The following defines an new math operator with:
\DeclareMathOperator{\OperatorCommand}{OperatorName} and\DeclareMathOperator*{\OperatorCommand}{OperatorName} if limits are to be used.
\documentclass[letterpaper]{article}
\usepackage{amsmath}
% We use the starred versions since we are interested in using limits.
% You can use \thinspace if you want to use a command. If you want to use more math oriented
% commands you can use \mspace{<length in mu>} like \mspace{2mu}
\DeclareMathOperator*{\esssup}{ess\,sup}
\begin{document}
\[\esssup_{x\in[0, 1]}\]
\end{document}
The \, and \! commands are the most useful for fine tuning math formulas.
Note that \thinspace is the same thing as \,.
Spaces which are very useful can be:
\thinspace = \,\medspace = \:\thickspace = \;
Negative spaces now:
\negthinspace = \!\negmedspace\negthickspace
If we get a little bit more technical, in the amsmath.sty, we can find the definitions of the above as:
\DeclareRobustCommand{\tmspace}[3]{%
\ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
\renewcommand{\,}{\tmspace+\thinmuskip{.1667em}}
\let\thinspace\,
\renewcommand{\!}{\tmspace-\thinmuskip{.1667em}}
\let\negthinspace\!
\renewcommand{\:}{\tmspace+\medmuskip{.2222em}}
\let\medspace\:
\newcommand{\negmedspace}{\tmspace-\medmuskip{.2222em}}
\renewcommand{\;}{\tmspace+\thickmuskip{.2777em}}
\let\thickspace\;
\newcommand{\negthickspace}{\tmspace-\thickmuskip{.2777em}}
\newcommand{\mspace}[1]{\mskip#1\relax}
Of course, \mspace{<length>} can be used for both positive and negative spaces. For example, \mspace{1mu} and \mspace{-1mu} where mu means math unit.
Note that \, is used extensively even to define \limsup command in the amsmath package in the following manner:
\def\limsup{\qopname\relax m{lim\,sup}}
The above can be found in amsopn.sty.
