Proposition 12

— Let $p$ and $q$ be two conjugate exponents ($1 \leq p \leq +\infty$). If $f \chi^{-1/q} \in L^1(G, \beta)$ and $g \in L^p(G, \beta)$, then $f$ and $g$ are convolvable, $f * g$ is equal locally $\beta$-almost everywhere to a function in $L^p(G, \beta)$, and

If $f \in L^p(G, \beta)$ and $g {\chi'}^{-1/q} \in L^1(G, \beta)$, then $f$ and $g$ are convolvable, $f * g$ is equal locally $\beta$-almost everywhere to a function in $L^p(G, \beta)$, and This follows from Props. 5 and 6 of No. 2 and the remarks in No. 4.