Exercise 23, § 3

Let $G =\mathbf{R}^*_+$; it is a locally compact commutative group and the measure $\mu=dx/x$ is a Haar measure on G. The group G and the group $i\mathbf{R}$ are in duality by the mapping $(x, it)\mapsto x^{it}$, the dual measure of $\mu$ being the Lebesgue measure on $i\mathbf{R}($cf. cor. 4 of II, p. 236). We denote by $\mathscr{F}$ the Fourier transform of G and by $\overline{\mathscr{F}}$ the Fourier cotransform of $\widehat{G}$.

We denote by X the identity function of G.

a) Let $\nu$ be a positive measure on G. We denote by $I_{\nu}$ the set of $\sigma \in \mathbf{R}$ such that the function $X^{\sigma}$ is $\nu$-integrable, and by $B_{\nu}$ the set of $s\in \mathbf{R}$ such that $\mathscr{R}(s)\in I_{\nu}$. The set $I_{\nu}$ is an interval of $\mathbf{R}$ and the function $s\mapsto \nu (X^s)$ is defined and holomorphic in the interior of $B_{\nu}$.

b) Let $f$ be a locally integrable function on G. We denote by $I_f= I_{f\cdot\mu}$ and $B_f= B_{f\cdot\mu}$. For every $s\in B_f$, we put

If there exist $\delta >0$ and $c\geqslant 0$ such that $|\widehat{f}(s)|\leqslant c(1 +|s|)^{-1-\delta}$ for $s\in B_f$, then for every $\sigma$ in the interior of $B_f$, one has

c) Let $f\in \mathscr{L}^2(G)$. Suppose that 0 belongs to the interior of $I_f$. One then has $\overline{\mathscr{F}}f(it) =\widehat{f}(it)$ for almost all $t\in G$. d) Let $f$ and $g$ be measurable functions on G. Let $\sigma$ be a real number such that $X^{-2\sigma}f\in L^2(G)$ and $X^{2\sigma}g\in L^2(G)$. One then has

e) Let $f$ and $g$ be measurable functions on G. Suppose that

(i) There exists $c\in \mathbf{R}$ such that $|f(x)|\leqslant c x/(1 +x)$ for every $x\in G$;

(ii) One has$]0,1[\subset I_g$ and for every $\sigma$ such that $0< \sigma <1$, the function $X^{\sigma}g$ is bounded ;

(iii) There exists $c'\in \mathbf{R}$ such that $|\widehat{g}(s)|\leqslant c'(1 +|s|)^{-2}$ for every $s\in \mathbf{C}$ such that $\mathscr{R}(s)\in ]0,1[$.

Then one has

for every $\sigma \in \mathbf{R}$ such that $0< \sigma <1$.

f) Let $\sigma \in \mathbf{R}$ and let $f\in \mathscr{C}(\sigma +i\mathbf{R})$. There exists a positive measure $\nu$ on G such that $\sigma \in I_{\nu}$ and $\widehat{\nu}(\sigma +it) =f(\sigma +it)$ for all $t\in \mathbf{R}$ if and only if for every integer $n\geqslant 1$, every family $(t_i)_{0\leqslant i\leqslant n}$ of complex numbers and every family $(z_i)_{0\leqslant i\leqslant n}$ of complex numbers such that $\mathscr{R}(z_i) =\sigma$ for all $i$, one has

$nn$

2

$i=0j=0$