I have the task to comupute the following expression:
\begin{align} \exp(-\vert x\vert^2) \partial^2_{x_1 x_2}\mathbb{1}_{x_1>0,x_2>0} \in \mathcal{D}'(\mathbb{R}^2). \end{align}
Anyone an idea what to do? I really do not understand what to compute... I mean a distribution is a function...?! Thanks in advance!
A distribution is a linear functional on the space $\mathcal{D}(\Bbb R^2)$ of $C^\infty$ function with compact support. Let $\phi\in\mathcal{D}(\Bbb R^2)$. Then $$\begin{align} \big\langle e^{-|x|^2}\partial_{x_1,x_2}\Bbb 1_{x_1>0,x_2>0},\phi\bigr\rangle&= \big\langle \partial_{x_1,x_2}\Bbb 1_{x_1>0,x_2>0},e^{-|x|^2}\phi\bigr\rangle\\ &=\big\langle \Bbb 1_{x_1>0,x_2>0},\partial_{x_1,x_2}\bigl(e^{-|x|^2}\phi\bigr)\bigr\rangle\\ &=\int_0^\infty\int_0^\infty\partial_{x_1,x_2}\bigl(e^{-|x|^2}\phi\bigr)\,dx_1dx_2\\ &=\phi(0,0). \end{align}$$ That is, $$ e^{-|x|^2}\partial_{x_1,x_2}\Bbb 1_{x_1>0,x_2>0}=\delta_{(0,0)}. $$