\documentclass[a4paper]{amsart}
%\documentclass[a4paper,12pt]{jsarticle}
\usepackage{nucoverwithwindow}

\begin{document}
% window setting: position and size.
\setWindowTopPosition{6.1cm}
\setWindowRightPosition{6.1cm}
\setWindowWidth{9.8cm}
\setWindowHeight{5cm}
% If you use jsarticle, then you should use true* as:
%\setWindowTopPosition[1truein]{6.1truecm}
%\setWindowRightPosition[1truein]{6.1truecm}
%\setWindowWidth{9.8truecm}
%\setWindowHeight{5truecm}

\begin{blockoverwindow}
\centering
.\hrule

\vfill 
AAAA.\\
This sentense is over the window.
The width is just textwidth.
\hrule.
\end{blockoverwindow}
% \begin{centerblockoverwindow}
% \centering
% .\hrule
% 
% \vfill 
% AAAA.\\
% This sentense is over the window.
% \hrule.
%\end{centerblockoverwindow}
\begin{blockofwindow}
\centering
.\hrule

\vfill 

BBBB.\\
This is in the window.
\hrule.
\end{blockofwindow}

\begin{center}
CCC.\\
This is bellow the window. 
The width is just textwidth.
\end{center}
% \begin{centerblockbellowwindow}
% \centering
% .\hrule

% \vfill 

% CCC.\\
% This is bellow the window.
% \hrule.
% \end{centerblockbellowwindow}

\end{document}
