Exercise 15, § 3

Let $\mu$ denote a Haar measure on G and $\mathfrak{B}$ the filter of neighbourhoods of $e$ in $\widehat{G}$.

a) Let U be an integrable open neighbourhood of $e$ in G. There exists a function $a\in L^1(G),a\geqslant 0$, with integral 1, vanishing outside U, such that $\mathscr{F}(a)\in L^1(\widehat{G})$, and

(Let V be a compact neighbourhood of $e$ such that $V\subset U,\mu(U)<\overline{2}\mu(V)$ ; let W be a symmetric compact neighbourhood of $e$ such that VW$^2\subset U$; take $a=\lambda f*g$, where $\lambda$ is a constant $>0$, and where $f, g$ are the characteristic functions of the sets VW and W.) b) If $f\in L^{\infty}(G)$, let $A(f)$ denote the set of elements of $\widehat{G}$ which belong to the weakly closed translation-invariant vector subspace of $L^{\infty}(G)$ generated by $f$. If $f, g\in L^{\infty}(G)$, one has

(Let U be a neighbourhood of $e$. Now prove that $f g$ is the weak limit of linear combinations of elements of $\widehat{G}$ belonging to $A(f$)UA($g$)$U$.)

c) Let $g\in L^{\infty}(G)$, K be a compact subset of $\widehat{G}$, $f$ a function of $L^1(G)$ such that $\mathscr{F}(f)$ vanishes on $A = A(g)$ and outside K. For every $V\in \mathfrak{B}$, let $\omega (V)$ be the least upper bound of $|\mathscr{F}(f)|$ on AV. If

lim inf$_{\mathfrak{B}}\omega (V)^2\mu$((AV$\mu(V)$- $A$)$\cap K$)$= 0$,

then $\langle f, g\rangle = 0$. (Let $\varepsilon >0$. There exists a compact subset H of G such that $\int_{G-H}|f|dx\leqslant \varepsilon$. Next there exists an open neighbourhood U of $e$ in $\widehat{G}$ such that $|\langle x,\widehat{x}\rangle -1|\leqslant \varepsilon$ for $x\in H,\widehat{x}\in U$. Applying a) to $\widehat{G}$ and U, one obtains a function $a\in L^1(\widehat{G})$. Let $b=\mathscr{F}(a)\in L^1(G)$. Using b), now prove that $\mathscr{F}(bg)$ vanishes outside AU. On the other hand,

and, since $f, gb\in L^2(G)$,

G $\widehat{G}$ (AU-A)$\cap K$

Majorize this last expression by :

$2\|g\|_{\infty}\omega (U)\surd\mu$((AU$\surd\overline{\mu(U)}-A$)$\cap K$)

to conclude.)

d) Let I be a closed ideal of $L^1(G)$, A = V(I)$,f$ a function of $L^1(G)$ such that $\mathscr{F}(f)$ vanishes on A, S the set of points where $\mathscr{F}(f)$ vanishes, $S'$ the boundary of S, B the largest perfect set contained in $S\cap A$. For $V\in \mathfrak{B}$, let $\omega (V)$ be the supremum of $|\mathscr{F}(f)|$ on AV. It is supposed that every point of B possesses a neighbourhood K in $\widehat{G}$ such that

lim inf$_{\mathfrak{B}}\omega (V)^2\mu$((AV$\mu(V)$- $A$)$\cap K$)$= 0$.

Then $f\in I$. (Use the fact that $L^1(G)$ satisfies Ditkin's condition, and reason as in Prop. 5 of I, p. 94; let $G'=\widehat{G}\cup \{\infty \}$ be the Alexandroff compactification of $\widehat{G}$, and let N be the set of $\chi \in G'$ such that $\mathscr{F}(f)$ does not belong to $\mathscr{F}(I)$ in a neighbourhood of $\chi$. First now prove that $N\subset B\cup \{\infty \}$. Then, using c), now prove that $N\subset \{\infty \}$. Finally, now prove that $N =\emptyset .)$ e) Let I be a closed ideal of $L^1(\mathbf{R}^n)$, A = V(I)$,f$ a function of $L^1(\mathbf{R}^n)$ such that $\mathscr{F}(f)$ vanishes on A. For $h >0$, let $A_h$ be the set of points of $\mathbf{R}^n$ outside A and at a distance from A less than $h$, and let $\omega (h) =$ sup$_{x\in A_h}|\mathscr{F}(f)(x)|$. If

lim inf$_{h\rightarrow 0}\frac{\omega(h)^2\mu(A_h)}{h^n}= 0$

then $f\in I$. (Use d).)

f) Let $f\in L^1(\mathbf{R}),\alpha \in ]0,1[$, and suppose $\int_{\mathbf{R}}|y|^{\alpha}|f(y)|dy <+\infty$. Then there exists a constant $k$ such that $|\mathscr{F}(f)(x+h)-\mathscr{F}(f)(x)|\leqslant kh^{\alpha}$ for every $x\in \mathbf{R}$ and every $h\in \mathbf{R}$.

g) Let I be a closed ideal of $L^1(\mathbf{R}),f$ a function of $L^1(\mathbf{R})$ such that $\mathscr{F}(f)$ vanishes on V(I). If

then one has $f\in I$. (Use e) and f).)