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I met the following definition of homogeneous Sobolev space: $\dot{H^s}(\mathbb{R}^n) = \left\{ f \in \mathcal{S}': \hat{f} \in L_{loc}^1(\mathbb{R}^n) \,\,\, \text{and} \,\,\, || f||_{\dot{H^s}} < \infty \right\}$,
where $s \geq 0$ (does not have to be an integar), $\hat{f}$ is the fourier transform of $f$,
$|| f||_{\dot{H^s}} =\left(\displaystyle\int_{\mathbb{R}^n}^{} |\xi|^{2s} |\hat{f}(\xi)|^2 \, d\xi \right)^\frac{1}{2}$ and
$\mathcal{S}'$ is the space of linear and continuous functionals from Schwarz Space $\mathcal{S}$ to $\mathbb{C}$ (called space of tempered distribiutions).
If $F \in \mathcal{S}'$ and $f \in \mathcal{S}$, then the fourier transform on it can be defined as $\hat{F}(f) = F(\hat{f}).$
And my question is: if $g \in \dot{H^s}(\mathbb{R}^n)$ then how $\hat{g} \in L_{loc}^1(\mathbb{R}^n)$, I mean how $\hat{g}$ can be a function if $g$ is a functional?
Similarly what does mean the integral defined in $||.||_{\dot{H^s}}$ provided that i do not know if $\hat{g}$ is a function?

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For simplicity let me assume that $s$ is a positive integer.

You should interpret $\int_{\mathbb{R}^n}^{} |\xi|^{2s} |\hat{f}(\xi)|^2 \, d\xi<\infty$ as meaning that $f$ is a tempered distribution with Fourier transform $\hat{f}$ so that $(\sum_{j=1}^n \xi_j^2)^s \hat{f}$ is a tempered distribution which is represented by a function $g\in L^2(\Bbb{R}^n)$, that is $\langle \hat{f},\phi \rangle= \int_{\Bbb{R}^n} g(\xi)\phi(\xi)d\xi$ for all $\phi \in S(\Bbb{R}^n)$.

So it is allowed to say that $|\xi|^{2s} \hat{f} \in L^2(\Bbb{R^n})\subset L^1_{loc}(\Bbb{R^n})$.

It doesn't have to be that $\hat{f}$ is $L^1_{loc}$.

What you can say is that taking $\psi\in C^\infty_c(\Bbb{R}^n), \psi=1$ on $|\xi|<1$ then $T(\phi)=|\xi|^{-2s}(\phi-\psi \sum_{|\alpha|<2s} \frac{\xi^\alpha}{\alpha!}\partial^\alpha \phi(0))$ defines a continuous map $\phi \to T(\phi), S(\Bbb{R}^n)\to S(\Bbb{R}^n)$, and the tempered distribution $$\langle h,\phi \rangle = \langle \hat{f},\phi \rangle - \langle g,T(\phi) \rangle$$

is of order $\le 2s$ and supported at $0$. So that $$ h = \sum_{|\alpha|\le 2m} c_\alpha \partial^\alpha \delta$$ and $$\langle \hat{f},\phi \rangle = \sum_{|\alpha|\le 2m} c_\alpha \partial^\alpha \phi(0) + \int_{\Bbb{R}^n} g(\xi) |\xi|^{-2s}(\phi(\xi)-\psi(\xi) \sum_{|\alpha|<2s} \frac{\xi^\alpha}{\alpha!}\partial^\alpha \phi(0)) d\xi$$

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  • $\begingroup$ So, if I understand properly, the condition $\hat{f} \in L_{loc}^1$ means $\hat{f}$ is represented by some $g \in L_{loc}^1?$ There is also a theorem saying that for a very specific s and p $\dot{H^s}(\mathbb{R}^n) \subset L^p(\mathbb{R}^n)$. Does it just say that $h \in \dot{H^s}$ has a representative in $L^p$? $\endgroup$
    – Maciej778
    Commented Mar 12, 2023 at 10:02
  • $\begingroup$ I also wonder if expressions like for example $\displaystyle\int_{A} e^{2 \pi i x \xi}\hat{f}(\xi)\ d\xi$ could be somehow sensible for $f \in \dot{H^s}$ and some sensible set $A$, because in some works I saw people writing similar things and i am still not sure what "$\hat{f}(\xi)$" means. $\endgroup$
    – Maciej778
    Commented Mar 12, 2023 at 10:19

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