I have frequently used \hookrightarrow and I don't find it particularly horrible. Anyway, that's a matter of taste.
I don't think that the combination of > and \subset may yield good results to obtain your symbol. Why do not try to draw the symbol by yourself? Here you have a bit of code using the simplest drawing environment in LaTeX. I'm sure that you can get much better results with ps-tricks or Tikz/pgf.
Code: Select all
\documentclass[a4paper,12pt]{article}
\usepackage{pict2e}
\begin{document}
% With \hookrightarrow
$H^{1}(\Omega)\hookrightarrow C^0(\overline{\Omega})$
% First try
\newcommand{\injection}{\mathrel{%
\setlength{\unitlength}{2pt}
\begin{picture}(6,4)(0,-1.5)
\qbezier(2,2)(0,2)(0,1)
\qbezier(0,1)(0,0)(2,0)
\put(2,0){\vector(1,0){4}}
\end{picture}}}
$H^{1}(\Omega)\injection C^0(\overline{\Omega})$
% Second try
\renewcommand{\injection}{\mathrel{%
\setlength{\unitlength}{2pt}
\begin{picture}(5,4)(0,-1.5)
\qbezier(2,2)(0,2)(0,1)
\qbezier(0,1)(0,0)(2,0)
\put(2,0){\line(1,0){3}}
\qbezier(3.8,1.2)(4,0.5)(4.95,0)
\qbezier(3.8,-1.2)(4,-0.5)(4.95,0)
\end{picture}}}
$H^{1}(\Omega)\injection C^0(\overline{\Omega})$
% With \subset (just to compare sizes)
$H^{1}(\Omega)\subset C^0(\overline{\Omega})$
\end{document}
By the way, to denote the clausure of a set, I find better \overline than \bar.