On maximizers of convolution operators in $L_p$ spaces

A convolution operator in ${\mathbb{R}}^d$ with kernel in $L_q$ acts from $L_p$ to $L_s$, where $1\mathrm{/}p+1\mathrm{/}q=1+1\mathrm{/}s$. The main theorem states that if $1<q,p,s<\mathrm{\infty }$, then there exists an $L_p$ function of unit norm on which the $s$-norm of the convolution is attained. A number of questions, solved and open, related to the statement and proof of the main theorem, are discussed. The problem of computing best constants in the Hausdorff-Young inequality for the Laplace transform, which prompted this research, is considered.