Exercise 15, § 3

Let $G$ be a locally compact group, $\beta$ a left Haar measure on G; by abuse of language, in what follows we identify the numerical functions which are $\beta$-integrable and their classes in $L^1(G, \beta)$; the same abuse for the $L^p(G, \beta)$.

a) Let $A$ be a continuous endomorphism of the Banach space $L^1(G, \beta)$ such that, for every $s \in G$, one has $A(f * \varepsilon_s) = A(f) * \varepsilon_s$ for every $f \in L^1(G, \beta)$.

EXERCISES

Now prove that, for every function $g \in \mathscr{K}(G)$, one then has $A(f * g) = A(f) * g$ (notice that $s \to g(s)A(f * \varepsilon_s)$ is a $\beta$-integrable mapping of $G$ into $L^1(G, \beta)$); converse. Deduce that there exists a bounded measure and one only $\mu$ on $G$ such that $A(f) = \mu * f$ for every $f \in L^1(G, \beta)$, and that $\|A\| = \|\mu\|$. (With the notations of Prop. 19 of No. 7, consider the limit of $A(f_V * g)$ following an ultrafilter finer than the filter of sections of $\mathscr{B}$, using the compactness of the unit ball of $\mathscr{M}^1(G)$ for the weak topology $\sigma(\mathscr{M}^1(G), \mathscr{K}(G))$.)

b) Let $\mu$ be a bounded measure on $G$; prove directly that the norm of the continuous endomorphism $\gamma(\mu): f \to \mu * f$ of $L^\infty(G, \beta)$ is equal to $\|\mu\|$. (Reduce to the case where $\mu$ has compact support, and where $\mu$ has a continuous density with respect to $|\mu|$.) Deduce again that the continuous endomorphism $\gamma(\mu): f \to \mu * f$ of $L^1(G, \beta)$ has a norm equal to $\|\mu\|$.

c) Suppose $G$ compact and $\mu$ positive. Prove that for $1 < p < +\infty$, the norm of the continuous endomorphism $\gamma(\mu): f \to \mu * f$ of $L^p(G, \beta)$ is equal to $\|\mu\|$.

d) Take for $G$ the cyclic group of order 3. Give an example of a measure $\mu$ on $G$ such that the norm of the endomorphism $\gamma(\mu)$ of $L^p(G, \beta)$ is strictly less than $\|\mu\|$ for $1 < p < +\infty$.