Exercise 17, § 3

The notation and conventions are those of Exercise 15.

a) Let $\mu$ be a bounded measure on $G$; for the endomorphism $\gamma(\mu): f \to \mu * f$ of $L^p(G, \beta)$ to be surjective ($1 \leq p < +\infty$), it is necessary and sufficient that there exist $c_p > 0$ such that $\|\tilde{\mu} * g\|_q \geq c_p \|g\|_q$ for all $g \in L^q(G, \beta)$ (where $\frac{1}{p} + \frac{1}{q} = 1$) (cf. Esp. vect. top., chap. IV, § 5, exerc. 11).

b) If $\mu$ is a base measure $\beta$, if $G$ is not discrete, now prove that $\gamma(\mu)$ is never surjective for $1 \leq p \leq +\infty$ (for $p < +\infty$, use a)), by reducing to the case where the density of $\mu$ with respect to $\beta$ belongs to $\mathscr{K}(G)$; for $p = +\infty$, reason 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 commutative, $1 \leq p \leq 2$ and if the endomorphism $\gamma(\mu)$ of $L^p(G, \beta)$ is surjective, now prove that it is bijective (using regularization, prove that if $\gamma(\mu)$ is not injective, the endomorphism $\gamma(\tilde{\mu})$ of $L^q(G, \beta)$ is not injective: one will use chap. IV, § 6, No. 5, cor. of prop. 4).

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