text:
Welcome To PCA
Our School Profile
Regent Board, Faculty & Staff
Regent Board
Preschool & Begindergarten
Variable Tuition
Set up a Shadow Day
Apply to PCA
Begindergarten
K-6 Elementary
7-12 Secondary
Secondary Counseling
9th-12th Grade Class Page
Home School Partnership
Secondary ExPlore Week
Clubs, Organizations & Activities
Elementary Club Connection
Secondary Clubs & Organizations
Athletics Booster Board
Athletic Spirit Wear Store
Band & Strings
Fine Arts Booster Board
Giving Tuesday & Year-End Gifts
The Eagle Fund
Christmas Town Auction 2019
Parent Plus Portal
Student Plus Portal
Forms, Documents & Lists
Food Service & Lunch Menus
New Family Mentor Program
Tuition Reduction Incentive Program (TRIP)
Alumni Happenings
PCA Yearbooks
Open House Spring 2020
Preschool & Elementary Office
734-459-3505, Press 1
734-459-3505, ext. 3190
If you have any further questions, please complete the form below.
|Admissions|International Student Program
International Student Program | Plymouth Christian Academy
Plymouth Christian Academy welcomes students from all cultures around the world to participate in our International Student Program. We are committed to providing a caring and supportive environment that will help prepare international students to successfully achieve their educational goals.
Established in 1976, Plymouth Christian Academy is located on a beautiful, 45-acre campus in Canton, Michigan. PCA has a rich and respected history as an excellent Christian private school characterized by an integrated Christian worldview, academic rigor, a rich fine arts program, a competitive athletic program, and a variety of leadership and service opportunities.
We offer a dual-enrollment program with surrounding colleges and universities as well as many Advanced Placement courses. Graduates of Plymouth Christian Academy are enrolled in highly competitive universities around the country.
International student admission to PCA is open to academically qualified students who are seeking a nurturing Christian atmosphere. We accept students of all races, nationalities and ethnic origins. Plymouth Christian Academy issues I-20 status to students who meet all general admissions requirements for the program.
Additionally, our International Homestay Program matches students with English-speaking, Christian families who provide housing, support, and help students acclimate to the language and culture in a nurturing environment. See the homestay section below on this page for more information on our International Homestay Program.
PCA International Student Program Tuition
International Student Tuition Rates (Grades 9-12)
Tuition plus Housing
Tuition without Housing
* International Student Application Fee: $500 (non-refundable)
International Student Program FAQ's
In what grade levels do you take international students?
We accept international students entering our school in grades 8, 9,10,11, or 12.
What percentage of PCA graduates attend universities and where do they attend?
98% of our graduates go on to attend colleges and universities of their choice around the country: Abilene Christian University, Adrian College, Albion College, Azusa Pacific University, Barry University, Belmont University, Calvin College, Cedarville College, Central Michigan University, Cornerstone University, Dartmouth College, Eastern Michigan University, Ferris State University, Grand Valley State University, Henry Ford College, Hillsdale College, Hope College, Indiana Wesleyan University, Kalamazoo College, Kendall College of Art & Design, Kettering University, Lawrence Technological University, LeTourneau University, Liberty University, Lipscomb University, Loyola University, Madonna University, Marietta College, Michigan State University, Michigan Technological University, Moody Bible Institute, North Park University, Oklahoma Christian University, Oral Roberts University, Purdue University, Rochester College, Schoolcraft College, Spring Arbor University, Taylor University, United States Air Force Academy, University of Alabama, University of California, Berkley University of California, Riverside University of Detroit Mercy, University of Michigan, University of Notre Dame, Vanderbilt University, Wayne State University, Western Michigan University and Wheaton College.
What are the expectations and responsibilities for international students attending PCA?
To learn more about expectations and responsibilities for international students attending PCA, please download the International Student Handbook HERE.
When is the application due?
The date to return completed applications for any given school year, along with the $500 application fee (non-refundable), is March 1 for priority consideration.
Documents that need to be submitted online are:
Application, filled out by the parent, guardian, or student, along with the application fee of $500.
Student Information Form, Parent/Guardian Agreement Statement, Education History
Health immunization records
Photocopy of the applicant's passport
Student photo
Applicant must submit a minimum Test of English as a Foreign Language (TOEFL) score of 50 for 8th grade, 60 for 9th-10th grade, and 70 for 11th & 12th grade.
After all applications have been submitted and received initial approval, an interview is scheduled. Final admissions decisions are made by May 1 if the application is received by the priority deadline.
What is the cost for an international student to attend PCA?
Tuition + Housing = $28,760
Tuition without Housing = $19,760
How does the applicant obtain an F-1 visa and an I-20 form?
International students must obtain an F-1 visa from their local US Embassy to attend school in the U.S. In order to obtain an F-1 visa, the international student must first be accepted at a SEVIS-approved school, such as Plymouth Christian Academy. After the student has been accepted to PCA, we will issue the student an I-20 form. PCA will send the I-20 by certified mail courier to the student in their home country. The student must login to the SEVIS website to activate their record prior to the visa interview.
Does PCA have an ESL Program?
PCA does have an English as a second language program for our international students. Because our international students are combined with our American students in all the college preparatory classes, each student accepted to PCA needs to demonstrate strong abilities in written and spoken English. Students who obtain a TOEFL score of 50 or higher should generally do well in classes at PCA. In addition, students must be able to listen and speak comfortably during the in-person Skype interview with our Admissions Team.
Does PCA board or find HomeStay families for international students?
PCA does not board international students but we assist in finding homestay families. Some students stay with friends or relatives in the area. Our International Homestay Program allows students to live with English-speaking, PCA families who provide support and help students acclimate to the language and culture in a safe and nurturing environment.
What are the responsibilities of the Host Family?
Host families must fill out an application to serve as a HomeStay family. Communication, attention and integration are the key ingredients of a successful HomeStay program. Host Families welcome their young guest like a member of the family, talk to them frequently and do all they can to make sure their educational experience at PCA is a success. On a basic level, the host family provides food and accommodation as well as sharing their everyday activities with their guest (meals, family life, family outings).
Does the student have to be a Christian to attend PCA?
We accept students who, along with their parents or guardians, may not be Christians but are very open to learning more about Jesus Christ and the Bible. Although each student is free to make his/her own choice about the Lord, we will present what the Bible says about a relationship with Jesus Christ.
What is the weather like in Canton, Michigan?
Weather in the U.S. Midwest (Canton, MI) can vary greatly between winter and summer. In the winter months, (November-February) temperatures range between 20 and 45 degrees Fahrenheit. Spring and autumn temperatures tend to be moderate, ranging from the 50s to 70s. Summers in the Midwest (June–August) tend to be warm. Temperatures in the 80s and 90s are common. No matter when you are visiting the Midwest, pack layers so you can adjust to changing temperatures.
What is the Plymouth-Canton area like?
Plymouth and Canton are considered suburbs of Detroit, Michigan and are located approximately 45 minutes by car from Detroit. Plymouth and Canton are also approximately 30 minutes by car from Ann Arbor, Michigan, where the University of Michigan is located. Both Plymouth and Canton are mid-size towns with spacious neighborhoods and safe and lively downtown areas. We are also five hours by car from Chicago and Niagara Falls.
How will students be transported to and from school?
HomeStay families provide transportation to and from Plymouth Christian Academy. The Plymouth-Canton area does not have public transportation.
Do students need health insurance to attend PCA?
Yes, all international students must have health insurance to attend our school. Families are required to provide verification that the insurance is in place. Learn more and set up your plan at International Student Insurance.
Can parents communicate with teachers and staff at PCA for updates on what students are learning and how they are doing?
Yes. PCA teachers and staff are happy to communicate with parents through email, phone calls, Skype sessions, or in other ways. Parents can make this request at the time of application or anytime after a student arrives at PCA.
International Student Program Admissions Process
Application, filled out by the parent, guardian, or student, along with the application fee of $500. Application fee is non-refundable.
Health Immunization Receords
Photocopy of the applicant's passport
Applicant must submit a minimum Test of English as a Foreign Language (TOEFL) score of 50 for 8th grade, 60 for 9th and 10th grade, and 70 for 11th and 12th grade.
* After all applications have been submitted and received initial approval, an interview is scheduled. Final admissions decisions are made by May 1 if the application is received by the priority deadline.
Submit $500 Application Fee (non-refundable).
Complete and submit the Application including photo, transcripts and visa.
You will receive a welcome email and we will set up a Skype interview with the Admissions Team.
If accepted, an acceptance letter, I-20 and admission forms will be sent to you to complete and return with full tuition. Tuition is due by August 1st.
International students must attend a New International Student Orientation in August to help them transition smoothly into PCA.
International Student Homestay Program
Do you feel called of God to use your home as a ministry? Would you like to open your home to a student from another culture? If you said yes to either of these questions, then we need you! We are looking for families who will offer the best possible environment for our international students while they are stateside. Learn more about the homestay experience by downloading our International Host Family Handbook HERE. Questions? Please contact Leanne Windle, International Host Family Coordinator, leanne.windle@plymouthchristian.org.
Interested in applying? Click the button below to download the Host Family Application. Once your application has been received, we will be in touch to set up a home visit.
Host Family Application
I'm ready to
pca@plymouthchristian.org
43065 Joy Road Canton, MI 48187
© 2019 Plymouth Christian Academy | All Rights Reserved
Website by Design Source Media


text:
\section{Introduction}

This paper concerns necessary and sufficient conditions for bounded point derivations on various function spaces. Given a compact subset $X$ in $\mathbb{C}$ and a Banach Space $B$ on $X$, a point $x_0 \in X$ is said to admit a bounded point derivation for $B$ if there exists a constant $C>0$ such that $|f'(x_0)| \leq C ||f||$, for all $f \in B$, where $||\cdot||$ is the norm of the Banach space. Bounded point derivations were originally studied in the case of the space $R(X)$, the uniform closure of rational functions with poles off $X$. An important problem in the theory of rational approximation was to determine conditions for which $R(X) = C(X)$, the space of continuous functions on $X$, because if $R(X) = C(X)$, then every continuous function on $X$ can be uniformly approximated by rational functions with poles off $X$. To solve this problem, the concepts of peak points and non-peak points were developed in papers such as \cite{Bishop, Curtis, Gonchar}. A point $x_0 \in X$ is said to be a peak point for a uniform algebra $A$ on $X$ if there exists $f \in A$ such that $f(x_0) = 1$ and $|f(x)| < 1$ for all other $x \in A$; otherwise it is a non-peak point. Bounded point derivations on $R(X)$ are generalizations of non-peak points, and thus provide information about approximation of derivatives of rational functions. Some of the earlier results in connection with this problem can be found in \cite{Browder} and \cite{Wermer}.  Hence it is of great value to determine necessary and sufficient conditions for bounded point derivations. Moreover, the conditions mentioned in this paper depend only on the point $x_0$ under consideration and the geometry of the set $X$.

\bigskip

Hallstrom was the first to determine necessary and sufficient conditions for bounded point derivations, which he determined for the space $R(X)$, the uniform closure of rational functions with poles outside $X$ \cite{Hallstrom}. These conditions are given in terms of a quantity known as analytic capacity, which is defined as follows. Let $X$ be a compact subset of $\mathbb{C}$. A function $f$ is said to be admissible on $X$ if

\begin{enumerate}
    \item $|f(z)| \leq 1$ for $z \in \hat{\mathbb{C}} \setminus X$
    \item $f(\infty) = 0$
    \item $f$ is analytic outside $X$
\end{enumerate}

\bigskip \noindent and the analytic capacity of $X$ is denoted by $\gamma(X)$ and defined by

\begin{equation*}
    \gamma(X) = \sup f'(\infty) = \sup \lim_{z \to \infty} zf(z)
\end{equation*}

\bigskip

\noindent where the supremum is taken over all admissible functions. Hallstrom's conditions for bounded point derivations are summarized in Theorem 1. In it, and throughout the rest of the paper $A_n$ denotes the annulus $\{z: 2^{-n-1} < |z-x_0| < 2^{-n}\}$.

\begin{theorem}

Let $X$ be a compact subset of the complex plane, $x_0 \in X$, and let $t$ be a non-negative integer. Then there exists a bounded point derivation on $R(X)$ at $x_0$ of order $t$ if and only if

\begin{equation*}
    \sum_{n=1}^{\infty} 2^{(t+1)n} \gamma(A_n \setminus X)< \infty.
\end{equation*}

\end{theorem}

\bigskip

In \cite{O'Farrell1974}, O'Farrell considered the problem of what happens if the integer $t$ in Hallstrom's theorem is replaced with a non-integer. He was able to show two results in opposite directions. The first result concerns sets that satisfy a cone condition. A set $X$ is said to satisfy a cone condition at a point $x_0$ if there exists a cone $\mathscr{C}$ with vertex at $x_0$ and midline $J$ that satisfies the following property: there exists a constant $C>0$ such that if $x\in J$ and $z$ is outside $\mathscr{C}$ then $|x-x_0| \leq C |z-x|$. The midline $J$ is also known as a non-tangential ray to $x_0$ and the limit as $x \to x_0$, $x \in J$ is called a non-tangential limit to $x_0$. O'Farrell's first result shows that replacing the $t$ in Hallstrom's theorem with $t + \lambda$, where $0 < \lambda < 1$, implies a H\"older condition for the $t$-th derivative of $f$, as long as the set $X$ satisfies a cone condition.

\begin{theorem}
\label{O'Farrell1}
Suppose $X$ is a compact subset of $\mathbb{C}$ which satisfies a cone condition at $x_0$ and $J$ is a non-tangential ray to $x_0$. Let $t$ be a non-negative integer and let $0 < \lambda< 1$. If

\begin{equation}
\label{O'Farrell}
    \sum_{n=1}^{\infty} 2^{(t+\lambda+1)n} \gamma(A_n \setminus X)< \infty
\end{equation}

\bigskip \noindent then there is a constant $C >0$ such that

\begin{equation*}
    \dfrac{|f^{(t)}(x)-f^{(t)}(x_0)|}{|x-x_0|^{\lambda}} \leq C ||f||_{\infty}
\end{equation*}

\bigskip \noindent for all $x$ in $J$ and $f \in R(X)$.

\end{theorem}
\bigskip

O'Farrell's second result involves representing measures and $\tau$-spikes. Let $\mu$ be a measure and let $ \displaystyle \mu^{\tau}= \int \dfrac{d|\mu(\zeta)|}{|\zeta-z|^{\tau}}$. Then $\mu$ is a representing measure on for a point $x_0 \in X$ on a Banach space $B$ if $\int f d\mu = f(x_0)$ for all $f$ in $B$ and a point $x_0 \in X$ is a $\tau$-spike for $B$ if $\mu^{\tau}(x_0) = \infty$ whenever $\mu$ is a representing measure for $x_0$ on $B$. On $R(X)$ a peak point is a $\tau$-spike for all $\tau>0$. O'Farrell's second result shows that if $x_0$ is a $(t+\lambda)$-spike for $R(X)$ only if \eqref{O'Farrell} holds.

\begin{theorem}
\label{O'Farrell2}
Suppose $X$ is a compact subset of $\mathbb{C}$, $x_0 \in X$,  $0<\lambda<1$ and let $t$ be a non-negative integer. If $\displaystyle \sum_{n=1}^{\infty} 2^{(t+\lambda+1)n} \gamma(A_n \setminus X) = \infty$ then $\mu^{t+\lambda} = \infty$ whenever $\mu$ is a representing measure for $x_0$ on $R(X)$.
\end{theorem}

\bigskip

For the remainder of the paper, we consider bounded point derivations on $A_{\alpha}(U)$, the space of functions that are analytic on an open set $U$ and belong to the ``little Lipschitz class''.  Let $U$ be an open subset in the complex plane and let $0 < \alpha < 1$. A function $f : U \to \mathbb{C}$ satisfies a Lipschitz condition with exponent $\alpha$ on $U$ if there exists $k>0$ such that for all $z,w \in U$
\begin{equation}
\label{lip condition}
|f(z) - f(w)| \leq k |z-w|^{\alpha}.
\end{equation}

\bigskip

Let Lip$\alpha(U)$ denote the space of functions that satisfy a Lipschitz condition with exponent $\alpha$ on $U$. Lip$\alpha(U)$ is a Banach space with norm given by $||f||_{Lip\alpha(U)} = \sup_{U} |f| + k(f)$, where $k(f)$ is the smallest constant that satisfies \eqref{lip condition}. If we let $||f||_{Lip\alpha(U)}' = k(f)$ then $||f||_{Lip\alpha(U)}'$ is a seminorm on Lip$\alpha(U)$.

\bigskip

 The little Lipschitz class, lip$\alpha(U)$, is the subspace of Lip$\alpha(U)$ which consists of those functions in Lip$\alpha(U)$ that also satisfy the additional property that for each $\epsilon >0$, there exists $\delta >0$ such that for all $z$, $w$ in $U$, $|f(z)-f(w)| \leq \epsilon |z-w|^{\alpha}$ whenever $|z-w| < \delta$. Lipschitz functions form an important class of functions and much work has been done on approximations of Lipschitz functions by rational functions in papers such as \cite{O'Farrell 1975, O'Farrell 1977a, O'Farrell 1977b}.

 \bigskip
 Let $A_{\alpha}(U)$ denote the space of functions that are analytic on $U$ and belong to lip$\alpha(U)$. A point $x_0\in \overline{U}$ is said to admit a bounded point derivation on $A_{\alpha}(U)$ if there exists a constant $C>0$ such that $|f'(x_0)| \leq C ||f||_{Lip\alpha(\mathbb{C})}$ for all $f \in A_{\alpha}(U)$. In \cite{Lord}, Lord and O'Farrell determined the necessary and sufficient conditions for bounded point derivations on $A_{\alpha}(U)$, which is given in terms of an appropriate Hausdorff content.

 The Hausdorff content of a set is defined using measure functions. A measure function is a monotone nondecreasing function $h : [0, \infty) \to [0, \infty)$. For example, $r^{\beta}$ is a measure function for $0 \leq  \beta < \infty$. If $h$ is a measure function then the Hausdorff content $M_h$ associated to $h$ is defined by

\[ M_h(E) = \inf \sum h(\textnormal{diam } B), \]

\bigskip

\noindent where the infimum is taken over all countable coverings of $E$ by balls and the sum is taken over all the balls in the covering. If $h(r) = r^{\beta}$ then we denote $M_h$ by $M^{\beta}$. The lower $(1+ \alpha)$-dimensional Hausdorff content is denoted by $M^{1+\alpha}_*(E)$ and defined by

\[ M^{1+\alpha}_*(E) = \sup M_h(E), \]

\bigskip

\noindent where the supremum is taken over all measurable functions $h$ such that $h(r) \leq r^{1+\alpha}$ and $r^{-1-\alpha}h(r)$ converges to $0$ as $r$ tends to $0$. The lower $(1+ \alpha)$-dimensional Hausdorff content is a monotone set function; i.e. if $E \subseteq F$ then $M^{1+\alpha}_*(E) \leq M^{1+\alpha}_*(F)$. The result of Lord and O'Farrell characterizing bounded point derivations on $A_{\alpha}(U)$ in terms of Hausdorff content is summarized in the following theorem.

\begin{theorem}
Let $U$ be an open subset of $\mathbb{C}$, $x_0 \in \partial U$ and let $t$ be a non-negative integer. Then there exists a bounded point derivation of order $t$ on $A_{\alpha}(U)$ if and only if

\begin{equation}
\label{Lord}
    \sum_{n=1}^{\infty} 2^{(t+1)n} M_*^{1+\alpha}(A_n \setminus U) < \infty.
\end{equation}
\end{theorem}

\bigskip

A natural question to ask is what is the significance of a non-integer $t$ in \eqref{Lord}. In analogy with $R(X)$, results similar to Theorem \ref{O'Farrell1} and Theorem \ref{O'Farrell2} should hold for $A_{\alpha}(U)$. In this paper we prove the following results.

\begin{theorem}
\label{main1}
Suppose $U$ is an open subset of $\mathbb{C}$ which satisfies a cone condition at $x_0$ and $J$ is a non-tangential ray to $x_0$. Let $t$ be a non-negative integer and let $0 < \lambda< 1$. If

\begin{equation*}
\sum_{n=1}^{\infty} 2^{(t+1+\lambda)n} M_*^{1+\alpha}(A_n \setminus U) < \infty
\end{equation*}

\bigskip

\noindent then there is a constant $C >0$ such that

\begin{equation*}
    \dfrac{|f^{(t)}(x)-f^{(t)}(x_0)|}{|x-x_0|^{\lambda}} \leq C ||f||_{Lip\alpha(\mathbb{C})}
\end{equation*}

\bigskip

\noindent for all $x \in J$ and $f \in A_{\alpha}(U)$.

\end{theorem}

\bigskip

\begin{theorem}
\label{main2}
Suppose $U$ is an open subset of $\mathbb{C}$, $x_0 \in \Bar{U}$, $0<\lambda<1$, and let $t$ be a non-negative integer. Also let $\mu^{\beta}(z)= \int \dfrac{d|\mu(\zeta)|}{|\zeta-z|^{\beta}}$. If $\displaystyle \sum_{n=1}^{\infty} 2^{(t+\lambda+1)n} M_*^{1+\alpha}(A_n \setminus U) = \infty$ then $\mu^{t+\lambda} = \infty$ whenever $\mu$ is a representing measure for $x_0$ on $A_{\alpha}(U)$.
\end{theorem}

\bigskip

The next section contains some preliminary lemmas that are used in the proofs of Theorems \ref{main1} and \ref{main2}. In Section 3 we prove Theorem \ref{main1} and in Section 4 we prove Theorem \ref{main2}.

\section{Preliminary lemmas}

Throughout the remainder of the paper, we make use of the following factorization lemma.

\begin{lemma}
\label{factor}
For complex numbers $a$ and $b$ and positive integer $n$,

\begin{equation*}
    a^n - b^n = (a-b)(a^{n-1}+a^{n-2}b + \cdots + ab^{n-2}+  b^{n-1}).
\end{equation*}

\end{lemma}

\bigskip

We will also make use of the following closely related lemma which is less well known.

\begin{lemma}
\label{factor2}
Let $a$ and $b$ be non-zero complex numbers and let $n$ be a negative integer. Then

\begin{equation*}
    a^n-b^n = \dfrac{b-a}{ab} \cdot (a^{n+1}+ a^{n+2}b^{-1} + \ldots + b^{n+1}).
\end{equation*}
\end{lemma}

\begin{proof}
Let $z=a^{-1}$ and $w = b^{-1}$. Then it follows from Lemma \ref{factor} that

\begin{align*}
    a^n-b^n &= z^{-n}-w^{-n}\\
    &=(z-w)(z^{-n-1}+ z^{-n-2}w + \ldots + w^{-n-1})\\
    &=\dfrac{b-a}{ab} \cdot (a^{n+1}+ a^{n+2}b^{-1} + \ldots + b^{n+1}).
\end{align*}

\end{proof}

\bigskip

Another key lemma is the following Cauchy type theorem for Lipschitz functions which appears in the paper of Lord and O'Farrell \cite[pg.110]{Lord}.

\begin{lemma}
\label{Cauchy2}

Let $\Gamma$ be a piecewise analytic curve bounding a region $\Omega \in \mathbb{C}$, and suppose that $\Gamma$ is free of outward pointing cusps. Let $0 < \alpha < 1$ and suppose that $f \in$ lip$\alpha(\mathbb{C})$ is analytic outside a closed region $S$. Then there exists a constant $\kappa >0$ such that

\begin{equation*}
\left| \int f(z)dz\right| \leq \kappa \cdot M_{*}^{1+\alpha}(\Omega \cap S) \cdot ||f||'_{Lip\alpha(\Omega)}.
\end{equation*}

\bigskip

\noindent The constant $\kappa$ only depends on $\alpha$ and the equivalence class of $\Gamma$ under the action of the conformal group of $\mathbb{C}$. In particular this means that $\kappa$ is the same for any curve obtained from $\Gamma$ by rotation or scaling.

\end{lemma}

\bigskip

The next lemma is an immediate corollary of the cone condition that is applicable in a wide variety of situations.

\begin{lemma}
\label{cone}
Suppose that $J$ is a non-tangential ray to $x_0$ in a cone $\mathscr{C}$, $x \in J$ and $z$ is outside $\mathscr{C}$, then for all positive integers $t$, there exists a constant $C >0$ depending only on $t$ such that

\begin{equation*}
    \dfrac{1}{|z-x|^t} \leq \dfrac{C}{|z-x_0|^{t}}.
\end{equation*}
\end{lemma}

\begin{proof}
Since $x$ lies on $J$, which is a non-tangential ray to $x_0$, there exists a constant $K>0$ such that for $z \notin \mathscr{C}$, $\dfrac{|x-x_0|}{|z-x|} \leq K$. Thus for $z \notin U$, $\dfrac{|z-x_0|}{|z-x|} \leq 1 + \dfrac{|x-x_0|}{|z-x|} \leq 1 + K$. Hence $\dfrac{1}{ |z-x|^{t}} \leq \dfrac{(1+ K)^t}{|z-x_0|^{t}}$.
\end{proof}

\bigskip

Finally, we will need the following decay lemma which was first proved by Lord and O'Farrell \cite[pg.109]{Lord}.

\begin{lemma}
\label{decay}
Let $\alpha$ be such that $0 < \alpha < 1$, let $K$ be a compact subset of $\mathbb{C}$ and let $f \in $ Lip$\alpha(\mathbb{C})$ be analytic outside $K$
and vanish at $\infty$. Then there is a constant $C$ depending on $\alpha$ but not on $K$ or $f$ such that the following estimates hold.

\begin{enumerate}
    \item $||f||_{\infty} \leq C ||f||'_{Lip\alpha(\mathbb{C})} \cdot M_*^{1+\alpha}(K)^{\frac{\alpha}{1+\alpha}}$
    \bigskip
    \item For $z \notin K$, $|f(z)| \leq \dfrac{C ||f||'_{Lip\alpha(\mathbb{C})} \cdot M_*^{1+\alpha}(K)}{\text{dist}(z,K)}$
\end{enumerate}

\end{lemma}

\section{A H\"older condition for derivatives}

In this section we present the proof of Theorem \ref{main1}.

\begin{proof}

To prove Theorem \ref{main1}, we first note that by translation invariance we may suppose that $x_0 = 0$. Moreover by replacing $f$ by $f-f(0)$ if needed, we may suppose that $f(0) = 0$. In addition, we may suppose that $U$ is contained in the ball $\{z:|z| < \frac{1}{2}\}$.

\bigskip

\begin{figure}
\begin{center}
\begin{tikzpicture}

\tikzset{->-/.style={decoration={
  markings,
  mark=at position .5 with {\arrow[scale=2]{>}}},postaction={decorate}}}

\draw [->-,thick,domain=-150:150] plot ({cos(\x)}, {sin(\x)});
   \draw [->-, thick,domain=150:210] plot ({4* cos(\x)}, { 4*sin(\x)});

   \draw [->-, thick](-0.866025, 0.5) to (-3.464102, 2);
   \draw [->-, thick] (-3.464102, -2) to (-0.866025, -0.5);

   \filldraw[fill=black, draw=black] (0,0) circle (.08 cm);
   \node [below] (x) at (0, 0){$0$};

   \node [left] (x) at (-4,0){C};
   \node [right] (x) at (1,0){$B_N$};
   \filldraw[fill=black, draw=black] (-2,0) circle (.08 cm);
   \node [below] (x) at (-2, 0){$x$};

\end{tikzpicture}
\begin{tikzpicture}

\tikzset{->-/.style={decoration={
  markings,
  mark=at position .5 with {\arrow[scale=2]{>}}},postaction={decorate}}}

\draw [thick,domain=-150:150] plot ({cos(\x)}, {sin(\x)});
   \draw [  thick,domain=150:210] plot ({4* cos(\x)}, { 4*sin(\x)});
    \draw [ ->-, thick,domain=-150:150] plot ({2* cos(\x)}, { 2*sin(\x)});
     \draw [ ->-, thick,domain=-150:150] plot ({2.5* cos(-\x)}, { 2.5*sin(-\x)});

     \draw [->-, thick](-1.732051, 1) to (-2.1650635, 1.25);

     \draw [->-, thick] (-2.1650635, -1.25) to (-1.732051, -1);
   \draw (-0.866025, 0.5) to (-3.464102, 2);
   \draw  (-3.464102, -2) to (-0.866025, -0.5);
   \filldraw[fill=black, draw=black] (0,0) circle (.08 cm);
   \node [below] (x) at (0, 0){$0$};
   \node [below] (x) at (0, -1.9){$D_n$};

   \node [left] (x) at (-4,0){C};
   \node [right] (x) at (1,0){$B_N$};
   \filldraw[fill=black, draw=black] (-2,0) circle (.08 cm);
   \node [below] (x) at (-2, 0){$x$};

\end{tikzpicture}

\caption{The contour of integration}
\label{fig3}
\end{center}
\end{figure}

\bigskip

  It is a result of Lord and O'Farrell \cite[Lemma 1.1]{Lord} that $A_{\alpha}(U \cup \{x_0\})$ is dense in $A_{\alpha}(U)$. (Note that in this paper what we refer to as $A_{\alpha}(U)$ is denoted by $a(U)$.) Hence we may suppose that $f$ is analytic at $0$ and thus there is a neighborhood $\Omega$ of $0$ such that $f$ is analytic on $\Omega$. We can further suppose that $U \subseteq \Omega$. Let $B_n$ denote the ball centered at $0$ with radius $2^{-n}$. Then there exists an integer $N>0$ such that $\Omega$ contains $B_N$ and hence $f$ is analytic inside the ball $B_N$. Since $J$ is a non-tangential ray to $0$, it follows that there is a sector in $U$ with vertex at $0$ that contains $J$. Let $C$ denote this sector. It follows from the Cauchy integral formula that

  \begin{equation*}
      f^{(t)}(x)-f^{(t)}(0) = \frac{t!}{2 \pi i} \int_{\partial(C \bigcup B_N)} \dfrac{f(z)}{(z-x)^{t+1}} - \dfrac{f(z)}{z^{t+1}}dz
  \end{equation*}

  \bigskip \noindent where the boundary is oriented so that the interior of $C \bigcup B_N$ lies always to the left of the path of integration. (See Figure \ref{fig3}.) We can factor an $x$ out of the integrand.

\begin{align*}
f^{(t)}(x)-f^{(t)}(0) &= \frac{t!}{2 \pi i} \int_{\partial(C \bigcup B_N)} \dfrac{f(z)}{(z-x)^{t+1}} - \dfrac{f(z)}{z^{t+1}}dz\\
&= \frac{t!}{2 \pi i} \int_{\partial(C \bigcup B_N)} \dfrac{f(z) \cdot (z^{t+1}-(z-x)^{t+1})}{z^{t+1}(z-x)^{t+1}} dz\\
&= \frac{t!}{2 \pi i} \int_{\partial(C \bigcup B_N)} \dfrac{f(z) \cdot (z-(z-x)) \cdot (z^{t} + z^{t-1}(z-x) + \ldots + (z-x)^{t}) }{z^{t+1}(z-x)^{t+1}} dz\\
&= \frac{t!}{2 \pi i} \int_{\partial(C \bigcup B_N)} f(z) \cdot x \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} dz.\\
\end{align*}

\bigskip

\noindent Thus

\begin{equation*}
    \dfrac{f^{(t)}(x)-f^{(t)}(0)}{x^{\lambda}} = \frac{t! x^{1-\lambda}}{2 \pi i} \int_{\partial(C \bigcup B_N)} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} dz.\\
\end{equation*}

\bigskip

\noindent  Let $D_n = A_n \setminus C$. Then

\begin{align*}
\dfrac{f^{(t)}(x)-f^{(t)}(0)}{x^{\lambda}} &= \frac{t! x^{1-\lambda}}{2 \pi i} \sum_{n=1}^{N} \int_{\partial D_n} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} dz\\
&+ \frac{t! x^{1-\lambda}}{2 \pi i} \int_{|z|=\frac{1}{2}} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} dz.
\end{align*}

\bigskip

\noindent We first bound the second integral. Since $x$ lies on $J$, which is a non-tangential ray to $x_0$, it follows from Lemma \ref{cone} that $|z-x|^{-k-1} \leq C |z|^{-k-1}$. Thus by the triangle inequality,

\begin{align}
\label{Mterm}
    \left|\frac{t! x^{1-\lambda}}{2 \pi i} \int_{|z|=\frac{1}{2}} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} dz \right| &\leq \dfrac{t! |x|^{1-\lambda}}{2 \pi} \int_{|z|=\frac{1}{2}}\sum_{k=0}^{t}|z|^{-t-2}dz \\
    &\leq C \dfrac{t! |x|^{1-\lambda}}{2 \pi} (t+1) 2^{t+2} \sup_{U} f \leq C \sup_{U} f. \nonumber
\end{align}

\bigskip

\noindent To bound the sum, we note that since $ \displaystyle x^{1-\lambda} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1}$ is analytic on $D_n \setminus U$ for $M \leq n \leq N$, an application of Lemma \ref{Cauchy2} shows that

\begin{align}
\label{intbound2}
\left| x^{1-\lambda} \int_{\partial D_n} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} dz \right| \leq \kappa M^{1+\alpha}(D_n \setminus U) \cdot \left \Vert x^{1-\lambda} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1}  \right \Vert _{Lip\alpha(D_n)}'.
\end{align}

\bigskip

\noindent Recall that the constant $\kappa$ is the same for curves in the same equivalence class. Since the regions $D_n$ differ from each other by a scaling it follows that $\kappa$ doesn't depend on $n$ in \eqref{intbound2}. The remainder of the proof is to show that $\displaystyle \left \Vert x^{1-\lambda} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} \right \Vert _{Lip\alpha(D_n)}'$ can be bounded by a constant independent of $f$ and $x$. It follows from the definition of the Lipschitz seminorm that

\begin{align}
\label{lipnorm}
  &\left  \Vert x^{1-\lambda}  f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1}  \right \Vert _{Lip\alpha(D_n)}'\\
  &= \sup_{z, w \in D_n; z \neq w} \dfrac{|x^{1-\lambda} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} - x^{1-\lambda} f(w) \sum_{k=0}^{t}  w^{k-t-1} (w-x)^{-k-1}| }{|z-w|^{\alpha}} \nonumber
\end{align}

\bigskip

\noindent and it follows from an application of the triangle inequality that \eqref{lipnorm} is bounded by

\begin{align}
    & \sup_{z, w \in D_n; z \neq w} \dfrac{| x^{1-\lambda} (f(z)-f(w)) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1}|}{|z-w|^{\alpha}}+ \label{long1}\\
    & \sup_{z, w \in D_n; z \neq w}  \dfrac{|x^{1-\lambda}|\cdot|f(w)| \cdot| \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1}-   w^{k-t-1} (w-x)^{-k-1}| }{|z-w|^{\alpha}}\label{long2}.
\end{align}

\bigskip

\noindent We can determine upper bounds for both \eqref{long1} and \eqref{long2}. To bound \eqref{long1} we recall that

\begin{equation*}
    ||f||'_{Lip_{\alpha}(D_n)}= \sup_{z, w \in D_n; z \neq w} \dfrac{|f(z)-f(w)|}{|z-w|^{\alpha}}.
\end{equation*}

\bigskip

\noindent It follows from Lemma \ref{cone} that since $x$ is on a non-tangential ray and $z \in D_n$ and $|z-x|^{-k-\lambda} \leq C |z|^{-k-\lambda}$, and it follows from the cone condition that $|x^{1-\lambda}| \leq C |z-x|^{1-\lambda}$. Hence

\begin{align}
\label{1bound}
\dfrac{| x^{1-\lambda} (f(z)-f(w)) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1}|}{|z-w|^{\alpha}} &\leq C \dfrac{|f(z)-f(w)|\cdot |z-x|^{1-\lambda} \cdot |\sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1}|}{|z-w|^{\alpha}} \nonumber\\
&\leq C ||f||'_{Lip_{\alpha}(D_n)} |z|^{-t-1-\lambda} \leq C 2^{n(t+1+ \lambda)} ||f||'_{Lip_{\alpha}(\mathbb{C})}.
\end{align}

\bigskip

Now we bound \eqref{long2}. By the triangle inequality, this is bounded by

\begin{align}
    &\sup_{z, w \in D_n; z \neq w}  \dfrac{|x^{1-\lambda} f(w)| \cdot| \sum_{k=0}^{t}  (z^{k-t-1} -   w^{k-t-1}) (z-x)^{-k-1}| }{|z-w|^{\alpha}} \label{short1} \\ &+   \dfrac{|x^{1-\lambda} f(w)| \cdot| \sum_{k=0}^{t}  w^{k-t-1} ((z-x)^{-k-1}-  (w-x)^{-k-1})| }{|z-w|^{\alpha}} \label{short2}.
\end{align}

\bigskip

 We first obtain a bound for \eqref{short1}. Since $z,w \in D_n$, $|z|\leq 2 |w|$ and $|w| \leq 2 |z|$. Also it follows from Lemma \ref{cone} that for all integers $m<0$, $|z-x|^{m} \leq C |z|^{m}$ and it follows from the cone condition that $|x^{1-\lambda}| \leq C |z-x|^{1-\lambda}$. Hence it follows from Lemma \ref{factor2} that

\begin{align*}
   &\sup_{z, w \in D_n; z \neq w}  \dfrac{|x^{1-\lambda}f(w)| \cdot| \sum_{k=0}^{t}  (z^{k-t-1} -   w^{k-t-1}) (z-x)^{-k-1}| }{|z-w|^{\alpha}} \\
   &\leq \sup_{z, w \in D_n; z \neq w} \dfrac{|x^{1-\lambda}f(w)| \cdot| \sum_{k=0}^{t}  (w-z)(z^{k-t}+z^{k-t+1}w^{-1} + \ldots + w^{k-t}) (z-x)^{-k-1}| }{|z|\cdot|w| \cdot |z-w|^{\alpha}} \\
    &\leq \sup_{z, w \in D_n; z \neq w} |f(w)| \cdot|z-w|^{1-\alpha} \sum_{k=0}^{t}  (|z|^{k-t-1}|w|^{-1}+|z|^{k-t}|w|^{-2} + \ldots + |z|^{-1}|w|^{k-t-1}) |z-x|^{-k-\lambda}  \\
    &\leq \sup_{z, w \in D_n; z \neq w} C |f(w)| \cdot|z-w|^{1-\alpha} |w|^{-t-2-\lambda}.
\end{align*}

\bigskip

\noindent Since $f(0) =0$, it follows that for $w \in \mathbb{C}$, $ \dfrac{|f(w)|}{|w|^{\alpha}} \leq ||f||'_{Lip\alpha(\mathbb{C})}$. Hence

\begin{align*}
  \sup_{z, w \in D_n; z \neq w} |f(w)| \cdot|z-w|^{1-\alpha} |w|^{-t-2-\lambda} \leq \sup_{z, w \in D_n; z \neq w} ||f||'_{Lip\alpha(\mathbb{C})} \cdot|z-w|^{1-\alpha} |w|^{-t-2-\lambda+\alpha}. \\
\end{align*}

\bigskip

\noindent Since $z$ and $w$ both belong to $D_n$, $|z-w| \leq C |w|$ and hence

\begin{align*}
    \sup_{z, w \in D_n; z \neq w} ||f||'_{Lip\alpha(\mathbb{C})} \cdot|z-w|^{1-\alpha} |w|^{-t-2-\lambda+\alpha} \leq  C 2^{n(t+1+\lambda)} ||f||'_{Lip\alpha(\mathbb{C})}.
\end{align*}

\bigskip

\noindent Thus

\begin{equation}
\label{short1bound}
    \sup_{z, w \in D_n; z \neq w}  \dfrac{|x^{1-\lambda}f(w)| \cdot| \sum_{k=0}^{t}  (z^{k-t-1} -   w^{k-t-1}) (z-x)^{-k-1}| }{|z-w|^{\alpha}} \leq C 2^{n(t+1+\lambda)}||f||'_{Lip\alpha(\mathbb{C})}.
\end{equation}

\bigskip

 We next obtain a bound for \eqref{short2}. Since $z,w \in D_n$, $|z|\leq 2 |w|$, and $|w|\leq 2|z|$. Also it follows from Lemma \ref{cone} that for all integers $m <0$,  $|z-x|^{m} \leq C |z|^{m}$ and $|w-x|^{m} \leq C |w|^{m}$ and it follows from the cone condition that $|x^{1-\lambda}|\leq C |z-x|^{1-\lambda}$. Hence it follows from Lemma \ref{factor2} that

\begin{align*}
   &\sup_{z, w \in D_n; z \neq w}  \dfrac{|x^{1-\lambda} f(w)| \cdot| \sum_{k=0}^{t}  w^{k-t-1} ((z-x)^{-k-1}-  (w-x)^{-k-1})| }{|z-w|^{\alpha}} \\
   &\leq  \dfrac{|f(w)|\cdot |z-x|^{1-\lambda} \cdot| \sum_{k=0}^{t} w^{k-t-1} (z-w)\cdot[(z-x)^{-k}+(z-x)^{-k+1}(w-x)^{-1} + \ldots + (w-x)^{-k}] | }{|z-x|\cdot |w-x|\cdot |z-w|^{\alpha}} \\
    &\leq  \dfrac{|f(w)|}{|z-w|^{\alpha-1}}  \sum_{k=0}^{t} |w|^{k-t-1} (|z-x|^{-k-\lambda}|w-x|^{-1}+|z-x|^{-k+1-\lambda}|w-x|^{-2} + \ldots + |z-x|^{-\lambda}|w-x|^{-k-1})   \\
    &\leq \sup_{z, w \in D_n; z \neq w} C |f(w)| \cdot|z-w|^{1-\alpha} |w|^{-t-2-\lambda}
\end{align*}

\bigskip

\noindent Since $f(0) =0$ it follows that for $w \in \mathbb{C}$, $ \dfrac{|f(w)|}{|w|^{\alpha}} \leq ||f||'_{Lip\alpha(\mathbb{C})}$. Hence

\begin{align*}
  \sup_{z, w \in D_n; z \neq w} |f(w)| \cdot|z-w|^{1-\alpha} |w|^{-t-2-\lambda} \leq \sup_{z, w \in D_n; z \neq w} ||f||'_{Lip\alpha(\mathbb{C})} \cdot|z-w|^{1-\alpha} |w|^{-t-2-\lambda+\alpha}. \\
\end{align*}

\bigskip

\noindent Since $z$ and $w$ both belong to $D_n$, $|z-w| \leq C |w|$ and hence

\begin{align*}
   \sup_{z, w \in D_n; z \neq w} ||f||'_{Lip\alpha(\mathbb{C})} \cdot|z-w|^{1-\alpha} |w|^{-t-2-\lambda+\alpha} \leq C 2^{n(t+1+\lambda)}||f||'_{Lip\alpha(\mathbb{C})}.
\end{align*}

\bigskip

\noindent Thus

\begin{equation}
    \label{short2bound}
    \sup_{z, w \in D_n; z \neq w}  \dfrac{|x^{1-\lambda} f(w)| \cdot| \sum_{k=0}^{t}  w^{k-t-1} ((z-x)^{-k-1}-  (w-x)^{-k-1})| }{|z-w|^{\alpha}} \leq 2^{n(t+1+\lambda)}||f||'_{Lip\alpha(\mathbb{C})}.
\end{equation}

\bigskip

\noindent By applying \eqref{short1bound} and \eqref{short2bound} it follows that  \eqref{long2} is bounded by $C 2^{n(t+1+\lambda)}||f||'_{Lip\alpha(\mathbb{C})}$ and it follows from this and \eqref{1bound} that

\begin{equation*}
    \left|x^{1-\lambda} \int_{\partial D_n} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} dz \right| \leq C 2^{n(t+1+\lambda)}M_*^{1+\alpha}(D_n \setminus U)||f||'_{Lip\alpha(\mathbb{C})}.
\end{equation*}

\bigskip

\noindent Since Hausdorff content is monotone, $M_*^{1+\alpha}(D_n \setminus U) \leq M_*^{1+\alpha}(A_n \setminus U)$ and hence by the hypothesis of the theorem,

\begin{align*}
    \left|x^{1-\lambda} \sum_{n=M}^N\int_{\partial D_n} f(z) \sum_{k=0}^{t}  z^{k-t-1} (z-x)^{-k-1} dz \right| &\leq C \sum_{n=1}^{\infty} 2^{n(t+1+\lambda)}M_*^{1+\alpha}(A_n \setminus U)||f||'_{Lip\alpha(\mathbb{C})}\\
    &\leq C||f||'_{Lip\alpha(\mathbb{C})}.
\end{align*}

\noindent and thus it follows from this and \eqref{Mterm} that

\begin{align*}
    \dfrac{|f^{(t)}(x) -f^{(t)}(0)|}{|x|^{\lambda}} \leq C||f||_{Lip\alpha(\mathbb{C})}
\end{align*}

\bigskip

\noindent for all $f\in A_{\alpha}(U)$ and $x \in J$.

\end{proof}
\section{Representing measures and bounded point derivations}

We now prove Theorem \ref{main2}.

\begin{proof}
As before we will assume that $x_0 = 0$ and $X$ is contained in the ball $\{z: |z| < \frac{1}{2}\}$. Choose a sequence $\epsilon_n \to 0$ such that $\displaystyle \sum_{n=1}^{\infty} 2^{(t+\lambda+1)n} \epsilon_n M_*^{1+\alpha}(A_n \setminus U) = \infty$ and $2^{(t+\lambda+1)n} \epsilon_n M_*^{1+\alpha}(A_n \setminus U) \leq 1$ for all $n$. Then for each integer $N$, there exists an integer $M >N$ such that

\begin{equation*}
    1 \leq \sum_{n=N}^M 2^{(t+\lambda+1)n}\epsilon_n M_*^{1+\alpha}(A_n \setminus U) \leq 2.
\end{equation*}

\bigskip

\noindent By Frostman's Lemma \cite[\textbf{2.2}]{Lord}, for each integer $n$ with $N \leq n \leq M$ there exists a positive measure $\nu_n$ with support on $A_n \setminus U$ such that $\int  \nu_n = C \epsilon_n M_*^{1+\alpha}(A_n \setminus U)$, where the constant $C>0$ and does not depend on $n$ or $U$. Now define a function $f_n$ by

\begin{equation*}
    f_n(z) = \int  \dfrac{d\nu_n(\zeta)}{z-\zeta}.
\end{equation*}

\bigskip

\noindent $f_n$ belongs to $A_{\alpha}(U)$ and is analytic off $A_n$. In addition, it follows from Lemma \ref{decay} that $|f_n(z)| \leq C 2^{-\lambda n}$ and if $z\notin A_{n-1} \cup A_n \cup A_{n+1}$, then $|f_n(z)| \leq \dfrac{C M_*^{1+\alpha}(A_n \setminus U)}{dist(z, A_n)}$.

\bigskip

Now define $g_N(z)$ by

\begin{equation*}
    g_N(z) = |z|^{\lambda} z^{t+1} \sum_{n=N}^{M} 2^{(t+ \lambda +1)n} f_n(z)
\end{equation*}

\bigskip

\noindent and consider the sequence $\{g_N\}_{1}^{\infty}$. We will show that this sequence is uniformly bounded on the unit disk.

\bigskip

Suppose that $z \in A_j$ for some positive integer $j$. By Lemma \ref{decay}, if $n \neq j-1, j, j+1$ then

\begin{equation*}
    |f_n(z)| \leq C 2^j M_*^{1+\alpha}(A_n \setminus U)
\end{equation*}

\bigskip

\noindent and if $n = j-1, j, j+1$ then

\begin{equation*}
    |f_n(z)| \leq C 2^{-\lambda n} \leq C
\end{equation*}

\bigskip

\noindent since $n$ is a positive integer. From these estimates it follows that

\begin{equation*}
    |g_N(z)| \leq |z|^{\lambda + t +1} \left(3C \cdot 2^{(t+\lambda+1)j}+ \sum_{n=N}^{M} C 2^{(t+\lambda+1)n} 2^j M_*^{1+\alpha}(A_n \setminus U)\right).
\end{equation*}

\bigskip \noindent Since $z \in A_j$, $|z|^{\lambda +t+1} \leq C 2^{-(t+\lambda+1)j}$ and

\begin{align*}
    |g_N(z)| & \leq 3C+ |z|^{\lambda + t} 2C.
\end{align*}

\bigskip

\noindent Thus $\{g_N(z)\}$ is uniformly bounded on the unit disk. Next define $h_N(z)$ by $h_N(z) = |z|^{-\lambda}z g_N(z)$, and consider the sequence $\{h_N(z)\}$. Because $g_N(z)$ is uniformly bounded on the unit disk, so also is $h_N(z)$ and since $h_N(z)$ is analytic outside the ball $\{|z| \leq 2^{-N}\}$, a subsequence (also denoted $\{h_N\}$) converges pointwise on $\mathbb{C}\setminus \{0\}$ to a function $h$ which is also analytic on $\mathbb{C}\setminus \{0\}$. Moreover, $g_N$ is uniformly bounded near $0$ for each $N$ and hence $h(0) = 0$. Thus $h$ is entire.

\bigskip

Let $k_N(z)$ be defined by $k_N(z) = z^{-t-2}h_N(z)$. Then

\begin{align*}
    k_N'(\infty) &= \lim_{z \to \infty} z k_N(z)\\
    &= \lim_{z \to \infty} z^{-t-1} h_N(z)\\
    &= \lim_{z \to \infty} |z|^{-\lambda} z^{-t} g_N(z)\\
    &= \lim_{z \to \infty} z \sum_{n=N}^M 2^{(t+\lambda+1)n}f_n(z).
\end{align*}

\bigskip

\noindent Since $\displaystyle \lim_{z \to \infty} zf_n(z) = \int \nu_n = C \epsilon_n M_*^{1+\alpha}(A_n \setminus U)$, it follows that $0 < C \leq |k_N'(\infty)| \leq 2C$. Thus passing to a second subsequence, still denoted by $\{k_N'\}$, we find that $\{k_N'(\infty)\} $ converges to $\beta$ for some $C \leq \beta \leq 2C$. Thus $\displaystyle \lim_{z \to \infty} z k_N(z) = \beta$ and hence $g_N(z)$ converges pointwise to $\beta |z|^{\lambda}z^{t}$.

\bigskip

 Assume that there exists a measure $\mu$ which represents $0$ on $A_{\alpha}(U)$ such that $\mu^{t+\lambda} <  \infty$. Then $|z|^{-\lambda}z^{-t}\mu$ is a finite measure. Let $L_N(z) = |z|^{-\lambda}g_N(z)$. Then $L_N(z)$ is analytic in a neighborhood of $0$ and belongs to $A_{\alpha}(U)$. Hence

\begin{align*}
    0 = L_N^{(t)}(0) &= t! \int \dfrac{L_N(z)}{z^t} d\mu(z) \\
    &= t! \int \dfrac{g_N(z)}{|z|^{\lambda}z^t} d\mu(z) \\
    &\to t! \int \beta d\mu(z) = t! \cdot \beta,
\end{align*}

\bigskip

\noindent which is a contradiction. Hence $\mu^{t+\lambda} = \infty$.

\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Les Trente Glorieuses (; 'The Glorious Thirty') was a thirty-year period of economic growth in France between 1945 and 1975, following the end of the Second World War. The name was first used by the French demographer Jean Fourastié, who coined the term in 1979 with the publication of his book Les Trente Glorieuses, ou la révolution invisible de 1946 à 1975 ('The Glorious Thirty, or the Invisible Revolution from 1946 to 1975'). The term is derived from Les Trois Glorieuses ('The Glorious Three'), the three days of revolution on 27–29 July 1830 in France.

As early as 1944, Charles de Gaulle introduced a dirigiste economic policy, which included substantial state-directed control over a capitalist economy. This was followed by thirty years of unprecedented growth, known as the Trente Glorieuses. Over this thirty-year period, France's economy grew rapidly like economies of other developed countries within the framework of the Marshall Plan, such as West Germany, Italy and Japan.

From 1946 to 1950, France, paralyzed by an obsolete economy and infrastructures, did not achieve real growth and living conditions remained very difficult after the war and the penury which resulted from it: the cost of living rose. Rationing, present until 1947-1948, and the housing crisis, accentuated the problems of a people still scarred by World War II.

These decades of economic prosperity combined high productivity with high average wages and high consumption, and were also characterized by a highly developed system of social benefits. According to various studies, the real purchasing power of the average French worker's salary went up by 170% between 1950 and 1975, while overall private consumption increased by 174% in the period 1950–74.

The French standard of living, which had been damaged by both World Wars, became one of the world's highest. The population also became far more urbanized; many rural départements experienced a population decline while the larger metropolitan areas grew considerably, especially that of Paris. Ownership of various household goods and amenities increased considerably, while the wages of the French working class rose significantly as the economy became more prosperous. As noted by the historians Jean Blondel and Donald Geoffrey Charlton in 1974,

If it is still the case that France lags in the number of its telephones, working-class housing has improved beyond recognition and the various 'gadgets' of the consumer society–from television to motor cars–are now purchased by the working class on an even more avid basis than in other Western European countries.

Up until the 1973 oil crisis, the French economy had been faring smoothly.  under François Mitterrand and Jacques Chirac. The oil shock had signified the end of this period of explosive growth.

In his book Capital in the Twenty-First Century, French economist Thomas Piketty describes the Trente Glorieuses as an exceptional 'catch up' period following the two world wars. He cites statistics showing that normal growth in wealthy countries is about 1.5–2%, whereas in Europe growth dropped to 0.5% between 1913 and 1950, and then 'caught up' with a growth rate of 4% between 1950 and 1970, until settling back to 1.5–2% from 1970 onward.

See also
Economic history of France
Japanese post-war economic miracle
Miracolo economico
Post–World War II economic expansion
Record years
Spanish miracle
Wirtschaftswunder

References

Further reading

 Volkmar Lauber, The political economy of France: from Pompidou to Mitterrand (Praeger Publishers, 1983).

1940s economic history
1950s economic history
1960s economic history
1970s economic history
1940s in France
1950s in France
1960s in France
1970s in France
Contemporary French history
Economic history of France
Economic booms
Post–World War II economic booms

es:Treinta gloriosos

redpajama_set_name: RedPajamaWikipedia


1970s Stephen Burrows Silk Chiffon Ensemble Top – D. Matt Smith, Inc.
Own a piece of fashion history with this Stephen Burrows blouse! This sea foam green blouse features Burrow's signature red zig zag stitching and the tiered, triple layered silk chiffon technique that Burrows made famous in 1973. Most likely part of his trademark single color ensemble sets, this rare blouse is in excellent vintage condition.