  In this article, we establish various facts about extremizers for
$L^p$-improving convolution operators $T\colon L^p \rightarrow L^q$ associated
with compactly-supported probability measures on either $\mathbb{R}^d$ or
$\mathbb{T}^d$ . If $\sigma$ has positive Fourier decay, we prove that
extremizers exist and extremizing sequences are precompact modulo translation
for all "non-endpoint" $(p,q)$. These extremizers also satisfy an interesting
positivity property and belong to $C_{loc}^\infty \cap L^\infty$.
