Exercise 15, § 4

Let G be a locally compact group, $\beta$ a left Haar measure on G; by an abuse of language, in what follows one identifies $\beta$-integrable numerical functions with their classes in $L^1(G, \beta)$; 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 all $f \in L^1(G, \beta)$. Show that for every function $g \in \mathscr{K}(G)$, one then has $A(f * g) = A(f) * g$ (observe that $s \mapsto g(s)A(f * \varepsilon_s)$ is a $\beta$-integrable mapping of G into $L^1(G, \beta)$); converse. From this, deduce that there exists one and only one bounded measure $\mu$ on G such that $A(f) = \mu * f$ for all $f \in L^1(G, \beta)$, and that $\|A\| = \|\mu\|$. (With the notations of Prop. 19 of No. 7, consider the limit of $A(f_V * g)$ with respect to an ultrafilter finer than the section filter of $\mathscr{B}$, making use of the compactness of the unit ball of $\mathscr{M}^1(G)$ for the weak topology $\sigma(\mathscr{M}^1(G), \overline{\mathscr{K}(G)})$.)

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

c) Assume that G is compact and $\mu$ is positive. Show that for $1 < p < +\infty$, the norm of the continuous endomorphism $\gamma(\mu): f \mapsto \mu * f$ 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 $1 < p < +\infty$.