% Macros
\definecolor{darkred}{rgb}{.7,0,0}
\newcommand{\real}[0]{\mathbf R} 
\newcommand{\nat}[0]{\mathbf N} 
\newcommand{\op}[0]{\mathcal} 
\newcommand{\grad}[0]{\nabla}
\newcommand{\dive}[0] {\,\nabla\,\cdot\,} 
\newcommand{\ip}[2]{\left \langle #1,#2 \right\rangle} 
\newcommand{\norm}[1]{\left\| #1 \right\|} 
\newcommand{\pard}[2]{\partial_{#2} #1} 
\newcommand{\pardl}[2]{\frac{\partial #1}{\partial #2}} 
\newcommand{\pardd}[2]{\partial_{#2}^2 #1} 
\newcommand{\parddl}[2]{\frac{\partial^2 #1}{\partial #2^2}} 
\newcommand{\dd}[0]{\,\mathrm d}
\newcommand{\hermite}[0]{\mathcal}
\newcommand{\gaussian}[1]{g_{#1}} 
\newcommand{\stdgaussian}[0]{g} 
\newcommand{\test}[1]{C^\infty_c(#1)} 
\newcommand{\flp}[2]{L^{#1}(#2)}
\newcommand{\wlp}[3]{L^{#1}(#2,#3)}
\newcommand{\sobolev}[2]{H^{#1}(#2)}
\newcommand{\wsobolev}[3]{H^{#1}(#2, #3)}
\newcommand{\red}[1]{\textcolor{darkred}{#1}}
\newcommand{\HRule}{\rule{\linewidth}{0.5mm}}
\newcommand{\smooth}[1]{C^\infty(#1)}
\newcommand{\cont}[2]{C^{#1}(#2)}
\newcommand{\Space}[0]{\mathscr}