What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm? How can I understand the fact $$\|f\|_{H^{-k}(\mathbb{R}^n)}=\|(I-\triangle)^{-k}f\|_{H^{k}(\mathbb{R}^n)}.$$
2026-04-23 21:34:15.1776980055
What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm?
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By definition, $\|f,H^s\| =\|(1+|\xi|^2)^{s/2}\hat f(\xi),L_2\| $, where $\hat f$ is a Fourier transform of $f$.
In addition, $\hat{\Delta f} = -|\xi|^2\hat f$ (up to a constant factor depending on you definition of Fourier transform), so if $k\in \mathbb N$ $$\|(I-\Delta)^k g,H^{-k}\|=\|(1+|\xi|^2)^{-k/2}\times(1+|\xi|^2)^k\hat g(\xi),L_2 \|$$ $$=\|(1+|\xi|^2)^{k/2} \hat g(\xi),L_2 \|=\|g,H^k\|,$$ which replies to your second question.