Poisson integrals and boundary behavior

Suppose $a, b \in \mathbb {C}, f \in L^{1}(\partial \mathbf{D}), z \in \partial \mathbf{D}, \lim_{t \downarrow 0} f\left(e^{i t} z\right) = a$, and $\lim_{t \uparrow 0} f\left(e^{i t} z\right) = b$. Prove that

Suppose $\zeta \in \partial \mathbf{D}$. Show that the function

is harmonic on $\mathbb {C} \backslash \left\{ \zeta \right\}$ by finding an analytic function on $\mathbb {C} \backslash \left\{ \zeta \right\}$ whose real part is the function above.

Suppose $f: \partial \mathbf{D} \rightarrow \mathbb {R}$ is the function defined by

for $(x, y) \in \mathbb {R}^{2}$ with $x^{2}+y^{2} = 1$. Find a polynomial $u$ of two variables $x, y$ such that $u$ is harmonic on $\mathbb {R}^{2}$ and $\left.u\right|_{\partial \mathbf{D}} = f$.

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 $u$ 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 $T$.

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

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).

Suppose $0 < p \leq \infty$ and $f \in H(U)$. Prove that $f \in H^{p}$ if and only if there is a harmonic function $u$ in $U$ such that $\left|f(z)\right|^{p} \leq u(z)$ for all $z \in U$. Prove that if there is one such harmonic majorant $u$ of $\left|f\right|^{p}$, then there is at least one, say $u_{f}$. (Explicitly, $\left|f\right|^{p} \leq u_{f}$ and $u_{f}$ is harmonic; and if $\left|f\right|^{p} \leq u$ and $u$ is harmonic, then $u_{f} \leq u$). Prove that $\left\| f\right\|_{p} = u_{f}(0)^{1 / p}$. Hint: Consider the harmonic functions in $D(0 ; R)$, $R < 1$, with boundary values $\left|f\right|^{p}$, and let $R \rightarrow 1$.

If $f \in H^{1}$ and $f^{*} \in L^{p}(T)$, prove that $f \in H^{p}$.

Suppose $f \in H(U)$ and $f(U)$ is not dense in the plane. Prove that $f$ has finite radial limits at almost all points of $T$.

Let $\varphi$ be a nonconstant inner function with no zero in $U$.