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I'm studying some theory about the Sobolev Space $H^2(\mathbb{R}^N)$ and I was abble to understand the proof that $$H^2(\mathbb{R}^N)=\{f \in L^2(\mathbb{R}^N): |\xi|^2\hat{f}(\xi) \in L^2(\mathbb{R}^N)\}, $$ where $\hat{f}$ denotes the Fourier transform of $f$.

Next, the text says that the expression defined for $f \in H^2(\mathbb{R}^N)$ by $$\lVert f \rVert_{H_2}=\left(\int_{\mathbb{R}^N}(1+|\xi|^4)|\hat{f}(\xi)|^2 d \xi\right)^{\frac{1}{2}}$$ defines a norm in $H^2$. I don't have any trouble in prove all the properties of norm, except the triangle inequality. For the triangle inequality I've tried using the Holder inequality to help me and the triangle inequality of $|.|_2$ norm, but I'm having no success.

Anyone can give me please some hint on how to proceed in prove the triangle inequality? Any help will be very valuable, this is the last the part that I need to fully understand this beautiful theorem of my text :)

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I think it's just Minkowski's inequality for the following measure $d\mu =(1+|\xi|^4 ) d\xi.$

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