# What properties does this inequality use? [closed]

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: $$\tilde{g}(\xi) = \int_{\bf{R}^d} g(x)e^{-2\pi i\xi\cdot x}dx$$

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## closed as off-topic by Aqua, Shailesh, cmk, Adrian Keister, The CountJul 13 at 0:36

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• Sorry that please click the link to see the image.... – Xeh Deng Jul 12 at 10:13
• Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures. – José Carlos Santos Jul 12 at 10:15
• Thanks for your advise and I have edited it. – Xeh Deng Jul 12 at 13:01
• @ José Carlos Santos – Xeh Deng Jul 12 at 13:01
• – cmk Jul 12 at 13:30