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\theoremstyle{definition}
\newtheorem{exercise}{Exercise}
\newtheorem{defn}{Definition}
\newtheorem{note}{Note}
\newtheorem*{requirements}{Requirements}
\newtheorem*{objectives}{Objectives}

\newcommand{\eqdef}{\coloneqq}
\newcommand{\PositiveRealNumbers}{\mathbb{R}_+}
\newcommand{\ComplexNumbers}{\mathbb{C}}
\newcommand{\UpperHalfPlane}{\Pi}
\newcommand{\eps}{\varepsilon}
\newcommand{\hsp}[1]{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}
\newcommand{\ubrace}[1]{\underbrace{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}_?}
\DeclareMathOperator{\enumber}{e}
\DeclareMathOperator{\VSO}{VSO}

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\begin{document}

\mytitle[Vertical Toeplitz operators and VSO]{Vertical Toeplitz operators\\on the Bergman space\\
on the upper half-plane\\
and very slowly oscillating functions\\
on the positive half-line}


\medskip\noindent
This theme is planned to be written in some weeks.
The actual text is just a rough draft.

\bigskip

\begin{objectives}
Prove that the set of functions
\[
\Gamma\eqdef\bigl\{\gamma_a\colon\ a\in L^\infty(\PositiveRealNumbers)\bigr\},
\]
where
\[
\gamma_a(x)\eqdef 2x\,\int_0^{+\infty}a(v)\,\enumber^{-2vx}\,dv,
\]
is dense in the algebra $\VSO(\PositiveRealNumbers)$
of the very slowly oscillating functions:
\[
\VSO(\PositiveRealNumbers)\eqdef\Bigl\{f\colon\PositiveRealNumbers\to\ComplexNumbers\colon\quad
\sup_{x\in\PositiveRealNumbers}|f(x)|<+\infty\quad\wedge\quad
\lim_{\frac{x}{y}\to1}|f(x)-f(y)|=0\Bigr\}.
\]
\end{objectives}

\begin{requirements}
Definition of $\VSO(\PositiveRealNumbers)$ through the logarithmic distance $\rho$:
\[
\rho(x,y)\eqdef\bigl|\ln(x)-\ln(y)\bigr|,
\]
basic properties of the functions $\phi_n$ and $\psi_n$ defined by the following formulas:
\[
\phi_n(v)\eqdef\frac{1}{\bigl((n-1)!\bigr)^2}\,\frac{d^{n-1}}{dv^{n-1}}\left(\enumber^{-v}\,v^{2n-1}\right),
\qquad
\psi_n(t)\eqdef\int_0^{+\infty}\enumber^{-vt}\,\phi_n(v)\,dv.
\]
\end{requirements}

\clearpage
\section*{Logarithmic distance on the positive half-line (review)}

\medskip

\begin{exercise}[logarithmic distance]
Recall the definition of $\rho(x,y)$:
\par\medskip
\[
\rho(x,y)\eqdef\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Prove that the logarithmic distance is dilation-invariant:
\par\medskip
\[
\rho(tx,ty)=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Compare the logarithmic distance with another
dilation-invariant distance:
\par\medskip
\[
\rho(x,y)\ \underbrace{\hsp{2em}}_{\le\ \text{or}\ \ge}\ \frac{|x-y|}{\max(x,y)}.
\]
\end{exercise}

\bigskip

\begin{exercise}[$\delta$-neighborhood of $1$]
Find the $\delta$-neighborhood of $1$ with respect to the distance $\rho$:
\par\medskip
\[
\rho(x,1)<\delta \qquad\Longleftrightarrow\qquad \hsp{15em}
\]
\end{exercise}

\bigskip

\section*{Very slowly oscillating functions\\on the positive half-line (review)}

\medskip

\begin{exercise}[uniform deviation of a function with respect to the logarithmic distance]
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$ and $\delta>0$.
Write the definition and various descriptions of $\Omega_{\rho,f}(\delta)$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}[very slowly oscillating functions]
Write the definition of $\VSO(\PositiveRealNumbers)$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\clearpage
\section*{Special approximate unit (review)}

\medskip

\begin{exercise}
Recall the definition of the function $\phi_n$:
\par\medskip
\[
\phi_n(v)=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Recall an explicit formula for the function $\phi_n$ (without derivatives):
\par\medskip
\[
\phi_n(v)=\hsp{6em}\sum\hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Calculate:
\par\medskip
\[
\lim_{v\to0^+}\phi_n(v)=\hsp{5em}
\qquad\qquad
\lim_{v\to+\infty}\phi_n(v)=\hsp{5em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Recall the explicit formula for the function $\psi_n$:
\par\medskip
\[
\psi_n(t)=\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Calculate the integral:
\par\medskip
\[
\int_0^{+\infty}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $\delta>0$.
Calculate the limit:
\par\medskip
\[
\lim_{n\to\infty}\int_0^{\enumber^{-\delta}}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $\delta>0$.
Make the change of variables $\displaystyle s=\frac{1}{t}$ in the following integral:
\par\medskip
\[
\int_{\enumber^{\delta}}^{+\infty}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $\delta>0$.
Calculate the limit:
\par\medskip
\[
\lim_{n\to\infty}\int_{\enumber^{\delta}}^{+\infty}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{defn}
Let $\delta>0$.
Denote by $R_{n,\delta}$ the following integral:
\par\medskip
\[
R_{n,\delta}\eqdef\int_{\substack{x>0\\\rho(x,1)>\delta}}\psi_n(t)\,dt.
\]
\end{defn}

\bigskip

\begin{exercise}
Write $R_{n,\delta}$ as a sum of some integrals with explicit limits of integration:
\par\medskip
\[
R_{n,\delta}=\int\ \psi_n(t)\,dt\ + \int\ \psi_n(t)\,dt.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $\delta>0$.
Calculate the limit:
\par\medskip
\[
\lim_{n\to\infty} R_{n,\delta}=\hsp{10em}
\]
\end{exercise}

\bigskip

\clearpage
\section*{Gammas belong to $\boldsymbol{\VSO(\PositiveRealNumbers)}$}

\medskip\noindent
The following two exercises show that the following set $\Gamma$
is a subset of $\VSO(\PositiveRealNumbers)$.
\[
\Gamma\eqdef\bigl\{\gamma_a\colon\ a\in L^\infty(\PositiveRealNumbers)\bigr\}
\]
In fact, these results motivated
the definition of the class $\VSO(\PositiveRealNumbers)$.

\bigskip

\begin{exercise}
Let $x\in\PositiveRealNumbers$. Calculate the integrals:
\par
\[
2x\int_{\PositiveRealNumbers}\enumber^{-2xv}\,dv=\hsp{10em}
\qquad
2x^2\int_{\PositiveRealNumbers}v\enumber^{-2xv}\,dv=\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $a\in L^\infty(\UpperHalfPlane)$.
Prove that $\gamma_a$ is bounded.
\par\medskip
\[
|\gamma_a(x)|\le\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $x,y\in\PositiveRealNumbers$, $x<y$.
Apply the Mean Value Theorem to the function $t\mapsto\enumber^{-t}$:
\par\smallskip
\[
\enumber^{-x}-\enumber^{-y}
\le\biggl(\,\sup_{x<t<y}\Bigl(\,\enumber^{-t}\Bigr)'\,\biggr)\,(y-x)
=\biggl(\,\sup_{x<t<y}\ubrace{3em}\,\biggr)\,(y-x)
=\,\ubrace{3em}\,(y-x).
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:gamma_a_is_Lipschitz_continuous_wrt_rho}
Let $a\in L^\infty(\UpperHalfPlane)$.
Prove that $\gamma_a$ is Lipschitz-continuous
with respect to the logarithmic distance $\rho$.
\end{exercise}

\bigskip




\clearpage
\section*{Gammas are dense in $\boldsymbol{\VSO(\PositiveRealNumbers)}$}

\medskip

\begin{exercise} \label{exer:gamma_Phi}
Let $n\in\{1,2,\ldots\}$, $x>0$, $y>0$.
Using an appropiate change of variables
express the following integral in terms of $\psi_n$:
\[
2x\,\int_0^{+\infty}\frac{\phi_n(2vy)}{y}\,\enumber^{-2xv}\,dv
=\hsp{15em}
%\frac{x}{y^2}\,\psi_n(x/y)
%=\frac{1}{\Beta(n,n)}\,\frac{x^n\,y^{n-1}}{(x+y)^{2n}}.
\]
\par\bigskip\noindent
Then use the formula for $\psi_n$ and write the result in an explicit form:
\par\medskip
\[
2x\,\int_0^{+\infty}\frac{\phi_n(2vy)}{y}\,\enumber^{-2xv}\,dv
=\hsp{15em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $\sigma\in\VSO(\PositiveRealNumbers)$ and $n\in\{1,2,\ldots\}$.
Define $a_n\in L^\infty(\PositiveRealNumbers)$ by
\begin{equation} \label{eq:a_n_through_sigma}
a_n(v)\eqdef\int_0^{+\infty}\sigma(y)\,\frac{\phi_n(2vy)}{y}\,dy.
\end{equation}
Applying Fubini's theorem and the result of Exercise \ref{exer:gamma_Phi} calculate $\gamma_{a_n}$:
\par\medskip
%\[
%\gamma_{a_n}(x)=\int_0^{+\infty}\sigma(y)\,\frac{x^n y^{n-1}}{\Beta(n,n)\,(x+y)^{2n}}\,dy.
%\]
\[
\gamma_{a_n}(x)=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $\sigma\in\VSO(\PositiveRealNumbers)$ and $n\in\{1,2,\ldots\}$.
Define $a_n$ by \eqref{eq:a_n_through_sigma}.
Find an upper bound for $\|\gamma_{a_n}-\sigma\|_\infty$
in terms of $\Omega_{\rho,\sigma}$, $\|\sigma\|_\infty$ and $R_{n,\delta}$.
\[
|\gamma_{a_n}(x)-\sigma(x)|\le\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $\sigma\in\VSO(\PositiveRealNumbers)$ and $\eps>0$.
Choose $n\in\{1,2,\ldots\}$ such that the function $a_n$
defined by \eqref{eq:a_n_through_sigma}
fulfill $\|\gamma_{a_n}-\sigma\|_\infty<\eps$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip
\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip
\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip
\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}[main result]
Prove that the set $\bigl\{\gamma_a\colon\ a\in L^\infty(\PositiveRealNumbers)\bigr\}$
is dense in $\VSO(\PositiveRealNumbers)$.
\end{exercise}


\label{endlabel}
\end{document}
