Consider $\mathbb{R}^{n}$ with the Lebesgue measure. Denote by $\mathcal{F} f(\xi)=\int_{\mathbb{R}^{n}} e^{-2 i \pi x \cdot \xi} f(x) d x$ the Fourier transform of $f \in L^{1}\left(\mathbb{R}^{n}\right)$ and by $\hat{f}$ the Fourier-Plancherel transform of $f \in L^{2}\left(\mathbb{R}^{n}\right)$. Let $\chi_{R}(\xi):=\left(1-\frac{|\xi|}{R}\right) \chi_{|\xi| \leqslant R}$ for $R>0$ and define for $s \in \mathbb{R}_{+}$

$$H^{s}\left(\mathbb{R}^{n}\right):=\left\{f \in L^{2}\left(\mathbb{R}^{n}\right) \mid\left(1+|\cdot|^{2}\right)^{s / 2} \hat{f}(\cdot) \in L^{2}\left(\mathbb{R}^{n}\right)\right\}$$

(i) Prove that $H^{s}\left(\mathbb{R}^{n}\right)$ is a vector subspace of $L^{2}\left(\mathbb{R}^{n}\right)$, and is a Hilbert space for the inner product $\langle f, g\rangle:=\int_{\mathbb{R}^{n}}\left(1+|\xi|^{2}\right)^{s} \hat{f}(\xi) \overline{\hat{g}(\xi)} d \xi$, where $\bar{z}$ denotes the complex conjugate of $z \in \mathbb{C}$.

(ii) Construct a function $f \in H^{s}(\mathbb{R}), s \in(0,1 / 2)$, that is not almost everywhere equal to a continuous function.

(iii) For $f \in L^{1}\left(\mathbb{R}^{n}\right)$, prove that $F_{R}: x \mapsto \int_{\mathbb{R}^{n}} \mathcal{F} f(\xi) \chi_{R}(\xi) e^{2 i \pi x \cdot \xi} d \xi$ is a well-defined function and that $F_{R} \in L^{1}\left(\mathbb{R}^{n}\right)$ converges to $f$ in $L^{1}\left(\mathbb{R}^{n}\right)$ as $R \rightarrow+\infty$.

[Hint: Prove that $F_{R}=K_{R} * f$ where $K_{R}$ is an approximation of the unit as $R \rightarrow+\infty .]$

(iv) Deduce that if $f \in L^{1}\left(\mathbb{R}^{n}\right)$ and $\left(1+|\cdot|^{2}\right)^{s / 2} \mathcal{F} f(\cdot) \in L^{2}\left(\mathbb{R}^{n}\right)$ then $f \in H^{s}\left(\mathbb{R}^{n}\right)$.

[Hint: Prove that: (1) there is a sequence $R_{k} \rightarrow+\infty$ such that $K_{R_{k}} * f$ converges to $f$ almost everywhere; (2) $K_{R} * f$ is uniformly bounded in $L^{2}\left(\mathbb{R}^{n}\right)$ as $R \rightarrow+\infty$.]