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\begin{document}
\title[Relative Angular Derivatives]{Relative Angular Derivatives}
\author{Jonathan E. Shapiro}
\address{Mathematics Department, California Polytechnic State University, San Luis
Obispo, CA 93407}
\email{jshapiro@calpoly.edu}
\date{}
\subjclass{Primary 46E22, 46E30}
\keywords{angular derivative, Hardy space, Aleksandrov measure, de Branges-Rovnyak space}
\maketitle
\begin{abstract}We generalize the notion of the angular derivative of a holomorphic self-map,
$b$, of the unit disk by replacing the usual difference quotient
$\frac{b(z)-b(z_{0})}{z-z_{0}}$ with a difference quotient relative to an
inner function $u$, $\frac{1-b(z)}{1-u(z)}$. We relate properties of this
generalized difference quotient to properties of the Aleksandrov measures
associated with the functions $b$ and $u$. Six conditions are shown to be
equivalent to each other, and these are used to define the notion of a
relative angular derivative. We see that this generalized derivative can be
used to reproduce some known results about ordinary angular derivatives, and
the generalization is shown to obey a form of the product rule.
\end{abstract}
\section{Introduction}
In this paper, we will define and analyze the notion of an angular derivative
of a holomorphic self-map of the unit disk relative to a nonconstant inner function.
Let $b$ be a holomorphic self-map of the unit disk, that is, an analytic
function on the unit disk $\mathbb{D}$ of the complex plane with $|b|<1$ on
$\mathbb{D}$. We will take $u$ to be our nonconstant inner function --- a
holomorphic function on $\mathbb{D}$ with $|u|=1$ almost everywhere on
${\partial\mathbb{D}}$. This notation will remain fixed.
Our analysis, and even our definition, of relative angular derivatives will
come primarily from the viewpoint of the Aleksandrov measures $\mu_{\lambda}$
and $\nu_{\lambda}$ ($\lambda\in\partial\mathbb{D}$), which we derive from our
functions $b$ and $u$. These measures are defined and discussed in Section
\ref{aleksandrovsec}. Throughout this paper, we will use $m$ to denote the
usual normalized Lebesgue measure on the unit circle. We will also use the
notation $\mu^{a.c.}$ and $\mu^{s}$ to denote the absolutely continuous and
singular parts of the measure $\mu$ ($=\mu_{1}$) with respect to $m$. For any
function $f$ on the unit disk $\mathbb{D}$, $f_{r}$ will denote the function
on the boundary ${\partial\mathbb{D}}$ such that $f_{r}(e^{i\theta
})=f(re^{i\theta})$ for $r<1$.
The relationship between Aleksandrov measures and angular derivatives has been
developed by many people recently. The most direct connection comes from
\begin{theorem}
The function $b$ has angular derivative at a point $z_{0}\in\partial
\mathbb{D}$ (where $\left| b(z_{0})\right| =$ $1$) exactly where its
corresponding Aleksandrov measure $\mu_{\lambda}$ has an atom, and
$\mu_{\lambda}(\left\{ z_{0}\right\} )=1/\left| b^{\prime}(z_{0})\right| .$
\end{theorem}
This theorem, which appears (somewhat hidden) in \cite[Ch. 7]{SDF} (see also
\cite[VI-7]{DS}), is discussed and even given a different proof in this paper
in Section \ref{scsec}. Indeed, Aleksandrov measures have even been used to
provide improvements on angular derivative conditions in several theorems
about composition operators. For example, Joel Shapiro and P. Taylor showed in
\cite{JHSPDT} that if $b$ has an angular derivative, then the corresponding
composition operator $C_{b}$ acting on the Hardy space $H^{2}$ is not compact.
This result was generalized in papers by D. Sarason \cite{DS2} and Shapiro and
C. Sundberg \cite{JHSCS}, which together prove that a composition operator
$C_{b}$ is compact on $H^{2}$ if an only if the corresponding Aleksandrov
measures $\mu_{\lambda}$ are all absolutely continuous, i.e., have not only no
atoms (which is equivalent to no angular derivative for $b$) but no component
singular with respect to Lebesgue measure. This theorem was proved in a
different way by J. Cima and A. Matheson \cite{CM}, who show that the square
of the essential norm of $C_{b}$ (operating on $H^{2}$) is equal to
$\sup_{\lambda\in\partial\mathbb{D}}\left\| \mu_{\lambda}^{s}\right\| $. The
author has also used Aleksandrov measures to generalize other theorems which
give properties of composition operators and composition operator differences
in terms of angular derivatives --- see \cite{JES}. In the language of this
paper, we will be able to restate the compactness condition for composition
operators as: $C_{b}$ is compact on $H^{2}$ if an only if for any $\zeta
\in\partial\mathbb{D}$, the function $\overline{\zeta}b$ has no angular
derivative relative to any inner function $u$.
In this paper we aim to develop further this type of useful generalization of
angular derivatives by studying the more broad category of relative angular
derivatives. We will find new perspectives from which to view the already
known relationships between Aleksandrov measures and angular derivatives, and
find new relationships by studying in detail the behavior of the relative
angular derivative from many perspectives, beginning with the generalization
of the difference quotient and using primarily the Aleksandrov measures, but
also relating their properties to the Hardy space $H^{2}$, and the de
Branges--Rovnyak spaces (as done by Sarason in \cite{DS}).
Section \ref{angularsec} contains some background material about angular
derivatives. In Section \ref{gensec}, we generalize the difference quotient,
$\frac{b(z)-b(z_{0})}{z-z_{0}}$, which appears in the definition of an angular
derivative, to $\frac{1-b}{1-u}$, which will be the primary object under
investigation throughout the paper. The study of this generalized difference
quotient allows us to define the notion of an angular derivative relative to
an inner function. It is in this section that we present the theorem which
lists six conditions, all of which will be shown to be equivalent, any one of
which can be used as a definition of a relative angular derivative.
Section \ref{dBsec} contains an introduction to the de Branges--Rovnyak
spaces, which will be useful in Section \ref{singularsec}, when we analyze the
boundary behavior of the generalized difference quotient with respect to
$\mu^{s}$. This, then, allows us to get an integral condition for the
existence of a relative angular derivative which is similar in form to that in
the definition of the Hardy spaces. In Section \ref{abscontsec}, we will
similarly analyze the behavior of the generalized difference quotient with
respect to $\mu^{a.c.}$. Section \ref{productsec} presents a use for our
characterization of relative angular derivatives to produce an analog of the
product rule for ordinary derivatives. Finally, in Section \ref{scsec}, we see
that many known results from the theory of angular derivatives can be obtained
easily by viewing an angular derivative as a special case of a relative
angular derivative.
\section{The Aleksandrov Measures\label{aleksandrovsec}}
For $\lambda\in{\partial\mathbb{D}}$, the function $\operatorname{Re}\left(
\frac{\lambda+b}{\lambda-b}\right) $ is positive, and, as the real part of an
analytic function, harmonic (on the disk, $\mathbb{D}$). It is thus the
Poisson integral of a positive measure on ${\partial\mathbb{D}}$, which we
will call ${\mu_{\lambda}}$. We have, then,
\[
\operatorname{Re}\left( \frac{\lambda+b(z)}{\lambda-b(z)}\right)
=\int_{\partial\mathbb{D}}P(\theta,z)d{\mu_{\lambda}}(e^{i\theta}%
)=P{\mu_{\lambda}}(z)
\]
and the Herglotz integral representation,
\[
\frac{\lambda+b(z)}{\lambda-b(z)}=\int_{\partial\mathbb{D}}H(\theta
,z)d{\mu_{\lambda}}(e^{i\theta})+i\,\operatorname{Im}\frac{\lambda
+b(0)}{\lambda-b(0)}.
\]
Note that for $z\in\mathbb{D}$, the Poisson kernel, $P(\theta,z)=\frac
{1-|z|^{2}}{|e^{i\theta}-z|^{2}}$, is the real part of the Herglotz kernel,
$H(\theta,z)=\frac{e^{i\theta}+z}{e^{i\theta}-z}$. The measure $\mu_{1}$ we
shall simply call $\mu$. The measure $\nu$ is similarly defined to correspond
with the inner function $u$.
The following are some properties of the Aleksandrov measures defined above:
\begin{itemize}
\item All positive Borel measures on ${\partial\mathbb{D}}$ are associated
with functions in this way.
\item The absolutely continuous part of $\mu$ is given by $\frac{1-|b|^{2}%
}{|1-b|^{2}}$ times the normalized Lebesgue measure (on ${\partial\mathbb{D}}$).
\item The measure $\mu$ is singular if and only if $b$ is an inner function,
i.e., $|b|=1$ almost everywhere on ${\partial\mathbb{D}}$.
\item For $\mu_{\lambda}^{s}$-a.e. $\xi\in{\partial\mathbb{D}}$ we have
$P{\mu_{\lambda}}(\xi)=\infty$ and thus $b(\xi)=\lambda$.
\end{itemize}
\section{Angular Derivatives\label{angularsec}}
The following is an overview of some of the properties of angular derivatives.
Much of this material can be found in \cite[Sec. 299]{CC}. I use it primarily
as presented by Sarason in \cite[Chapter VI]{DS}. We use this material as a
starting point.
For a holomorphic function $b$, we can talk about its derivative, $b^{\prime
}(z)$ for $z\in\mathbb{D}$, or, looking at the boundary behavior, the angular
derivative of the function at a point $z_{0}\in{\partial\mathbb{D}}$.
\begin{theorem}
\label{angulardef} For a function $b$, holomorphic in $\mathbb{D}$, and a
point $z_{0}$ of ${\partial\mathbb{D}}$, the following are equivalent:
\begin{itemize}
\item The function $b$ has a nontangential limit, $b(z_{0})$, at the point
$z_{0}$, and the difference quotient $(b(z)-b(z_{0}))/(z-z_{0})$ has a
nontangential limit at $z_{0}$.
\item The derivative $b^{\prime}$ has a nontangential limit at $z_{0}$.
\end{itemize}
\end{theorem}
The theorem above is true for any holomorphic $b$, but if we restrict
ourselves to holomorphic self-maps of the disk, as we do here, and require
that the function $b$ have unit modulus at the boundary point $z_{0}$, then we
say that the function $b$ has an angular derivative in the sense of
Carath\'{e}odory at the point $z_{0}$.
\begin{theorem}
[Carath\'{e}odory]\label{cara} If $z_{0}$ is a point of ${\partial\mathbb{D}}$
and
\[
c=\liminf_{z\rightarrow z_{0}}\frac{1-|b(z)|}{1-|z|}<\infty,
\]
then $b$ has an angular derivative in the sense of Carath\'{e}odory at $z_{0}
$. The relation $b^{\prime}(z_{0})=c\,b(z_{0})/z_{0}$ holds, and
$\frac{1-|b(z)|}{1-|z|}$ tends to $c$ as $z$ tends nontangentially to $z_{0}$.
The number $c$ is positive.
\end{theorem}
It is this notion of angular derivative which we will generalize in this paper.
\section{Generalizations of Angular Derivatives\label{gensec}}
Now we will examine extended notions of the angular derivative of the function
$b$ by replacing the identity function $z$ by an arbitrary (nonconstant) inner
function $u$ in the denominator of the standard difference quotient,
$\frac{b(z)-b(z_{0})}{z-z_{0}}$. We will then examine the behavior of this
generalized difference quotient, $\frac{1-b(z)}{1-u(z)}$.
The main theorem which will provide the basis for our definition of the
relative angular derivative is:
\begin{theorem}
[Main Theorem]\label{mainthm} The following conditions are equivalent:
\begin{enumerate}
\item $\nu\ll\mu$ and $\frac{d\nu}{d\mu}\in L^{2}(\mu)$;\label{measurecond}
\item $\frac{1-b}{1-u}k_{0}^{u}\in\mathcal{H}(b)$;\label{quotconda}
\item $\frac{1-b}{1-u}k_{w}^{u}\in\mathcal{H}(b)$\ \ for all $w\in\mathbb{D}$;\label{quotcondb}
\item $\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}}{1-u_{r}}\right| d\nu$
stays bounded as $r\nearrow1$;\label{intcond}
\item $\frac{1-b}{1-u}\in H^{2}$ and $\frac{1-b}{1-u}\in H^{2}(\mu^{a.c.})$;\label{h2accond}
\item $\frac{1-b}{1-u}\in H^{2}$ and $\frac{1-b}{1-u}\in H^{2}(\mu)$;\label{h2cond}
\end{enumerate}
\end{theorem}
If any of the above hold, then we will say that $b$ has an angular derivative
relative to $u$.
Of these conditions listed above, the equivalence of (\ref{measurecond}%
),(\ref{quotconda}), and (\ref{h2accond}) were previously known, and shown, in
some form, in various parts of \cite{DS}.
Note that both the definition of an angular derivative and Ca\-ra\-th\'{e}%
o\-do\-ry's theorem about angular derivatives depend on the behavior of a
difference quotient near only one point. It is clear from the nature of the
conditions above that the notion of relative angular derivative depends on the
behavior of the generalized difference quotient at more than one point.
Condition (\ref{h2accond}), for example, is a condition on the boundary values
$m$-a.e., and condition (\ref{intcond}) is a condition $\nu$-a.e.. Since $\nu$
is singular with respect to $m$, the equivalence of these two conditions in
defining a relative angular derivative is somewhat unexpected. Also note that
in the definition of the angular derivative of a function $b$, we have a
value, $b^{\prime}(z_{0})$, to associate with this derivative. Theorem
\ref{dqconv} will provide us with the basis for defining the ``value'' of an
angular derivative of a function $b$ relative to a function $u$. In this case,
we can define the value $\mu^{s} $-a.e. on ${\partial\mathbb{D}}$, and that
value can be taken to be $\frac{d\nu}{d\mu}$. In Section \ref{scsec}, we will
see that this notion of the value of a relative angular derivative is a good
generalization of that for angular derivatives.
\smallskip\
\noindent\emph{Remark:} The definition of relative angular derivatives could
have been expanded to include cases where the functions $b$ and $u$ might not
meet the conditions above, but the functions $\bar{\xi}b$ (for some $\xi
\in{\partial\mathbb{D}}$) and $u$ do. The function $b$, then, ``almost'' has
an angular derivative relative to $u$. In fact, Theorem \ref{cara} shows
clearly that the condition for $b$ to have an angular derivative in the sense
of Carath\'{e}odory is not altered by the multiplication of $b$ by a constant
of unit modulus. A better generalization, perhaps, would maintain this
property. We can accomplish this by altering the definition to:
\begin{quote}
The holomorphic self-map of the unit disk $b$ has an angular derivative
relative to the inner function $u$ if there is some $\xi\in{\partial
\mathbb{D}}$ such that any of the nine conditions above hold for the functions
$\bar{\xi}b$ and $u$ and their corresponding measures.
\end{quote}
\noindent We will not use this modified definition, however, because of the
added complication and the fact that this change does not alter the
fundamental notion at all. We should, in any case, remember this method of
generalizing angular derivatives as it will show up again in Section
\ref{scsec}.
\section{The de Branges--Rovnyak Spaces\label{dBsec}}
The de Branges--Rovnyak spaces are defined as the ranges of certain operators
on $H^{2}$. For $\varphi\in L^{\infty}$ of the unit circle with $\Vert
\varphi\Vert_{\infty}\le1$, we can define the de Branges--Rovnyak space
$\mathcal{H}(\varphi)$. Since we are interested in holomorphic self-maps of
the disk $b$, we will talk about the spaces $\mathcal{H}(b)$ for such $b$. The
space $\mathcal{H}(b)$ is defined to be the range of the operator
$(1-T_{b}T_{\bar{b}})^{1/2}$, where $T_{b}$ denotes the Toeplitz operator with
symbol $b$ (a function, in our case, in the unit ball of $H^{\infty}$). The
Toeplitz operator is the multiplication operator followed by the projection
onto $H^{2}$, that is, $T_{\phi}f=P_{+}(\phi f)$ where $P_{+}$ is the
projection operator from $L^{2}$ to $H^{2}$. The space $\mathcal{H}(b)$
defined this way becomes a Hilbert space, with norm $\Vert\,\cdot\,\Vert_{b} $
and inner product $\langle\cdot,\cdot\rangle_{b}$, where the inner product is
defined by $\langle(1-T_{b}T_{\bar{b}})^{1/2}x,(1-T_{b}T_{\bar{b}}%
)^{1/2}y\rangle_{b}=\langle x,y\rangle_{H^{2}}$, for $x,y\perp\ker
(1-T_{b}T_{\bar{b}})$.
We will not here go into full detail on the properties of these de
Branges--Rovnyak spaces. What is presented is an overview of the properties
which we will find useful in relation to relative angular derivatives. Most of
this material, as well as a detailed study of these spaces, can be found in
the works of Sarason, particularly in \cite{DS}.
The space $H^{2}$ has kernel functions $k_{w}$ for $w\in\mathbb{D}$, where
$k_{w}(z)=(1-\bar{w}z)^{-1}$ are such that for $f$ in $H^{2}$, we have
$f(w)=\langle f,k_{w}\rangle$. Similarly, there are functions $k_{w}^{b}$ in
$\mathcal{H}(b)$ which have the property that for $f$ in $\mathcal{H}(b)$,
$f(w)=\langle f,k_{w}^{b}\rangle_{b}$. These are given by $k_{w}%
^{b}(z)=(1-T_{b}T_{\bar{b}})k_{w}(z)=\frac{(1-\overline{b(w)}b(z))}{1-\bar
{w}z}$. Note that we can calculate the norms of these kernel functions in
$\mathcal{H}(b)$:
\[
\Vert k_{w}^{b}\Vert_{b}^{2}=k_{w}^{b}(w)=\frac{1-|b(w)|^{2}}{1-|w|^{2}}.
\]
The space $H^{2}(\mu)$ can be transformed into the space $\mathcal{H}(b)$ by
an operator $V_{b}$, which we will make use of in this paper. In order to
define $V_{b}$, we will consider the Cauchy integral of a complex Borel
measure $\rho$ on ${\partial\mathbb{D}}$, which is defined (for our purposes,
for $z\in\mathbb{D}$) by
\[
K_{\rho}f(z)=Kf\rho(z)=\int_{\partial\mathbb{D}}\frac{f(e^{i\theta}%
)}{1-e^{-i\theta}z}d\rho(e^{i\theta}).
\]
This lets us define the operator $V_{b}$ on the space $L^{2}(\mu)$ by
\[
V_{b}f(z)=(1-b(z))K_{\mu}f(z)\ \ \text{for}\ \ z\in\mathbb{D}.
\]
This operator has the properties that $V_{b}k_{w}=(1-\overline{b(w)}%
)^{-1}k_{w}^{b}$, and also $\langle k_{w},k_{z}\rangle_{\mu}=\langle
V_{b}k_{w},V_{b}k_{z}\rangle_{b}$. (The inner product on $H^{2}(\mu)$ is
$\langle\cdot,\cdot\rangle_{\mu}$, defined by $\langle f,g\rangle_{\mu}%
=\int_{\partial\mathbb{D}}f\bar{g}\,d\mu$ for $f,g\in H^{2}(\mu)$.) The
operator $V_{b}$, then, is an isometry of $H^{2}(\mu)$ onto $\mathcal{H}(b)$,
since it maps the kernel functions $k_{w} $ of $H^{2}$, the span of which is a
dense linear manifold in $H^{2}(\mu)$, to (a constant times) the kernel
functions of $\mathcal{H}(b)$, the span of which is a dense linear manifold in
$\mathcal{H}(b)$, and it preserves norms for linear combinations of kernel
functions. For more details about the transformation above, see \cite[III-6,7]%
{DS}.
The following theorem relates the existence of an angular derivative of $b$
relative to $u$ to the existence of a particular function in the de
Branges--Rovnyak space. It can be found in \cite[III-11]{DS}.
\begin{theorem}
\label{dBequiv} The following are equivalent:
\begin{enumerate}
\item $\nu\ll\mu$ and $\frac{d\nu}{d\mu}\in L^{2}(\mu)$;
\item the function $\left( \frac{1-b}{1-u}\right) k_{0}^{u}$ is in
$\mathcal{H}(b)$.
\end{enumerate}
\end{theorem}
We can extend this theorem to get
\begin{theorem}
\label{dBthird} The two conditions in Theorem \ref{dBequiv} are equivalent to
the following third condition:
\[
\left( \frac{1-b}{1-u}\right) k_{w}^{u}\in\mathcal{H}(b)\ \ \text{for
all}\ \ w\in\mathbb{D}.
\]
\end{theorem}
\noindent\emph{Proof: } First, notice that for $w=0$, this third condition is
part 2 of Theorem \ref{dBequiv}, so it implies part 2. Then, assuming part 1,
we (imitating the proof of the Theorem \ref{dBequiv} in \cite{DS}) consider
\begin{align*}
V_{b}\left( \frac{d\nu}{d\mu}k_{w}\right) & =(1-b)K_{\mu}\left(
\frac{d\nu}{d\mu}k_{w}\right) =(1-b)K_{\nu}(k_{w})\\
& =\left( \frac{1-b}{1-u}\right) V_{u}(k_{w})=(1-\overline{u(w)}%
)^{-1}\left( \frac{1-b}{1-u}\right) k_{w}^{u}.
\end{align*}
Since $\frac{d\nu}{d\mu}k_{w}$ is in $L^{2}(\mu)$, it is mapped by $V_{b}$ to
an element of $\mathcal{H}(b)$, so we have what we need to prove Theorem
\ref{dBthird}, and thus the equivalence of parts (\ref{measurecond}),
(\ref{quotconda}) and (\ref{quotcondb}) of our main theorem. \hfill$\square$
\section{The Singular Part of the Measure $\mu\label{singularsec}$}
We will now examine the boundary behavior of our generalized difference
quotient $\frac{1-b}{1-u}$ with respect to the singular part of the measure
$\mu$.
\begin{theorem}
\label{dqconv} The conditions $\nu\ll\mu$ and $\frac{d\nu}{d\mu}\in L^{2}%
(\mu)$ imply that $\frac{1-b_{r}}{1-u_{r}}\longrightarrow\frac{d\nu}{d\mu}$ in
$L^{2}(\mu^{s})$ as $r\nearrow1$.
\end{theorem}
\noindent\emph{Proof: }For this, we need to make use of the following theorem
by A. G. Pol\-to\-rat\-skii in \cite{AP}:
\begin{theorem}
\label{convbdry} For an element $h\in\mathcal{H}(b)$, we have
\[
h_{r}\longrightarrow V_{b}^{-1}h
\]
in $L^{2}(\mu^{s})$ norm, as $r\nearrow1$.
\end{theorem}
Assuming $\nu\ll\mu$ and $\frac{d\nu}{d\mu}\in L^{2}(\mu)$, we may take
$h=\frac{1-b}{1-u}k_{0}^{u}(1-\overline{u(o)})^{-1}$. We can see from our
proof of Theorem \ref{dBthird} that $V_{b}(\frac{d\nu}{d\mu})=h$, and thus
$h\in\mathcal{H}(b)$. Then, since $k_{0}^{u}=(1-\overline{u(o)}u)$, we have
\[
h_{r}=\frac{1-b_{r}}{1-u_{r}}\left( \frac{1-\overline{u(o)}u_{r}}%
{1-\overline{u(o)}}\right)
\]
and we use the theorem of Poltoratskii to get
\[
h_{r}\longrightarrow V_{b}^{-1}(h)=\frac{d\nu}{d\mu}%
\]
in $L^{2}(\mu^{s})$ as $r\nearrow1$. Now we see that
\begin{align*}
\frac{1-b_{r}}{1-u_{r}} & =h_{r}\left( \frac{1-\overline{u(o)}}%
{1-\overline{u(o)}u_{r}}\right) \\
& =h_{r}\left( 1-\frac{\overline{u(o)}-\overline{u(o)}u_{r}}{1-\overline
{u(o)}u_{r}}\right) \\
& =h_{r}-h_{r}\left( \frac{\overline{u(o)}(1-u_{r})}{1-\overline{u(o)}u_{r}%
}\right) \\
& \longrightarrow\frac{d\nu}{d\mu}.
\end{align*}
This last convergence is seen to be true because $h_{r}\rightarrow\frac{d\nu
}{d\mu}$, and we can now show that $h_{r}\left( \frac{\overline{u(o)}%
(1-u_{r})}{1-\overline{u(o)}u_{r}}\right) \rightarrow0$ in $L^{2}(\mu^{s})$.
We do this by noting that $\left| h_{r}\left( \frac{\overline{u(o)}%
(1-u_{r})}{1-\overline{u(o)}u_{r}}\right) \right| ^{2}$ is both uniformly
integrable (with respect to $\mu^{s} $) and tends to zero $\mu^{s}$-a.e.. It
is uniformly integrable since $\left( \frac{\overline{u(o)}(1-u_{r}%
)}{1-\overline{u(o)}u_{r}}\right) $ is bounded (by $\frac2{1-\overline{u(o)}%
}$) and $|h_{r}|^{2}$ converges in $L^{1}(\mu^{s})$, and it tends to zero
$\mu^{s}$-a.e. since $h_{r}\rightarrow\frac{d\nu}{d\mu}$ and $\left(
\frac{\overline{u(o)}(1-u_{r})}{1-\overline{u(o)}u_{r}}\right) \rightarrow0$
$\nu$-a.e.. This proves Theorem \ref{dqconv}. \hfill$\square$
We now continue to prove parts of the main theorem.
\begin{theorem}
\label{dqintconv} The conditions $\nu\ll\mu$ and $\frac{d\nu}{d\mu}\in
L^{2}(\mu)$ imply that
\[
\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}}{1-u_{r}}\right| ^{2}d\mu
^{s}\longrightarrow\left\| \frac{d\nu}{d\mu}\right\| _{L^{2}(\mu)}^{2}%
\]
and
\[
\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}}{1-u_{r}}\right|
d\nu\longrightarrow\left\| \frac{d\nu}{d\mu}\right\| _{L^{2}(\mu)}^{2}%
\]
as $r\nearrow1$.
\end{theorem}
\noindent\emph{Proof:} The first part of this theorem is true since it just
expresses the fact that, for the functions $\frac{1-b_{r}}{1-u_{r}}$ and
$\frac{d\nu}{d\mu}$, Hilbert space convergence (in $L^{2}(\mu^{s})$ --- by
Theorem \ref{dqconv}) implies convergence of norms (where $\left\| \frac
{d\nu}{d\mu}\right\| _{L^{2}(\mu^{s})}^{2}=\left\| \frac{d\nu}{d\mu
}\right\| _{L^{2}(\mu)}^{2} $).
The second part expresses the fact that, for the functions $\frac{1-b_{r}%
}{1-u_{r}}$ and $\frac{d\nu}{d\mu}$, the norm convergence implies weak
convergence. This can give us
\[
\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}}{1-u_{r}}\right| \frac{d\nu
}{d\mu}\,d\mu\longrightarrow\int_{\partial\mathbb{D}}\left( \frac{d\nu}{d\mu
}\right) ^{2}d\mu,
\]
which is equivalent to what we want. \hfill$\square$
We can express this limit, which appeared in both parts of the theorem, in
several ways:
\[
\left\| \frac{d\nu}{d\mu}\right\| _{L^{2}(\mu)}^{2}=\left\| V_{b}\left(
\frac{d\nu}{d\mu}\right) \right\| _{b}^{2}=\left\| \frac{1-b}{1-u}k_{0}%
^{u}\right\| _{b}^{2}|1-\overline{u(o)}|^{-2}%
\]
and
\[
\left\| \frac{d\nu}{d\mu}\right\| _{L^{2}(\mu)}^{2}=\int_{\partial
\mathbb{D}}\left( \frac{d\nu}{d\mu}\right) ^{2}d\mu=\int_{\partial
\mathbb{D}}\frac{d\nu}{d\mu}d\nu=\left\| \frac{d\nu}{d\mu}\right\|
_{L^{1}(\nu)}.
\]
Now we will prove the converse of the second part of Theorem \ref{dqintconv}
above, which establishes the equivalence of conditions (\ref{intcond}) and
(\ref{measurecond}) in the main theorem.
\begin{theorem}
\label{intcondpart} If $\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}%
}{1-u_{r}}\right| d\nu$ is bounded as $r\nearrow1$, then we have $\nu\ll\mu$
and $\frac{d\nu}{d\mu}\in L^{2}(\mu)$.
\end{theorem}
\noindent\emph{Proof:} We begin the proof by rewriting the relative difference
quotient in terms of Herglotz integrals:
\[
\frac{1-b}{1-u}=\left( \frac{1+u}{1-u}\right) \left( \frac{1+b}%
{1-b}\right) ^{-1}\left( \frac{1+b}{1+u}\right) ,
\]
where we have $\frac{1+u}{1-u}=\int_{\partial\mathbb{D}}H(\theta
,z)d\nu(e^{i\theta})+i\,\operatorname{Im}\frac{1+u(0)}{1-u(0)}$, and the
similar formula for $\frac{1+b}{1-b}$.
We can write
\[
\left( \frac{1+u}{1-u}\right) \left( \frac{1+b}{1-b}\right) ^{-1}%
=\frac{H\nu}{H\mu}=\frac{H\nu^{s}+H\nu^{a.c.}}{H\mu}.
\]
We will now show
\begin{lemma}
\label{Cconv} The following are true:
\begin{enumerate}
\item $\frac{H\nu^{s}(z)}{H\mu(z)}\rightarrow\infty$ as $z\rightarrow\xi$
nontangentially, for $\nu^{s}$ almost all $\xi\in{\partial\mathbb{D}}$.
\item $\frac{H\nu^{a.c.}(z)}{H\mu(z)}\rightarrow\frac{d\nu^{a.c.}}{d\mu}(\xi)$
as $z\rightarrow\xi$ nontangentially, for $\mu^{s}$ almost all $\xi
\in{\partial\mathbb{D}}$.
\end{enumerate}
\end{lemma}
To do this, we will need a lemma by Poltoratskii in \cite{AP}.
\begin{lemma}
For $\rho$ a positive Borel measure, and $f\in L^{1}(\rho)$, the nontangential
limit $\lim_{z\rightarrow\xi}\frac{Kf\rho(z)}{K\rho(z)}$ exists for $\rho$
almost every $\xi\in{\partial\mathbb{D}}$ and is equal to $f(\xi)$ $\rho^{s}%
$-a.e., where $\rho^{s}$ is the singular part of $\rho$ w.r. to Lebesgue measure.
\end{lemma}
For $K(\theta,z)=\frac1{1-e^{-i\theta}z}$, the Cauchy kernel, we easily see
that
\[
H(\theta,z)=\frac{e^{i\theta}+z}{e^{i\theta}-z}=2K(\theta,z)-1,
\]
so this lemma by Poltoratskii works just as well with the Herglotz kernel in
place of the Cauchy kernel. Part 2 of Lemma \ref{Cconv} above is an almost
immediate consequence of this (modified version of the) lemma by Poltoratskii,
with $\rho=\mu$, and $f=\frac{d\nu^{a.c.}}{d\mu}$. We have $\frac{d\nu^{a.c.}%
}{d\mu}\in L^{1}(\mu)$, and
\[
\frac{H\nu^{a.c.}(z)}{H\mu(z)}=\frac{H\frac{d\nu^{a.c.}}{d\mu}\mu(z)}{H\mu
(z)},
\]
so the result follows.
To prove part 1 of Lemma \ref{Cconv}, we consider a function $f$ defined on
${\partial\mathbb{D}}$ so that $f=0\ \ \ \mu$-a.e., $f=1\ \ \nu^{s}$-a.e. (as
done by Poltoratskii in a similar situation in \cite{AP}). Then we consider
the following nontangential limits:
\[
\lim_{z\rightarrow\xi}\frac{Hf(\mu+\nu^{s})(z)}{H(\mu+\nu^{s})(z)}%
=\lim_{z\rightarrow\xi}\frac{H\nu^{s}(z)}{H(\mu+\nu^{s})(z)}=1\ \ \ \nu
^{s}\text{-a.e.}\ \ \xi
\]
by using the above lemma by Poltoratskii with the function $f$ and $\rho
=\mu+\nu^{s}$. (Note that $\nu$ is already singular w.r. to Lebesgue measure,
and here we are only interested in the behavior of the above limit $\nu^{s}$-a.e..)
We continue to examine other nontangential limits:
\[
\lim_{z\rightarrow\xi}\frac{H\mu(z)}{H\nu^{s}(z)}=\lim_{z\rightarrow\xi
}\left( \frac{H(\mu+\nu^{s})(z)}{H\nu^{s}(z)}-1\right) =0\ \ \ \nu
^{s}\text{-a.e.}\ \ \xi.
\]
Finally, we get
\[
\lim_{z\rightarrow\xi}\frac{H\nu^{s}(z)}{H\mu(z)}=\infty\ \ \ \nu
^{s}\text{-a.e.}\ \ \xi,
\]
which is what we wanted to show.
Now we can get around to proving the theorem. To do this we assume that
$\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}}{1-u_{r}}\right| d\nu$ is
bounded as $r\nearrow1$. We also assume first that $\nu\not \ll\mu$, i.e.,
that $\nu^{s}$ is nonzero, and we derive a contradiction. This will give us
$\nu\ll\mu$. We will then show that $\frac{d\nu}{d\mu}\in L^{2}(\mu)$.
Assume $\nu^{s}$ is nonzero. To get our contradiction, we would like to show
that the integrand, $\left| \frac{1-b_{r}(\xi)}{1-u_{r}(\xi)}\right| $,
approaches $\infty$ for all $\xi$ on a set of positive $\nu$-measure. Note
that if $b_{r}(\xi)$ stays bounded away from 1 for all $\xi$ on a set of
positive $\nu$-measure, then the condition above is certainly true, since
$u_{r}(\xi)\rightarrow1\ \ \ \nu$-a.e. $\xi$.
Otherwise, we see that the function $\frac{1+b_{r}(\xi)}{1+u_{r}(\xi)}$ is
bounded away from zero, in fact, it approaches 1 $\nu$-a.e.. We can then
write
\begin{align*}
\frac{1-b_{r}(\xi)}{1-u_{r}(\xi)} & =\left( \frac{1+b_{r}(\xi)}{1+u_{r}%
(\xi)}\right) \left( \frac{1+u_{r}(\xi)}{1-u_{r}(\xi)}\right) \left(
\frac{1+b_{r}(\xi)}{1-b_{r}(\xi)}\right) ^{-1}\\
& =\left( \frac{1+b_{r}(\xi)}{1+u_{r}(\xi)}\right) \left( \frac{H\nu
^{s}(r\xi)+H\nu^{a.c.}(r\xi)}{K\mu(r\xi)}\right) .
\end{align*}
The right side $\rightarrow\infty$ as $r\nearrow1$, since the first factor
$\rightarrow1$, and the second factor $\rightarrow\infty$, for $\nu^{s}$-a.e.
$\xi$. Hence if $\nu^{s}$ is not the zero-measure, we do have $\left|
\frac{1-b_{r}(\xi)}{1-u_{r}(\xi)}\right| \rightarrow\infty$ for all $\xi$ on
a set of positive $\nu$-measure. Now, by a standard measure-theory argument,
we must have $\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}}{1-u_{r}}\right|
d\nu\rightarrow\infty$ as $r\nearrow1$. This is the contradiction we were
looking for, so we must have $\nu^{s}$ equal zero, and thus we must have
$\nu\ll\mu$. This means that $\nu^{a.c.}=\nu$, so we can write
\[
\frac{1-b_{r}(\xi)}{1-u_{r}(\xi)}=\left( \frac{1+b_{r}(\xi)}{1+u_{r}(\xi
)}\right) \left( \frac{H\nu(r\xi)}{H\mu(r\xi)}\right) .
\]
For $\nu$-a.e. $\xi$, $H\nu(r\xi)$ and $H\mu(r\xi)\rightarrow\infty$ as
$r\nearrow1$. So for $\nu$-a.e. $\xi$,
\begin{align*}
\lim_{r\nearrow1}\frac{1-b_{r}(\xi)}{1-u_{r}(\xi)} & =\lim_{r\nearrow1}%
\frac{H\nu(r\xi)}{H\mu(r\xi)}\left( \frac{1+b_{r}(\xi)}{1+u_{r}(\xi)}\right)
\\
& =\frac{d\nu}{d\mu}(\xi)
\end{align*}
by part 2 of Lemma \ref{Cconv}, the fact that $\nu\ll\mu^{s}$, and the fact
that $b(r\xi)$ and $u(r\xi)\rightarrow1\ \ \nu$-a.e..
We put everything together now to see that since $\int_{\partial\mathbb{D}%
}\left| \frac{1-b_{r}}{1-u_{r}}\right| d\nu$ is bounded as $r\nearrow1$, we
have $\frac{1-b}{1-u}\in L^{1}(\nu)=H^{1}(\nu)$ since $\nu$ is singular. This
tells us that for $f(e^{i\theta})$ the boundary values of the function
$\frac{1-b}{1-u}$ (defined $\nu$-a.e.), we have $\int_{\partial\mathbb{D}%
}|f(e^{i\theta})|d\nu(e^{i\theta})<\infty$, and, in fact, equal to the
$H^{1}(\nu)$ norm of $\frac{1-b}{1-u}$, or the $L^{1}(\nu)$ norm of $f$.
From the above, we see that the boundary function, $f(e^{i\theta})$, is, $\nu
$-a.e., $\frac{d\nu}{d\mu}(e^{i\theta})$, so $\int_{\partial\mathbb{D}}%
\frac{d\nu}{d\mu}\,d\nu<\infty$. Since $d\nu=\frac{d\nu}{d\mu}\,d\mu$, we get,
finally, $\int_{\partial\mathbb{D}}\left( \frac{d\nu}{d\mu}\right) ^{2}%
d\mu<\infty$, or $\frac{d\nu}{d\mu}\in L^{2}(\mu)$. This completes the proof
of the theorem. \hfill$\square$
\smallskip\
\noindent\emph{Remark:} We might want to add the following to the list of
equivalent statements in the main theorem:
\[
\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}}{1-u_{r}}\right| ^{2}d\mu
^{s}\ \ \text{stays bounded as}\ \ r\nearrow1.
\]
We did, in fact, prove that if $b$ has an angular derivative relative to $u$,
then the above must hold. This was part of Theorem \ref{dqintconv}. The
converse, however, is not true. We can find $b$ and $u$ such that the above
holds, but such that we do not have $\nu\ll\mu$. Take $b(z)=z$ and
$u(z)=z^{2}$, for example. We get $\mu=\delta_{1}$ and $\nu$ will have atoms
at both $1$ and $-1$. The condition, $\int_{\partial\mathbb{D}}\left|
\frac{1-b_{r}}{1-u_{r}}\right| ^{2}d\mu^{s}\ \ \text{remains bounded
as}\ \ r\nearrow1$ is satisfied, since
\[
\int_{\partial\mathbb{D}}\left| \frac{1-rz}{1-(rz)^{2}}\right| ^{2}%
d\delta_{1}=\left( \frac{1-r}{1-r^{2}}\right) ^{2}\rightarrow\frac14
\]
as $r\nearrow1$, but we do not have $\nu\ll\mu$. We can, however, imitate the
proofs given in this section to prove
\begin{theorem}
The function $b$ has an angular derivative relative to $u$ if and only if
$\nu\ll\mu$ and $\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}}{1-u_{r}%
}\right| ^{2}d\mu^{s}\ \ \text{stays bounded as}\ \ r\nearrow1$.
\end{theorem}
\section{The Absolutely Continuous Part of $\mu\label{abscontsec}$}
Here we will discuss the equivalence of parts (\ref{measurecond}),
(\ref{h2accond}) and (\ref{h2cond}) in the main theorem. The equivalence of
part (\ref{h2accond}) with part (\ref{measurecond}) comes from a consolidation
of two separate theorems of Sarason in \cite{DS}. The reason for this is that
the proof given of the equivalence of the condition $\nu\ll\mu$ and
$\frac{d\nu}{d\mu}\in L^{2}(\mu)$ and the condition $\frac{1-b}{1-u}\in H^{2}$
and $H^{2}(\mu^{a.c.})$ depends on whether $b$ is a nonextreme or extreme
point in the unit ball of $H^{\infty}$, i.e., whether or not the function
$\log(1-|b|^{2})$ is integrable (on ${\partial\mathbb{D}}$).
In the case $b$ is nonextreme, i.e., $\log(1-|b|^{2})$ is integrable, Sarason
shows in \cite[IV-8]{DS}:
\begin{theorem}
[Comparison of Measures]For $b$ nonextreme, the following are equivalent:
\begin{enumerate}
\item $\nu\ll\mu$ and $\frac{d\nu}{d\mu}\in L^{2}(\mu)$;
\item $\frac{1-b}{1-u}$ and $\frac{a}{1-u}$ are in $H^{2}$.
\end{enumerate}
\end{theorem}
Here the function $a$ is defined to be the outer function whose boundary
values have modulus $(1-|b|^{2})^{1/2}$ and which is positive at the origin.
The condition, then, that $\frac{a}{1-u}\in H^{2}$ is equivalent to
$\frac{1-b}{1-u}\in H^{2}(\mu^{a.c.})$ (remember that $\mu^{a.c.}%
=\frac{1-|b|^{2}}{|1-b|^{2}}m$).
Note that both of these conditions, that $\frac a{1-u}\in H^{2}$ and
$\frac{1-b}{1-u}\in H^{2}(\mu^{a.c.})$ are, in fact, equivalent to
\[
\int_{\partial\mathbb{D}}\frac{1-|b|^{2}}{|1-u|^{2}}dm<\infty,
\]
since this integral condition is the same as the assertion that the boundary
function of $\frac{1-b}{1-u}$ is in $L^{2}(\mu^{a.c.})$. Also, we have the
function $\frac{1-b}{1-u}$, as well as the function $\frac a{1-u}$ in the
Nevanlinna class $N^{+}$, since they are both the quotients of outer
functions. (The functions $1-b$ and $1-u$ are outer since they both have real
parts which are positive everywhere in the disk - see \cite[page 51]{PD}.)
Functions in $N^{+}$ which are in $L^{2}$ are also in $H^{2}$ (see \cite[page
28]{PD}).
For the case where $b$ is an extreme point of the unit ball of $H^{\infty}$,
i.e., $\log(1-|b|^{2})$ is not integrable, Sarason has in \cite[V-9]{DS}:
\begin{theorem}
[Comparison of Measures]If $b$ is extreme, then the following are equivalent:
\begin{enumerate}
\item $\nu\ll\mu$ and $\frac{d\nu}{d\mu}\in L^{2}(\mu)$;
\item the function $\frac{1-b}{1-u}$ is in $H^{2}$, and the function
$\frac{1}{1-u}$ is in $L^{2}(\rho)$.
\end{enumerate}
\end{theorem}
Here the measure $\rho$ is $(1-|b|^{2})m$, and, for $b$ extreme (and only for
$b$ extreme), we have $H^{2}(\rho)=L^{2}(\rho)$. Part 2 above, then, also is
easily seen to be equivalent to $\frac{1-b}{1-u}\in H^{2}$ and $\frac
{1-b}{1-u}\in H^{2}(\mu^{a.c.})$.
When we put the extreme and the nonextreme cases together, we get part
(\ref{h2accond}) of the main theorem.
We already proved earlier (Theorem \ref{dqintconv}) that if $b$ has an angular
derivative relative to $u$ then $\frac{1-b}{1-u}\in L^{2}(\mu^{s})$, and,
since $H^{2}(\mu^{s})=L^{2}(\mu^{s})$, we have $\frac{1-b}{1-u}\in H^{2}%
(\mu^{s}) $. This, together with the previous result gives us $\frac{1-b}%
{1-u}\in H^{2}(\mu)$, which then gives us part (\ref{h2cond}) of the main theorem.
\section{The Product Rule\label{productsec}}
We can use the characterization of when $b$ has an angular derivative relative
to $u$ from Section \ref{singularsec} to create an analog of the regular
product rule for derivatives. In this case, we will consider what happens when
we have two functions, $b_{1}$ and $b_{2}$, both holomorphic self-maps of the
disk, and the function $b=b_{1}b_{2}$. Under what conditions will this $b$
have an angular derivative relative to an inner function $u$? It will be shown
to be sufficient to assume that both $b_{1}$ and $b_{2}$ have angular
derivatives relative to $u$.
\begin{theorem}
\label{prodrule} If $b_{1}$ and $b_{2}$ are holomorphic self maps of the unit
disk, both with angular derivatives relative to $u$, then the function
$b=b_{1}b_{2}$ has an angular derivative relative to $u$.
\end{theorem}
\noindent\emph{Proof: }Under the assumptions of the theorem, we have, by
condition (\ref{intcond}) of our main theorem (proved in the Section
\ref{singularsec}):
\[
\int_{\partial\mathbb{D}}\left| \frac{1-b_{1r}}{1-u_{r}}\right|
d\nu\ \ \text{and}\ \ \int_{\partial\mathbb{D}}\left| \frac{1-b_{2r}}%
{1-u_{r}}\right| d\nu
\]
are both bounded as $r\nearrow1$. We now use
\[
1-b=1-b_{1}b_{2}=(1-b_{1})+(1-b_{2})-(1-b_{1})(1-b_{2})
\]
to get
\begin{align*}
\int_{\partial\mathbb{D}}\left| \frac{1-b_{r}}{1-u_{r}}\right| d\nu &
=\int_{\partial\mathbb{D}}\left| \frac{(1-b_{1r})}{1-u_{r}}+\frac{(1-b_{2r}%
)}{1-u_{r}}-\frac{(1-b_{1r})(1-b_{2r})}{1-u_{r}}\right| d\nu\\
& \le\int_{\partial\mathbb{D}}\left| \frac{(1-b_{1r})}{1-u_{r}}\right|
d\nu+\int_{\partial\mathbb{D}}\left| \frac{(1-b_{2r})}{1-u_{r}}\right|
d\nu\\
& \ +\int_{\partial\mathbb{D}}\left| \frac{(1-b_{1r})(1-b_{2r})}{1-u_{r}%
}\right| d\nu.
\end{align*}
We know that $\nu$-almost everywhere the integrand in the third term on the
right above tends to zero as $r\nearrow1$, and, by the same argument as used
in the proof of Theorem \ref{dqconv}, the integral, too, tends to zero. The
first two terms stay bounded (and even approach $\left\| \frac{d\nu}{d\mu
_{1}}\right\| _{L^{1}(\nu)}$ and $\left\| \frac{d\nu}{d\mu_{2}}\right\|
_{L^{1}(\nu)}$ as $r\nearrow1$), so we get from this, and part (\ref{intcond})
of the main theorem, $b$ has an angular derivative relative to $u$.
\hfill$\square$
From the point of view of the measures, this theorem tells us that if two
measures, $\mu_{1}$ and $\mu_{2}$ both have a common singular measure $\nu$
satisfying $\nu\ll\mu_{1}$ and $\nu\ll\mu_{2}$, as well as $\frac{d\nu}%
{d\mu_{1}}\in L^{2}(\mu_{1})$ and $\frac{d\nu}{d\mu_{2}}\in L^{2}(\mu_{2})$,
then the measure $\mu$ which corresponds to the function which is the product
of the two functions corresponding to $\mu_{1}$ and $\mu_{2}$ satisfies the
conditions $\nu\ll\mu$ and $\frac{d\nu}{d\mu}\in L^{2}(\mu)$.
In fact, because of Theorem \ref{dqconv}, we get the value, $\nu$-a.e., of the
angular derivative; $\frac{d\nu}{d\mu}$ is just given by the limit as
$r\nearrow1$ of the function $\frac{1-b_{r}}{1-u_{r}}=\frac{(1-b_{1r}%
)}{1-u_{r}}+\frac{(1-b_{2r})}{1-u_{r}}-\frac{(1-b_{1r})(1-b_{2r})}{1-u_{r}}$.
The first term on the right has limit $\mu_{1}^{s}$-a.e. of $\frac{d\nu}%
{d\mu_{1}}$, and the second term has limit $\mu_{2}^{s}$-a.e. of $\frac{d\nu
}{d\mu_{2}}$, and third term has limit zero $\nu$-a.e.. Thus, since $\nu\ll
\mu_{1}$ and $\nu\ll\mu_{2}$, we get
\[
\frac{d\nu}{d\mu}(\xi)=\frac{d\nu}{d\mu_{1}}(\xi)+\frac{d\nu}{d\mu_{2}}(\xi)
\]
for $\nu$-a.e. $\xi$.
\section{Relative Angular Derivatives as Generalizations of Angular
Derivatives\label{scsec}}
We can now examine the special case of a relative angular derivative which we
have when our inner function $u$ is a multiple of the identity function,
$u(z)=\overline{z_{0}}z$ for some $z_{0}\in{\partial\mathbb{D}}$. Since we are
interested in the behavior of a holomorphic function $b$ near the point
$z_{0}$, we will let $\zeta=b(z_{0})$ and analyze the angular derivative of
$\bar{\zeta}b$ relative to $u$. Our generalized difference quotient is now
\begin{equation}
\frac{1-\bar{\zeta}b(z)}{1-u(z)}=\frac{1-\bar{\zeta}b(z)}{1-\overline{z_{0}}%
z}=\frac{z_{0}}\zeta\left( \frac{\zeta-b(z)}{z_{0}-z}\right) ,\label{gdq}%
\end{equation}
which is just ($z_{0}/\zeta$ times) a regular difference quotient for a
function $b$. We should thus expect that in our theorems, if we take this
choice for $u$, we will get results that apply to angular derivatives (in the
sense of Carath\'{e}odory - assuming, as we shall, that $|b(z_{0})|=1$).
For this choice of $u$, we can easily see that $\nu=\delta_{z_{0}}$, since
\[
\frac{1+u(z)}{1-u(z)}=\frac{z_{0}+z}{z_{0}-z}=H(z_{0},z)=\int_{\partial
\mathbb{D}}H(\theta,z)d\delta_{z_{0}}.
\]
Thus if we are to have a function $b$ such that $\bar{\zeta}b$ has an angular
derivative relative to this $u$, we must have $\delta_{z_{0}}\ll\mu_{\zeta}$,
that is, $\mu_{\zeta}$ must have an atom at the point $z_{0}$. Note that the
second condition, that $\frac{d\nu}{d\mu_{\zeta}}\in L^{2}(\mu) $ is then automatic.
This gives us, as in \cite[VI-7]{DS},
\begin{theorem}
[Special Case]For $b$ a holomorphic self-map of the disk, with $\zeta
=b(z_{0})$, $\bar{\zeta}b$ has an angular derivative at a point $z_{0}$ of
${\partial\mathbb{D}}$ if and only if the corresponding measure $\mu_{\zeta}$
has an atom at $z_{0}$.
\end{theorem}
If $z\rightarrow z_{0}$ nontangentially, then the limit of the left side in
equation (\ref{gdq}) above must be $\frac{d\nu}{d\mu_{\zeta}}(z_{0})$, and the
limit of the right side must be $z_{0}b^{\prime}(z_{0})/\zeta$. Thus
\[
\frac{d\nu}{d\mu_{\zeta}}(z_{0})=z_{0}b^{\prime}(z_{0})/\zeta.
\]
Since $\frac{d\nu}{d\mu_{\zeta}}$ is real and positive, we get
\begin{theorem}
[Special Case]For a holomorphic function $b$ with an angular derivative in the
sense of Carath\'{e}odory at $z_{0}$, we must have $z_{0}b^{\prime}%
(z_{0})/b(z_{0})=|b^{\prime}(z_{0})|$.
\end{theorem}
\noindent(This result can be found in slightly different form in \cite[Sec.
299]{CC} and is part of \cite[VI-3]{DS}, presented here as Theorem \ref{cara}.)
As we have already mentioned, $\mu_{\zeta}$ must have an atom at $z_{0}$, and
since $\nu=\delta_{z_{0}}=\frac{d\nu}{d\mu_{\zeta}}\mu_{\zeta}$ we must have
$\frac{d\nu}{d\mu_{\zeta}}(z_{0})=1/\mu_{\zeta}(\{z_{0}\})$, so this gives us
\begin{theorem}
[Special Case]If $b$ has an angular derivative at a point $z_{0}$, and
$\zeta=b(z_{0})$, then $\mu_{\zeta}(\{z_{0}\})=1/|b^{\prime}(z_{0})|$.
\end{theorem}
This is a result proved in a different way in \cite[VI-7]{DS}.
Now let us examine the case where, for some point $z_{0}\in{\partial
\mathbb{D}}$, both $b$ and $u$ have angular derivatives at $z_{0}$. The
measures $\mu_{b(z_{0})}$ and $\nu_{u(z_{0})}$ will then both have atoms at
$z_{0}$, with
\[
\mu_{b(z_{0})}(\{z_{0}\})=\frac{b(z_{0})}{z_{0}b^{\prime}(z_{0})}%
\ \ \text{and}\ \ \nu_{u(z_{0})}(\{z_{0}\})=\frac{u(z_{0})}{z_{0}u^{\prime
}(z_{0})}.
\]
We can now use $\nu_{u(z_{0})}=\frac{d\nu_{u(z_{0})}}{d\mu_{b(z_{0})}}%
\mu_{b(z_{0})}$ to get
\[
\frac{d\nu_{u(z_{0})}}{d\mu_{b(z_{0})}}(z_{0})=\frac{\nu_{u(z_{0})}%
(\{z_{0}\})}{\mu_{b(z_{0})}(\{z_{0}\})}=\frac{z_{0}b^{\prime}(z_{0})u(z_{0}%
)}{z_{0}b(z_{0})u^{\prime}(z_{0})}=\frac{b^{\prime}(z_{0})}{u^{\prime}(z_{0}%
)}\frac{u(z_{0})}{b(z_{0})}.
\]
Another way to get the above result is by considering
\begin{align*}
\frac{d\nu_{u(z_{0})}}{d\mu_{b(z_{0})}}(z_{0}) & =\lim_{r\nearrow1}%
\frac{1-\overline{b(z_{0})}b_{r}}{1-\overline{u(z_{0})}u_{r}}(z_{0})\\
& =\lim_{r\nearrow1}\left( \frac{u(z_{0})}{b(z_{0})}\right) \left(
\frac{b(z_{0})-b_{r}(z_{0})}{z_{0}-rz_{0}}\right) \left( \frac{z_{0}-rz_{0}%
}{u(z_{0})-u_{r}(z_{0})}\right) \\
& =\frac{b^{\prime}(z_{0})}{u^{\prime}(z_{0})}\frac{u(z_{0})}{b(z_{0})}.
\end{align*}
This gives us
\begin{theorem}
[Special Case]At any point $z_{0}$ where both $b$ and $u$ have angular
derivatives (in the sense of Carath\'{e}odory), the value of the angular
derivative of $\overline{b(z_{0})}b$ relative to $\overline{u(z_{0})}u$ at
$z_{0}$ is equal to the quotient of the angular derivatives of $b$ and of $u$
at $z_{0}$ divided by the quotient of the values of $b$ and $u$ at $z_{0}$.
\end{theorem}
Note that at any boundary point $z_{0}$ where $u$ has an angular derivative,
it is necessary for $b$ to have an angular derivative in the sense of
Carath\'{e}odory, too, if we are to have any $\zeta\in{\partial\mathbb{D}} $
such that $\bar{\zeta}b$ has an angular derivative relative to $\overline
{u(z_{0})}u$, and if this is the case, then we necessarily have $\zeta
=b(z_{0})$, and the theorem above holds.
From the Hilbert Space perspective, we get
\begin{theorem}
[Special Case]A holomorphic self-map of the disk $b$ has an angular derivative
at a point $z_{0}$ if and only if there is some $\zeta\in{\partial\mathbb{D}}$
such that the function $\frac{b(z)-\zeta}{z-z_{0}}$ lies in $\mathcal{H}(b)$.
\end{theorem}
This comes as a consequence of applying part 2 of our main theorem to the
function $\bar{\zeta}b$, where $\zeta=b(z_{0})$ (again, with $u(z)=\overline
{z_{0}}z$). The theorem then tells us that $b$ has an angular derivative at
$z_{0} $ if and only if $\frac{1-\bar{\zeta}b}{1-\overline{z_{0}}z}%
\in\mathcal{H}(b)$ (note: $k_{0}^{u}=1$ for this $u$) which is the same as
$\frac{b(z)-\zeta}{z-z_{0}}\in\mathcal{H}(b)$. We must choose $\zeta=b(z_{0}%
)$, by the way, since, for any other value of $\zeta$, the function
$\frac{b(z)-\zeta}{z-z_{0}}$ will not even be in $H^{2}$. This theorem is part
of \cite[VI-4]{DS}, in which it is further proved that the above are
equivalent to: Every function in $\mathcal{H}(b)$ has a nontangential limit at
the point $z_{0}$.
\begin{thebibliography}{99}
\bibitem{CC}C. Carath\'{e}odory, \emph{Theory of Functions of a Complex
Variable, Vol. 2}, Chelsea Publishing Co., New York, 1954.
\bibitem {CM}J. Cima and A. Matheson, \emph{Essential norms of composition
operators and Aleksandrov measures}, Pacific J. of Math., \textbf{179} (1997) 59-64.
\bibitem {PD}P. L. Duren, \emph{Theory of $H^{p}$ Spaces}, Academic Press, New
York and London, 1970.
\bibitem {SDF}S. D. Fisher, \emph{Function Theory on Planar Domains}, John
Wiley \& Sons, New York, 1983.
\bibitem {AP}A. G. Poltoratskii, \emph{Boundary behavior of pseudocontinuable
functions}, Algebra i Analiz, \textbf{5} (1993), 189-210.
\bibitem {DS}D. Sarason, \emph{Sub--Hardy Hilbert Spaces in the Unit Disk},
John Wiley \& Sons, Inc., New York, 1994.
\bibitem {DS2}D. Sarason, \emph{Composition operators as integral operators},
Analysis and Partial Differential Equations, (1990), 545-565.
\bibitem {JES}J. E. Shapiro, \emph{Aleksandrov measures used in essential norm
inequalities for composition operators}, J. of Operator Theory, \textbf{40}
(1998), 133-146.
\bibitem {JHSCS}J. H. Shapiro and C. Sundberg, \emph{Compact composition
operators on }$L^{1}$, Proc. Amer. Math. Soc., \textbf{108} (1990), 443--449.
\bibitem {JHSPDT}J. H. Shapiro and P. D. Taylor, \emph{Compact, nuclear, and
Hilbert-Schmidt composition operators on }$H^{2}$, Indiana Univ. Math. J.
\textbf{23} (1973), 471-496.
\end{thebibliography}
\end{document}