exist real functions, harmonic in the whole plane, that are 0 at all points of $L_{1} \cup L_{2}$ without vanishing identically?

Suppose $u$ is a positive harmonic function in $U$, and $u\left(r e^{i \theta }\right) \rightarrow 0$ as $r \rightarrow 1$, for every $e^{i \theta } \neq 1$. Prove that there is a constant $c$ such that

Here is an example of a harmonic function in $U$ which is not identically 0 but all of whose radial limits are 0 :

Prove that this $u$ is not the Poisson integral of any measure on $T$ and that it is not the difference of two positive harmonic functions in $U$.

Let $\Phi$ be the set of all positive harmonic functions $u$ in $U$ such that $u(0) = 1$. Show that $\Phi$ is a convex set and find the extreme points of $\Phi$. (A point $x$ in a convex set $\Phi$ is called an extreme point of $\Phi$ if $x$ lies on no segment both of whose end points lie in $\Phi$ and are different from $x$). Hint: If $C$ is the convex set whose members are the positive Borel measures on $T$, of total variation 1, show that the extreme points of $C$ are precisely those $\mu \in C$ whose supports consist of only one point of $T$.

Let $X^{*}$ be the dual space of the Banach space $X$. A sequence $\left\{ \Lambda_{n}\right\}$ in $X^{*}$ is said to converge weakly to $\Lambda \in X^{*}$ if $\Lambda_{n} x \rightarrow \Lambda x$ as $n \rightarrow \infty$, for every $x \in X$. Note that $\Lambda_{n} \rightarrow \Lambda$ weakly whenever $\Lambda_{n} \rightarrow \Lambda$ in the norm of $X^{*}$. (See Exercise 8, Chap. 5). The converse need not be true. For example, the functionals $f \rightarrow \hat{f}(n)$ on $L^{2}(T)$ tend to 0 weakly (by the Bessel inequality), but each of these functionals has norm 1.

Prove that $\left\{ \left\| \Lambda_{n}\right\| \right\}$ must be bounded if $\left\{ \Lambda_{n}\right\}$ converges weakly.

Suppose $E \subset T, m(E) = 0$. Prove that there is an $f \in H^{\infty }$, with $f(0) = 1$, that has

at every $e^{i \theta } \in$ E. Suggestion: Find a lower semicontinuous $\psi \in L^{1}(T), \psi > 0, \psi = +\infty$ at every point of $E$. There is a holomorphic $g$ whose real part is $P[\psi ]$. Let $f = 1 / g$.

Define $f \in H(U)$ and $g \in H(U)$ by $f(z) = \exp \left\{ (1+z) /(1-z)\right\} , g(z) = (1-z) \exp \left\{ -f(z)\right\}$. Prove that

exists at every $e^{i \theta } \in T$, that $g^{*} \in C(T)$, but that $g$ is not in $H^{\infty }$.

Suggestion: Fix $s$, put

For certain values of $s,\left|g\left(z_{t}\right)\right| \rightarrow \infty$ as $t \rightarrow \infty$.

Suppose $u$ is harmonic in $U$, and $\left\{ u_{r}: 0 \leq r < 1\right\}$ is a uniformly integrable subset of $L^{1}(T)$. (See Exercise 10, Chap. 6). Modify the proof of Theorem 11.30 to show that $u = P[f]$ for some $f \in L^{1}(T)$.

Put $\theta_{n} = 2^{-n}$ and define

for $z \in U$. Show that $u$ is the Poisson integral of a measure on $T$, that $u(x) = 0$ if $-1 < x < 1$, but that

is unbounded, as $\epsilon$ decreases to 0. (Thus $u$ has a radial limit, but no nontangential limit, at 1). Hint: If $\epsilon = \sin \theta$ is small and $z = 1-\epsilon +i \epsilon$, then

Let $D_{n}(t)$ be the Dirichlet kernel, as in Sec. 5.11, define the Fejér kernel by

put $L_{N}(t) = \min \left(N, \pi^{2} / N t^{2}\right)$. Prove that

and that $\int_{T} L_{N} d \sigma \leq 2$. Use this to prove that the arithmetic means

of the partial sums $s_{n}$ of the Fourier series of a function $f \in L^{1}(T)$ converge to $f\left(e^{i \theta }\right)$ at every Lebesgue point of $f$. (Show that $\sup \left|\sigma_{N}\right|$ is dominated by $M f$, then proceed as in the proof of Theorem 11.23).

If $1 \leq p \leq \infty$ and $f \in L^{p}\left(\mathbb {R}^{1}\right)$, prove that $\left(f * h_{\lambda }\right)(x)$ is a harmonic function of $x+i \lambda$ in the upper half plane. ($h_{\lambda }$ is defined in Sec. 9.7 ; it is the Poisson kernel for the half plane).