I'm attempting to understand Wikipedia's definition of the Schwarz Space, in particular, the sentence
$$\sup_{x\in \mathbb{R}^n}\left|x^{\beta}D^{\alpha}\right|<\infty.$$
Given $\alpha, \beta\in \mathbb{N}^n$ I understand the meaning of the notations $x^{\beta}$ and $D^{\alpha}=\partial ^{\alpha}$ individually, but I do not undestand the meaning of $$x^{\beta}D^{\alpha}.$$
What is its meaning?
I can tell you what $$ \sup_{x\in \mathbb{R}^n}\left|x^{\beta}D^{\alpha}f(x)\right|<\infty. $$ means. Start with a function $f$. Take a certain partial derivative of it. Multiply by a monomial in $x$. Absolute value of the result. Take the supremum of that over all $x$.
Example $$ x^{(2,4)}D^{(3,7)}f(x) = x_1^2 x_2^4 \frac{\partial^3}{\partial x_1^3}\frac{\partial^7}{\partial x_2^7}f(x_1,x_2) $$