Existence and smoothness of extremizers for convolution with compactly supported measures

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 $C_{loc}^\infty \cap L^\infty$.