Exercise 17, § 4

Notations and conventions are those of Exer. 15.

a) Let $\mu$ be a bounded measure on G; for the endomorphism $\gamma(\mu): f \mapsto \mu * f$ of $L^p(G, \beta)$ to be surjective ($1 \leq p < +\infty$), it is necessary and sufficient that there exist a $c_p > 0$ such that $\| \nu * g \|_q \geq c_p \| g \|_q$ for all $g \in L^q(G, \beta)$ (where $\frac{1}{p} + \frac{1}{q} = 1$) (cf. TVS, IV, §4, No. 2, remarks following Cor. 3 of Th. 1).

b) If $\mu$ is a measure with base $\beta$, and G is not discrete, show that $\gamma(\mu)$ is never surjective for $1 \leq p \leq +\infty$ (for $p < +\infty$, use a), reducing to the case that the density of $\mu$ with respect to $\beta$ belongs to $\mathscr{K}(G)$; for $p = +\infty$, argue directly by observing that for $f \in L^1(G, \beta)$ and $g \in L^\infty(G, \beta)$, $f * g$ is uniformly continuous for the right uniform structure).

c) If G is abelian, $1 \leq p \leq 2$ and the endomorphism $\gamma(\mu)$ of $L^p(G, \beta)$ is surjective, show that it is bijective (using regularization, show that if $\gamma(\mu)$ is not injective then the endomorphism $\gamma(\nu)$ of $L^q(G, \beta)$ is not injective: one makes use of Ch. IV, §6, No. 5, Cor. of Prop. 4).

d) Take $G = \mathbf{Z}$ and $\mu = \varepsilon_1 - \varepsilon_0$; show that $\gamma(\mu)$ is injective in the $L^p(\mathbf{Z}, \beta)$ for which $p \neq +\infty$, but not in $L^\infty(\mathbf{Z}, \beta)$, and it is not surjective for any value of $p$.