today I encountered an inequality like this: $$\langle\xi\rangle = \sqrt{1+|\xi|^2}$$ $$C\|g\|_{W^{m,2}(\bf{R}^d)}\leq \|\langle\cdot\rangle^m|{\tilde{g}}|\|_{L^{2}(\bf{R}^d)}\leq {\tilde{C}} \|g\|_{W^{m,2}(\bf{R}^d)}$$
In the paper, the author did not give the reasons. Since I do not major in Math, could anyone help me to give a simple illustration for it? It seems like Sobolev space function's property. PS: $\tilde{g}(\xi) = \int_{\bf{R}^d} g(x)e^{-2\pi i\xi\cdot x}dx$