$H^p$ Spaces

Prove Theorems 17.4 and 17.5 for upper semicontinuous subharmonic functions.

Assume $f \in H(\Omega )$ and prove that $\log (1+\left|f\right|)$ is subharmonic in $\Omega$.

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 $R \rightarrow 1$.

Prove likewise that $f \in N$ if and only if $\log^{+}\left|f\right|$ has a harmonic majorant in $U$.

Suppose $f \in H^{p}, \varphi \in H(U)$, and $\varphi (U) \subset U$. Does it follow that $f \circ \varphi \in H^{p}$? Answer the same question with $N$ in place of $H^{p}$.

If $0 < r < s \leq \infty$, show that $H^{s}$ is a proper subclass of $H^{r}$.

Show that $H^{\infty }$ is a proper subclass of the intersection of all $H^{p}$ with $p < \infty$.

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 $T$.

Fix $\alpha \in U$. Prove that the mapping $f \rightarrow f(\alpha )$ is a bounded linear functional on $H^{2}$. Since $H^{2}$ is a Hilbert space, this functional can be represented as an inner product with some $g \in H^{2}$. Find this $g$.

Fix $\alpha \in U$. How large can $\left|f^{\prime }(\alpha )\right|$ be if $\left\| f\right\|_{2} \leq 1$? Find the extremal functions. Do the same for $f^{(n)}(\alpha )$.

Suppose $p \geq 1, f \in H^{p}$, and $f^{*}$ is real a.e. on $T$. Prove that $f$ is then constant. Show that this result is false for every $p < 1$.

Suppose $f \in H(U)$, and suppose there exists an $M < \infty$ such that $f$ maps every circle of radius $r < 1$ and center 0 onto a curve $\gamma_{r}$ whose length is at most $M$. Prove that $f$ has a continuous extension to $\bar{U}$ and that the restriction of $f$ to $T$ is absolutely continuous.

Suppose $\mu$ is a complex Borel measure on $T$ such that

Prove that then either $\mu = 0$ or the support of $\mu$ is all of $T$.

Suppose $K$ is a proper compact subset of the unit circle $T$. Prove that every continuous function on $K$ can be uniformly approximated on $K$ by polynomials. Hint: Use Exercise 14.

Complete the proof of Theorem 17.17 for the case $0 < p < 1$.

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

Suppose $\varphi$ is a nonconstant inner function, $\left|\alpha \right| < 1$, and $\alpha \notin \varphi (U)$. Prove that $\lim_{r \rightarrow 1} \varphi \left(r e^{i \theta }\right) = \alpha$ for at least one $e^{i \theta } \in T$.

Suppose $f \in H^{1}$ and $1 / f \in H^{1}$. Prove that $f$ is then an outer function.

Suppose $f \in H^{1}$ and $\mathrm{Re}\left(f(z)\right) > 0$ for all $z \in U$. Prove that $f$ is an outer function.

Prove that $f \in N$ if and only if $f = g / h$, where $g$ and $h \in H^{\infty }$ and $h$ has no zero in $U$.

Prove the following converse of Theorem 15.24: If $f \in H(U)$ and if

then $f$ is a Blaschke product. Hint: $\left({ }^{*}\right)$ implies

Since $\log^{+}\left|f\right| \geq 0$, it follows from Theorems 17.3 and 17.5 that $\log^{+}\left|f\right| = 0$, so $\left|f\right| \leq 1$. Now $f = B g, g$ has no zeros, $\left|g\right| \leq 1$, and $\left({ }^{*}\right)$ holds with $1 / g$ in place of $f$. By the first argument, $\left|1 / g\right| \leq 1$. Hence $\left|g\right| = 1$.

Find the conditions mentioned in Sec. 17.22.

Section 17.22 (Remark): Let $S$ be the shift operator on $H^{2}$, $(S f)(z) = z f(z)$ $(f \in H^{2}, z \in U)$; a closed subspace $Y$ of $H^{2}$ with $S(Y) \subset Y$ is called $S$-invariant. By Beurling's theorem (Theorem 17.21), every closed $S$-invariant subspace of $H^{2}$ other than $\left\{ 0\right\}$ is of the form $\varphi H^{2} = \left\{ \varphi f: f \in H^{2}\right\}$ for an inner function $\varphi$, and $\varphi$ is determined by the subspace up to a constant factor; by Theorem 17.15, every inner function has the form

where $c$ is a constant, $\left|c\right| = 1$, $B$ is a Blaschke product, and $\mu$ is a finite positive Borel measure on $T$ which is singular with respect to Lebesgue measure. Combining these, the $S$-invariant subspaces of $H^{2}$ are characterized by the following data: a sequence of complex numbers $\left\{ \alpha_{n}\right\}$ (possibly finite, or even empty) such that $\left|\alpha_{n}\right| < 1$ and $\Sigma \left(1-\left|\alpha_{n}\right|\right) < \infty$ (the zeros of $B$), and a positive Borel measure $\mu$ on $T$, singular with respect to Lebesgue measure. The book states that it is easy to find conditions, in terms of $\left\{ \alpha_{n}\right\}$ and $\mu$, which ensure that one $S$-invariant subspace of $H^{2}$ contains another.

The conformal mapping of $U$ onto a vertical strip shows that M. Riesz's theorem on conjugate functions cannot be extended to $p = \infty$. Deduce that it cannot be extended to $p = 1$ either.

Suppose $1 < p < \infty$, and associate with each $f \in L^{p}(T)$ its Fourier coefficients

Deduce the following statements from Theorem 17.26:

Show that there is a much simpler proof of Theorem 17.26 if $p = 2$, and find the best value of $A_{2}$.

Suppose $f(z) = \sum_{0}^{\infty } a_{n} z^{n}$ in $U$ and $\sum \left|a_{n}\right| < \infty$. Prove that

for all $\theta$.

Prove that the following statements are correct if $\left\{ n_{k}\right\}$ is a sequence of positive integers which tends to $\infty$ sufficiently rapidly. If

then $\left|f^{\prime }(z)\right| > n_{k} /(10 k)$ for all $z$ such that

Hence

for every $\theta$, although

exists (and is finite) for almost all $\theta$. Interpret this geometrically, in terms of the lengths of the images under $f$ of the radii in $U$.

Use Theorem 17.11 to obtain the following characterization of the boundary values of $\mathrm{H}^{p^{-}}$ functions, for $1 \leq p \leq \infty$ :

A function $g \in L^{p}(T)$ is $f^{*}$ (a.e). for some $f \in H^{p}$ if and only if

for all negative integers $n$.