I'm using the following codes:
\documentclass[12pt,a4paper]{amsart}
%------------------------------------------------------------
\usepackage{amsmath,amssymb,amsthm}
\usepackage{color}
\usepackage{url}
%------------------------------------------------------------
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
%------------------------------------------------------------
\usepackage{geometry}
\geometry{left=2cm,right=2cm,top=2cm,bottom=2cm}
% ------------------------------------------------------------
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{definitions}{Definitions}[section]
\newtheorem{notation}{Notation}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
\numberwithin{equation}{section}
%%%%%%%%%%%%% hyperref %%%%%%%%%%%%%%%%%%%%
\usepackage[colorlinks=true,pagebackref=true]{hyperref}
\hypersetup{urlcolor=blue, citecolor=red, linkcolor=blue}
\begin{document}
\title[Paper1]{Paper1}
\author[First author, Second author] {First author$^{1}$, Second
author$^{2}$}
\address{$^{1}$ University 1.}
\email{\textcolor[rgb]{0.00,0.00,0.84}{first@gmail.com}}
\address{$^{2}$ Sfax University 2.}
\email{\textcolor[rgb]{0.00,0.00,0.84}{first@hotmail.com}}
\subjclass[2010]{46C05, 47A05}
\keywords{operator, spectrum.}
\maketitle
\begin{abstract}
In this article, we ..
\end{abstract}
\section{One}
\begin{thebibliography}{10} \addcontentsline{toc}{chapter}{References}
\footnotesize
\bibitem{acg1}{G. Corach,} {Partial isometries,} Linear Algebra Appl.
\end{thebibliography}
\end{document}
I hope to get after compilation a title and the name of authors in lowercase letters and not in capital letters.

\newpageafter\section{One}and make compilation you will see in page 2 in the bottom ''2 FIRST AUTHOR, SECOND AUTHOR'' .But I hope to get in the bottom of every pages like this ''First Author, Second Author''. For example also in page 3 I get ''PAPER1 3'' but I hope to get ''Paper1 3''.. Thanks a lot – Student Jun 08 '18 at 10:40\usepackage{geometry}\geometry{left=2cm,right=2cm,top=2cm,bottom=2cm}– Student Jun 08 '18 at 19:50amsartand the other ams document classes apply what is traditional for the ams publishing history. other publishers have other traditions. and individual mathematicians have their own opinions. i can say, however, thatamsartis not really meant to be modified using thegeometrypackage. – barbara beeton Jun 08 '18 at 20:41