(a) State the Riemann-Lebesgue lemma. Show that the Fourier transform maps $\mathscr{S}\left(\mathbb{R}^{n}\right)$ to itself continuously.

(b) For some $s \geqslant 0$, let $f \in L^{1}\left(\mathbb{R}^{3}\right) \cap H^{s}\left(\mathbb{R}^{3}\right)$. Consider the following system of equations for $\mathbf{B}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$

$$\nabla \cdot \mathbf{B}=f, \quad \boldsymbol{\nabla} \times \mathbf{B}=\mathbf{0}$$

Show that there exists a unique $\mathbf{B}=\left(B_{1}, B_{2}, B_{3}\right)$ solving the equations with $B_{j} \in$ $H^{s+1}\left(\mathbb{R}^{3}\right)$ for $j=1,2,3$. You need not find $\mathbf{B}$ explicitly, but should give an expression for the Fourier transform of $B_{j}$. Show that there exists a constant $C>0$ such that

$$\left\|B_{j}\right\|_{H^{s+1}} \leqslant C\left(\|f\|_{L^{1}}+\|f\|_{H^{s}}\right), \quad j=1,2,3$$

For what values of $s$ can we conclude that $B_{j} \in C^{1}\left(\mathbb{R}^{n}\right)$ ?